diff --git a/Fig2_Functions.m b/Fig2_Functions.m
new file mode 100644
index 0000000..9fc4d32
--- /dev/null
+++ b/Fig2_Functions.m
@@ -0,0 +1,36 @@
+function out = Fig1_Functions
+out{1} = @init;
+out{2} = @fun_eval;
+out{3} = [];
+out{4} = [];
+out{5} = [];
+out{6} = [];
+out{7} = [];
+out{8} = [];
+out{9} = [];
+
+% --------------------------------------------------------------------------
+function dydt = fun_eval(t,y,b,gma,n,RT,delta)
+dydt=[(b+gma*y(1)^n/(1+y(1)^n))*(RT-y(1)) - delta*y(1);(b+gma*y(2)^n/(1+y(2)^n))*(RT-y(1)) - delta*y(2);];
+
+% --------------------------------------------------------------------------
+function [tspan,y0,options] = init
+handles = feval(PosFeedMutInhib);
+y0=[0];
+options = odeset('Jacobian',[],'JacobianP',[],'Hessians',[],'HessiansP',[]);
+tspan = [0 10];
+
+% --------------------------------------------------------------------------
+function jac = jacobian(t,y,b,gma,n,RT,delta)
+% --------------------------------------------------------------------------
+function jacp = jacobianp(t,y,b,gma,n,RT,delta)
+% --------------------------------------------------------------------------
+function hess = hessians(t,y,b,gma,n,RT,delta)
+% --------------------------------------------------------------------------
+function hessp = hessiansp(t,y,b,gma,n,RT,delta)
+%---------------------------------------------------------------------------
+function Tension3 = der3(t,y,b,gma,n,RT,delta)
+%---------------------------------------------------------------------------
+function Tension4 = der4(t,y,b,gma,n,RT,delta)
+%---------------------------------------------------------------------------
+function Tension5 = der5(t,y,b,gma,n,RT,delta)
diff --git a/Fig2a_driver.m b/Fig2a_driver.m
new file mode 100644
index 0000000..0200ecd
--- /dev/null
+++ b/Fig2a_driver.m
@@ -0,0 +1,323 @@
+bettercolors;
+
+global cds sys
+
+sys.gui.pausespecial=0; %Pause at special points
+sys.gui.pausenever=1; %Pause never
+sys.gui.pauseeachpoint=0; %Pause at each point
+
+syshandle=@Fig2_Functions; %Specify system file
+
+SubFunHandles=feval(syshandle); %Get function handles from system file
+RHShandle=SubFunHandles{2}; %Get function handle for ODE
+
+b = 0.01;
+gma = 5;
+n = 6;
+RT = 2;
+delta = 2;
+
+local = 1;
+codim = 1;
+
+xinit=[0;0]; %Set ODE initial condition
+
+%Specify ODE function with ODE parameters set
+RHS_no_param=@(t,x)RHShandle(t,x,b,gma,n,RT,delta);
+
+%Set ODE integrator parameters.
+options=odeset;
+options=odeset(options,'RelTol',1e-8);
+options=odeset(options,'maxstep',1e-2);
+
+%Integrate until a steady state is found.
+[tout xout]=ode45(RHS_no_param,[0,200],xinit,options);
+
+xinit=xout(size(xout,1),:);
+
+pvec=[b,gma,n,RT,delta]'; % Initialize parameter vector
+
+ap=1;
+
+[x0,v0]=init_EP_EP(syshandle, xinit', pvec, ap); %Initialize equilibrium
+
+
+opt=contset;
+opt=contset(opt,'MaxNumPoints',1500); %Set numeber of continuation steps
+opt=contset(opt,'MaxStepsize',.01); %Set max step size
+opt=contset(opt,'Singularities',1); %Monitor singularities
+opt=contset(opt,'Eigenvalues',1); %Output eigenvalues
+opt=contset(opt,'InitStepsize',0.01); %Set Initial stepsize
+
+[x1,v1,s1,h1,f1]=cont(@equilibrium,x0,v0,opt);
+
+opt=contset(opt,'backward',1);
+[x1b,v1b,s1b,h1b,f1b]=cont(@equilibrium,x0,v0,opt);
+
+
+if local == 1
+%% LOCAL BRANCHES
+
+ind = 2;
+xBP=x1(1:2,s1(ind).index); %Extract branch point
+pvec(ap)=x1(3,s1(ind).index); %Extract branch point
+[x0,v0]=init_BP_EP(syshandle, xBP, pvec, s1(ind), 0.05);
+opt=contset(opt,'backward',1);
+[x3b,v3b,s3b,h3b,f3b]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+opt = contset(opt,'backward',0);
+[x3,v3,s3,h3,f3]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+
+ind = 7;
+xBP=x1(1:2,s1(ind).index); %Extract branch point
+pvec(ap)=x1(3,s1(ind).index); %Extract branch point
+[x0,v0]=init_BP_EP(syshandle, xBP, pvec, s1(ind), 0.05);
+opt = contset(opt,'backward',1);
+[x4b,v4b,s4b,h4b,f4b]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+opt = contset(opt,'backward',0);
+[x4,v4,s4,h4,f4]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+
+end
+% CODIM 1 plots
+
+
+width=5.2/2;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 10;
+f = figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+Fig2a = subplot(1,1,1);
+xlabel(Fig2a,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+ylabel(Fig2a,{'$R^\ell$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+Fig2aTitle = title(Fig2a,{'(a)'},'FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica');
+set(Fig2a,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+grid
+Fig2a.XLim = [0 4];
+Fig2a.YLim = [0 5];
+Fig2a.Box = 'on';
+set(gca,'LineWidth',1.5)
+hold on
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%GLOBAL curves
+curves = {'1','1b'};
+
+for i = 1:length(curves)
+ xeqcurve=eval(['x' curves{i}]);
+ minevaleq=eval(['f' curves{i} '(2,:)']); %This is the last eigenvalue. That is the one that determines stability
+
+ L=length(xeqcurve(1,:));
+
+ curveind=1;
+ lengthind=0;
+ maxlengthind=0;
+ evalstart=floor(heaviside(minevaleq(1)));
+ datamateq=zeros(4,L);
+
+ for i=1:L
+ evalind=floor(heaviside(minevaleq(i)));
+ if evalstart~=evalind
+ curveind=curveind+1;
+ i;
+ evalstart=evalind;
+ maxlengthind=max(lengthind,maxlengthind);
+ lengthind=0;
+ end
+ datamateq(1,i)=xeqcurve(3,i); % This is the parameter that is varied.
+ datamateq(2,i)=xeqcurve(1,i); % This is the dependent axis of the bifurcation plot. The one you wish to plot
+ datamateq(3,i)=evalind;
+ datamateq(4,i)=curveind;
+
+ lengthind=lengthind+1;
+ end
+
+ maxlengthind=max(maxlengthind,lengthind);
+
+ curveindeq=curveind;
+
+ for i=1:curveindeq
+ index=find(datamateq(4,:)==i);
+ eval(['curve' num2str(i) 'eq' '=datamateq(1:3,index);']);
+ end
+
+ for i=1:curveindeq
+ stability=eval(['curve' num2str(i) 'eq(3,1)']);
+ if stability==0
+ plotsty='-';
+ else
+ plotsty=':';
+ end
+
+ plotcolor='k';
+
+ plotstr=strcat(plotcolor,plotsty);
+
+ plot(eval(['curve' num2str(i) 'eq(1,:)']),eval(['curve' num2str(i) 'eq(2,:)']),plotstr,'Linewidth',3)
+ hold on
+ end
+end
+
+if local == 1
+%%%LOCAL
+curves = {'3','3b'};
+
+for i = 1:length(curves)
+ xeqcurve=eval(['x' curves{i}]);
+ minevaleq=eval(['f' curves{i} '(2,:)']); %This is the last eigenvalue. That is the one that determines stability
+
+ L=length(xeqcurve(1,:));
+
+ curveind=1;
+ lengthind=0;
+ maxlengthind=0;
+ evalstart=floor(heaviside(minevaleq(1)));
+ datamateq=zeros(4,L);
+
+ for i=1:L
+ evalind=floor(heaviside(minevaleq(i)));
+ if evalstart~=evalind
+ curveind=curveind+1;
+ i;
+ evalstart=evalind;
+ maxlengthind=max(lengthind,maxlengthind);
+ lengthind=0;
+ end
+ datamateq(1,i)=xeqcurve(3,i); % This is the parameter that is varied.
+ datamateq(2,i)=xeqcurve(2,i); % This is the dependent axis of the bifurcation plot. The one you wish to plot
+ datamateq(3,i)=evalind;
+ datamateq(4,i)=curveind;
+
+ lengthind=lengthind+1;
+ end
+
+ maxlengthind=max(maxlengthind,lengthind);
+
+ curveindeq=curveind;
+
+ for i=1:curveindeq
+ index=find(datamateq(4,:)==i);
+ eval(['curve' num2str(i) 'eq' '=datamateq(1:3,index);']);
+ end
+
+ for i=1:curveindeq
+ stability=eval(['curve' num2str(i) 'eq(3,1)']);
+ if stability==0
+ plotsty='-';
+ else
+ plotsty=':';
+ end
+
+ mycolor = highcontrast(2,:);
+ plotstr=plotsty;
+
+ plot(eval(['curve' num2str(i) 'eq(1,:)']),eval(['curve' num2str(i) 'eq(2,:)']),plotstr,'Color',mycolor,'Linewidth',2)
+ hold on
+ end
+end
+
+end
+
+opt=contset;
+opt=contset(opt,'MaxNumPoints',5500); %Set numeber of continuation steps
+opt=contset(opt,'MaxStepsize',1); %Set max step size
+opt=contset(opt,'Singularities',1); %Monitor singularities
+opt=contset(opt,'Eigenvalues',1); %Output eigenvalues
+opt=contset(opt,'InitStepsize',0.5); %Set Initial stepsize
+aps = [1,5];
+%%%WP
+ind = 4;
+xtmp = x3(1:end-1,s3(ind).index);
+pvec(ap) = x3(end,s3(ind).index);
+[x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x5,v5,s5,h5,f5] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x5b,v5b,s5b,h5b,f5b] = cont(@limitpoint,x0,v0,opt);
+
+ind = 5
+xtmp = x3b(1:end-1,s3b(ind).index);
+pvec(ap) = x3b(end,s3b(ind).index);
+[x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x6,v6,s6,h6,f6] = cont(@limitpoint,x0,v0,opt);
+ opt = contset(opt,'backward',1);
+[x6b,v6b,s6b,h6b,f6b] = cont(@limitpoint,x0,v0,opt);
+
+%BPs
+ind = 2;
+xtmp = x3b(1:end-1,s3b(ind).index);
+pvec(ap) = x3b(end,s3b(ind).index);
+[x0,v0] = init_BP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x7,v7,s7,h7,f7] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x7b,v7b,s7b,h7b,f7b] = cont(@limitpoint,x0,v0,opt);
+
+ind = 8;
+xtmp = x3b(1:end-1,s3b(ind).index);
+pvec(ap) = x3b(end,s3b(ind).index);
+[x0,v0] = init_BP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x8,v8,s8,h8,f8] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x8b,v8b,s8b,h8b,f8b] = cont(@limitpoint,x0,v0,opt);
+
+%%%
+close 1
+
+width=5.2/2;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 10;
+
+figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+
+Fig1b = subplot(1,1,1);
+xlabel(Fig1b,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',12,'FontName','Helvetica','color','k')
+
+% Fig1bTitle = title(Fig1b,{'(b)'},'FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica');
+set(Fig1b,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+hold on
+grid
+Fig1b.XLim = [0 6];
+Fig1b.YLim = [0 10];
+Fig1b.Box = 'on';
+Fig1b.XColor = 'k';
+Fig1b.YColor = 'k';
+ylabel({'$\delta$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',12,'FontName','Helvetica','color','k')
+set(gca,'LineWidth',1.5)
+curves = {'5','6','7','8'};
+
+linecolor(5) = {highcontrast(2,:)};
+linecolor(6) = {highcontrast(2,:)};
+linecolor(7) = {highcontrast(3,:)};
+linecolor(8) = {highcontrast(3,:)};
+
+
+linestyle(5) = {'-'};
+linestyle(6) = {'-'};
+linestyle(7) = {'--'};
+linestyle(8) = {'--'};
+
+for i = 5:8
+ xeqcurve=eval(['x' num2str(i)]);
+ plot(xeqcurve(end-1,:),xeqcurve(end,:),'LineStyle',linestyle{i},'Color',linecolor{i},'LineWidth',4)
+ xeqcurve = eval(['x' num2str(i) 'b']);
+ plot(xeqcurve(end-1,:),xeqcurve(end,:),'LineStyle',linestyle{i},'Color',linecolor{i},'LineWidth',4)
+end
+
+scatter(0.1,7.5,'ko','filled')
+text(0.1,7.5,' (b)','FontSize',fontsize,'FontName','Helvetica')
+
+scatter(0.1,3,'ko','filled')
+text(0.1,3,' (d)','FontSize',fontsize,'FontName','Helvetica')
+
+scatter(4.5,3,'ko','filled')
+text(4.5,3,' (e)','FontSize',fontsize,'FontName','Helvetica')
+
+scatter(4.5,7.5,'ko','filled')
+text(4.5,7.5,' (c)','FontSize',fontsize,'FontName','Helvetica')
+
+
+print(1,'2a','-depsc')
diff --git a/Fig2b-e.py b/Fig2b-e.py
new file mode 100644
index 0000000..3aa3695
--- /dev/null
+++ b/Fig2b-e.py
@@ -0,0 +1,210 @@
+import matplotlib
+# matplotlib.use('Qt5Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+from scipy.integrate import odeint
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+
+plt.rc("text", usetex=False)
+plt.rc("font", **{'family':'sans-serif', 'sans-serif':['Helvetica'], 'size':10})
+
+def f(u, v, b, gamma, n, RT, delta):
+ return (b+gamma*u**n/(1+u**n))*v - delta*u
+
+def g(u, v, b, gamma, n, RT, delta):
+ return -f(u,v,b, gamma, n, RT, delta)
+
+def rdPDE(y, t, b, gamma, n, RT, delta, Du, Dv, dx):
+ """
+ The ODEs are derived using the method of lines.
+ https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#mol
+ """
+ # The vectors u and v are interleaved in y. We define
+ # views of u and v by slicing y.
+ u = y[::2]
+ v = y[1::2]
+
+ # dydt is the return value of this function.
+ dydt = np.empty_like(y)
+
+ dudt = dydt[::2]
+ dvdt = dydt[1::2]
+
+ #this is neumann:
+ dudt[0] = f(u[0], v[0], b, gamma, n, RT, delta) + Du * (-2.0*u[0] + 2.0*u[1]) / dx**2
+ dudt[1:-1] = f(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + Du * np.diff(u,2) / dx**2
+ dudt[-1] = f(u[-1], v[-1], b, gamma, n, RT, delta) + Du * (-2.0*u[-1] + 2.0*u[-2]) / dx**2
+ dvdt[0] = g(u[0], v[0], b, gamma, n, RT, delta) + Dv * (-2.0*v[0] + 2.0*v[1]) / dx**2
+ dvdt[1:-1] = g(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + Dv * np.diff(v,2) / dx**2
+ dvdt[-1] = g(u[-1], v[-1], b, gamma, n, RT, delta) + Dv * (-2.0*v[-1] + 2.0*v[-2]) / dx**2
+
+ dydt[::2] = dudt
+ dydt[1::2] = dvdt
+
+ return dydt
+
+# %%
+# %%time
+L = 1
+N = 1000
+x = np.linspace(0, L, N)
+dx = 1/N
+T = 100
+M = 1000
+print(M)
+t = np.linspace(0, T, M)
+
+
+b, gamma, n, RT, delta = 0.1, 5, 6, 2, 3
+Du = 0.01
+Dv = 10
+
+print(dx**2/2*Dv)
+dt = T/M
+print(dt)
+
+width = 0.1
+height = 5
+u0 = np.zeros(N)
+u0[x > 1-width] = height
+# u0ic = 0.05645
+# u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-width*height)*np.ones(np.size(x))
+y0 = np.zeros(2*N)
+y0[::2] = u0
+y0[1::2] = v0
+
+# NON POLAR LOW RAC
+b = 0.1
+delta = 7.5
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx), ml=2, mu=2)
+
+u = sol[:,::2]
+v = sol[:,1::2]
+
+plt.figure()
+plt.pcolormesh(u)
+plt.colorbar()
+
+fig = plt.figure("fig1",figsize=(4,2.6))
+ax = plt.subplot(221)
+ax.set_xlim(0,1)
+ax.set_ylim(-0.1,2.1)
+ax.grid(linewidth=1.5)
+# line_ic, = ax.plot(x,u[0,:],':',linewidth=4,color=(221/255,170/255,51/255),label=r'IC')
+line_ss, = ax.plot(x,u[-1,:],linewidth=4,color=(221/255,170/255,51/255),label=r'Steady-state')
+ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax.spines["left"].set_linewidth(1.5)
+ax.spines["top"].set_linewidth(1.5)
+ax.spines["right"].set_linewidth(1.5)
+ax.spines["bottom"].set_linewidth(1.5)
+
+# POLARIZABLE
+b = 0.1
+delta = 3
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx), ml=2, mu=2)
+
+u = sol[:,::2]
+v = sol[:,1::2]
+
+ax = plt.subplot(223)
+ax.set_xlim(0,1)
+ax.set_ylim(-0.1,2.1)
+ax.grid(linewidth=1.5)
+# line_ic, = ax.plot(x,u[0,:],':',linewidth=4,color=(221/255,170/255,51/255),label=r'IC')
+line_ss, = ax.plot(x,u[-1,:],linewidth=4,color=(221/255,170/255,51/255),label=r'Steady-state')
+ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax.spines["left"].set_linewidth(1.5)
+ax.spines["top"].set_linewidth(1.5)
+ax.spines["right"].set_linewidth(1.5)
+ax.spines["bottom"].set_linewidth(1.5)
+
+#TURING REGIME
+b = 4.5
+delta = 7.5
+
+width = 0.1
+height = 2
+u0ic = 1
+u0 = u0ic*np.ones(np.size(x))+0.1*np.sin(4*np.pi*x)
+v0 = (RT-1)*np.ones(np.size(x))
+y0 = np.zeros(2*N)
+y0[::2] = u0
+y0[1::2] = v0
+
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx), ml=2, mu=2)
+
+u = sol[:,::2]
+v = sol[:,1::2]
+
+ax = plt.subplot(222)
+ax.set_xlim(0,1)
+ax.set_ylim(-0.1,2.1)
+ax.grid(linewidth=1.5)
+# line_ic, = ax.plot(x,u[0,:],':',linewidth=4,color=(221/255,170/255,51/255),label=r'IC')
+line_ss, = ax.plot(x,u[-1,:],linewidth=4,color=(221/255,170/255,51/255),label=r'Steady-state')
+ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax.spines["left"].set_linewidth(1.5)
+ax.spines["top"].set_linewidth(1.5)
+ax.spines["right"].set_linewidth(1.5)
+ax.spines["bottom"].set_linewidth(1.5)
+
+#HIGH RAC, NON POLAR
+b = 4.5
+delta = 3
+width = 0.1
+height = 5
+u0 = np.zeros(N)
+u0[x > 1-width] = height
+# u0ic = 0.05645
+# u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-width*height)*np.ones(np.size(x))
+y0 = np.zeros(2*N)
+y0[::2] = u0
+y0[1::2] = v0
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx), ml=2, mu=2)
+
+u = sol[:,::2]
+v = sol[:,1::2]
+
+ax = plt.subplot(224)
+ax.set_xlim(0,1)
+ax.set_ylim(-0.1,2.1)
+ax.grid(linewidth=1.5)
+# line_ic, = ax.plot(x,u[0,:],':',linewidth=4,color=(221/255,170/255,51/255),label=r'IC')
+line_ss, = ax.plot(x,u[-1,:],linewidth=4,color=(221/255,170/255,51/255),label=r'Steady-state')
+ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax.spines["left"].set_linewidth(1.5)
+ax.spines["top"].set_linewidth(1.5)
+ax.spines["right"].set_linewidth(1.5)
+ax.spines["bottom"].set_linewidth(1.5)
+
+plt.tight_layout()
+plt.savefig('Fig2b.tiff',dpi=1200)
+plt.show()
+
+
+# %%
+# check mass conservation
+print(np.sum(dx*(u[0,:] + v[0,:])))
+print(np.sum(dx*(u[-1,:] + v[-1,:])))
+mass = []
+for i in range(len(t)):
+ mass.append(np.sum(dx*(u[i,:] + v[i,:])))
+
+fig = plt.figure("fig2",figsize=(4,3))
+ax1 = plt.subplot(111)
+plt.plot(t,mass,linewidth=4)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$t$')
+ax1.set_ylabel(r'$\int_\Omega R + R_i \, dx$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+ax1.set_xlim(0,T)
+ax1.grid(linewidth=1.5)
+plt.tight_layout()
+# plt.savefig('Fig1.png',dpi=600)
+plt.show()
diff --git a/Fig3a.m b/Fig3a.m
new file mode 100644
index 0000000..f44b2d8
--- /dev/null
+++ b/Fig3a.m
@@ -0,0 +1,325 @@
+bettercolors;
+
+global cds sys
+
+sys.gui.pausespecial=0; %Pause at special points
+sys.gui.pausenever=1; %Pause never
+sys.gui.pauseeachpoint=0; %Pause at each point
+
+syshandle=@Fig2_Functions; %Specify system file
+
+SubFunHandles=feval(syshandle); %Get function handles from system file
+RHShandle=SubFunHandles{2}; %Get function handle for ODE
+
+b = 0.01;
+gma = 5;
+n = 6;
+RT = 2;
+delta = 2;
+
+local = 1;
+codim = 1;
+
+xinit=[0;0]; %Set ODE initial condition
+
+%Specify ODE function with ODE parameters set
+RHS_no_param=@(t,x)RHShandle(t,x,b,gma,n,RT,delta);
+
+%Set ODE integrator parameters.
+options=odeset;
+options=odeset(options,'RelTol',1e-8);
+options=odeset(options,'maxstep',1e-2);
+
+%Integrate until a steady state is found.
+[tout xout]=ode45(RHS_no_param,[0,200],xinit,options);
+
+xinit=xout(size(xout,1),:);
+
+pvec=[b,gma,n,RT,delta]'; % Initialize parameter vector
+
+ap=1;
+
+[x0,v0]=init_EP_EP(syshandle, xinit', pvec, ap); %Initialize equilibrium
+
+
+opt=contset;
+opt=contset(opt,'MaxNumPoints',1500); %Set numeber of continuation steps
+opt=contset(opt,'MaxStepsize',.01); %Set max step size
+opt=contset(opt,'Singularities',1); %Monitor singularities
+opt=contset(opt,'Eigenvalues',1); %Output eigenvalues
+opt=contset(opt,'InitStepsize',0.01); %Set Initial stepsize
+
+[x1,v1,s1,h1,f1]=cont(@equilibrium,x0,v0,opt);
+
+opt=contset(opt,'backward',1);
+[x1b,v1b,s1b,h1b,f1b]=cont(@equilibrium,x0,v0,opt);
+
+
+if local == 1
+%% LOCAL BRANCHES
+
+ind = 2;
+xBP=x1(1:2,s1(ind).index); %Extract branch point
+pvec(ap)=x1(3,s1(ind).index); %Extract branch point
+[x0,v0]=init_BP_EP(syshandle, xBP, pvec, s1(ind), 0.05);
+opt=contset(opt,'backward',1);
+[x3b,v3b,s3b,h3b,f3b]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+opt = contset(opt,'backward',0);
+[x3,v3,s3,h3,f3]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+
+ind = 7;
+xBP=x1(1:2,s1(ind).index); %Extract branch point
+pvec(ap)=x1(3,s1(ind).index); %Extract branch point
+[x0,v0]=init_BP_EP(syshandle, xBP, pvec, s1(ind), 0.05);
+opt = contset(opt,'backward',1);
+[x4b,v4b,s4b,h4b,f4b]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+opt = contset(opt,'backward',0);
+[x4,v4,s4,h4,f4]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+
+end
+% CODIM 1 plots
+
+
+width=5.2/2;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 12;
+f = figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+Fig2a = subplot(1,1,1);
+xlabel(Fig2a,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+ylabel(Fig2a,{'$R^\ell$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+Fig2aTitle = title(Fig2a,{'(a)'},'FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Times');
+set(Fig2a,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+grid
+Fig2a.XLim = [0 4];
+Fig2a.YLim = [0 5];
+Fig2a.Box = 'on';
+set(gca,'LineWidth',1.5)
+hold on
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%GLOBAL curves
+curves = {'1','1b'};
+
+for i = 1:length(curves)
+ xeqcurve=eval(['x' curves{i}]);
+ minevaleq=eval(['f' curves{i} '(2,:)']); %This is the last eigenvalue. That is the one that determines stability
+
+ L=length(xeqcurve(1,:));
+
+ curveind=1;
+ lengthind=0;
+ maxlengthind=0;
+ evalstart=floor(heaviside(minevaleq(1)));
+ datamateq=zeros(4,L);
+
+ for i=1:L
+ evalind=floor(heaviside(minevaleq(i)));
+ if evalstart~=evalind
+ curveind=curveind+1;
+ i;
+ evalstart=evalind;
+ maxlengthind=max(lengthind,maxlengthind);
+ lengthind=0;
+ end
+ datamateq(1,i)=xeqcurve(3,i); % This is the parameter that is varied.
+ datamateq(2,i)=xeqcurve(1,i); % This is the dependent axis of the bifurcation plot. The one you wish to plot
+ datamateq(3,i)=evalind;
+ datamateq(4,i)=curveind;
+
+ lengthind=lengthind+1;
+ end
+
+ maxlengthind=max(maxlengthind,lengthind);
+
+ curveindeq=curveind;
+
+ for i=1:curveindeq
+ index=find(datamateq(4,:)==i);
+ eval(['curve' num2str(i) 'eq' '=datamateq(1:3,index);']);
+ end
+
+ for i=1:curveindeq
+ stability=eval(['curve' num2str(i) 'eq(3,1)']);
+ if stability==0
+ plotsty='-';
+ else
+ plotsty=':';
+ end
+
+ plotcolor='k';
+
+ plotstr=strcat(plotcolor,plotsty);
+
+ plot(eval(['curve' num2str(i) 'eq(1,:)']),eval(['curve' num2str(i) 'eq(2,:)']),plotstr,'Linewidth',3)
+ hold on
+ end
+end
+
+if local == 1
+%%%LOCAL
+curves = {'3','3b'};
+
+for i = 1:length(curves)
+ xeqcurve=eval(['x' curves{i}]);
+ minevaleq=eval(['f' curves{i} '(2,:)']); %This is the last eigenvalue. That is the one that determines stability
+
+ L=length(xeqcurve(1,:));
+
+ curveind=1;
+ lengthind=0;
+ maxlengthind=0;
+ evalstart=floor(heaviside(minevaleq(1)));
+ datamateq=zeros(4,L);
+
+ for i=1:L
+ evalind=floor(heaviside(minevaleq(i)));
+ if evalstart~=evalind
+ curveind=curveind+1;
+ i;
+ evalstart=evalind;
+ maxlengthind=max(lengthind,maxlengthind);
+ lengthind=0;
+ end
+ datamateq(1,i)=xeqcurve(3,i); % This is the parameter that is varied.
+ datamateq(2,i)=xeqcurve(2,i); % This is the dependent axis of the bifurcation plot. The one you wish to plot
+ datamateq(3,i)=evalind;
+ datamateq(4,i)=curveind;
+
+ lengthind=lengthind+1;
+ end
+
+ maxlengthind=max(maxlengthind,lengthind);
+
+ curveindeq=curveind;
+
+ for i=1:curveindeq
+ index=find(datamateq(4,:)==i);
+ eval(['curve' num2str(i) 'eq' '=datamateq(1:3,index);']);
+ end
+
+ for i=1:curveindeq
+ stability=eval(['curve' num2str(i) 'eq(3,1)']);
+ if stability==0
+ plotsty='-';
+ else
+ plotsty=':';
+ end
+
+ mycolor = highcontrast(2,:);
+ plotstr=plotsty;
+
+ plot(eval(['curve' num2str(i) 'eq(1,:)']),eval(['curve' num2str(i) 'eq(2,:)']),plotstr,'Color',mycolor,'Linewidth',2)
+ hold on
+ end
+end
+
+end
+
+opt=contset;
+opt=contset(opt,'MaxNumPoints',5500); %Set numeber of continuation steps
+opt=contset(opt,'MaxStepsize',1); %Set max step size
+opt=contset(opt,'Singularities',1); %Monitor singularities
+opt=contset(opt,'Eigenvalues',1); %Output eigenvalues
+opt=contset(opt,'InitStepsize',0.5); %Set Initial stepsize
+aps = [1,5];
+%%%WP
+ind = 4;
+xtmp = x3(1:end-1,s3(ind).index);
+pvec(ap) = x3(end,s3(ind).index);
+[x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x5,v5,s5,h5,f5] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x5b,v5b,s5b,h5b,f5b] = cont(@limitpoint,x0,v0,opt);
+
+ind = 5
+xtmp = x3b(1:end-1,s3b(ind).index);
+pvec(ap) = x3b(end,s3b(ind).index);
+[x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x6,v6,s6,h6,f6] = cont(@limitpoint,x0,v0,opt);
+ opt = contset(opt,'backward',1);
+[x6b,v6b,s6b,h6b,f6b] = cont(@limitpoint,x0,v0,opt);
+
+%BPs
+ind = 2;
+xtmp = x3b(1:end-1,s3b(ind).index);
+pvec(ap) = x3b(end,s3b(ind).index);
+[x0,v0] = init_BP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x7,v7,s7,h7,f7] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x7b,v7b,s7b,h7b,f7b] = cont(@limitpoint,x0,v0,opt);
+
+ind = 8;
+xtmp = x3b(1:end-1,s3b(ind).index);
+pvec(ap) = x3b(end,s3b(ind).index);
+[x0,v0] = init_BP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x8,v8,s8,h8,f8] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x8b,v8b,s8b,h8b,f8b] = cont(@limitpoint,x0,v0,opt);
+
+%%%
+close 1
+
+width=5.2/2;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 10;
+
+figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+
+Fig1b = subplot(1,1,1);
+xlabel(Fig1b,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',12,'FontName','Helvetica','color','k')
+
+% Fig1bTitle = title(Fig1b,{'(b)'},'FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica');
+set(Fig1b,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+hold on
+grid
+Fig1b.XLim = [0 6];
+Fig1b.YLim = [0 10];
+Fig1b.Box = 'on';
+Fig1b.XColor = 'k';
+Fig1b.YColor = 'k';
+ylabel({'$\delta$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',12,'FontName','Helvetica','color','k')
+set(gca,'LineWidth',1.5)
+curves = {'5','6','7','8'};
+
+linecolor(5) = {highcontrast(2,:)};
+linecolor(6) = {highcontrast(2,:)};
+linecolor(7) = {highcontrast(3,:)};
+linecolor(8) = {highcontrast(3,:)};
+
+
+linestyle(5) = {'-'};
+linestyle(6) = {'-'};
+linestyle(7) = {'--'};
+linestyle(8) = {'--'};
+
+for i = 5:8
+ xeqcurve=eval(['x' num2str(i)]);
+ plot(xeqcurve(end-1,:),xeqcurve(end,:),'LineStyle',linestyle{i},'Color',linecolor{i},'LineWidth',4)
+ xeqcurve = eval(['x' num2str(i) 'b']);
+ plot(xeqcurve(end-1,:),xeqcurve(end,:),'LineStyle',linestyle{i},'Color',linecolor{i},'LineWidth',4)
+end
+
+% scatter(0.1,7,'ko','filled')
+% % text(0.1,7,' (b)')
+%
+% scatter(0.1,3,'ko','filled')
+% % text(0.1,3,' (c)')
+%
+% scatter(4.5,6,'ko','filled')
+% % text(4.5,6,' (d)')
+%
+% scatter(4.5,3,'ko','filled')
+% text(4.5,3,' (e)')
+quiver(0.1,3,4.4,0,0,'k','LineWidth',3)
+quiver(4.5,3,0,4.5,0,'k','LineWidth',3)
+
+% print(1,'3a','-depsc')
+print(1,'3a.eps','-depsc','-painters')
diff --git a/Fig3b.py b/Fig3b.py
new file mode 100644
index 0000000..6ce873e
--- /dev/null
+++ b/Fig3b.py
@@ -0,0 +1,181 @@
+import matplotlib
+# matplotlib.use('Qt5Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+from scipy.integrate import odeint
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+
+plt.rc("text", usetex=False)
+plt.rc("font", **{'family':'sans-serif', 'sans-serif':['Helvetica'], 'size':10})
+
+def f(u, v, b, gamma, n, RT, delta):
+ return (b+gamma*u**n/(1+u**n))*v - delta*u
+
+def g(u, v, b, gamma, n, RT, delta):
+ return -f(u,v,b, gamma, n, RT, delta)
+
+def rdPDE(y, t, b, gamma, n, RT, delta, Du, Dv, dx):
+ """
+ The ODEs are derived using the method of lines.
+ https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#mol
+ """
+ # The vectors u and v are interleaved in y. We define
+ # views of u and v by slicing y.
+ u = y[::2]
+ v = y[1::2]
+
+ # dydt is the return value of this function.
+ dydt = np.empty_like(y)
+
+ dudt = dydt[::2]
+ dvdt = dydt[1::2]
+
+ if (t>50) & (t<=100):
+ # b = b + 4/50*(t-50)
+ b = 4.5
+ elif (t>100) & (t<=250):
+ b = 4.5
+ delta = delta+4.5/150*(t-100)
+ elif t>250:
+ b = 4.5
+ delta = delta+4.5/150*(250-100)
+
+ #this is neumann:
+ dudt[0] = f(u[0], v[0], b, gamma, n, RT, delta) + Du * (-2.0*u[0] + 2.0*u[1]) / dx**2
+ dudt[1:-1] = f(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + Du * np.diff(u,2) / dx**2
+ dudt[-1] = f(u[-1], v[-1], b, gamma, n, RT, delta) + Du * (-2.0*u[-1] + 2.0*u[-2]) / dx**2
+ dvdt[0] = g(u[0], v[0], b, gamma, n, RT, delta) + Dv * (-2.0*v[0] + 2.0*v[1]) / dx**2
+ dvdt[1:-1] = g(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + Dv * np.diff(v,2) / dx**2
+ dvdt[-1] = g(u[-1], v[-1], b, gamma, n, RT, delta) + Dv * (-2.0*v[-1] + 2.0*v[-2]) / dx**2
+
+ dydt[::2] = dudt
+ dydt[1::2] = dvdt
+
+ return dydt
+
+# %%
+# %%time
+L = 1
+N = 1000
+x = np.linspace(0, L, N)
+dx = 1/N
+T = 500
+M = 2500
+t = np.linspace(0, T, M)
+
+b, gamma, n, RT, delta = 0.1, 5, 6, 2, 3
+Du = 0.01
+Dv = 10
+
+# width = 0.1
+# height = 2
+# u0 = np.zeros(N)
+# u0[x > 1-width] = height
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+y0 = np.zeros(2*N)
+y0[::2] = u0
+y0[1::2] = v0
+
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx), ml=2, mu=2)
+
+# %%
+
+u = sol[:,::2]
+v = sol[:,1::2]
+
+fig = plt.figure("fig1",figsize=(2.6,2.6))
+ax1 = plt.subplot(111)
+pmesh = plt.pcolormesh(x,t,u,cmap=cm.hot)
+cbar = fig.colorbar(pmesh,ax=ax1)
+plt.axhline(y=50,linestyle='--',linewidth=2,color='0.5')
+plt.axhline(y=100,linestyle='--',linewidth=2,color='0.5')
+plt.axhline(y=250,linestyle='--',linewidth=2,color='0.5')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+
+plt.tight_layout()
+plt.savefig('Fig3b.tiff',dpi=1200)
+plt.show()
+
+
+# %%
+# check mass conservation
+print(np.sum(dx*(u[0,:] + v[0,:])))
+print(np.sum(dx*(u[-1,:] + v[-1,:])))
+mass = []
+for i in range(len(t)):
+ mass.append(np.sum(dx*(u[i,:] + v[i,:])))
+
+fig = plt.figure("fig2",figsize=(2.6,2.6))
+ax1 = plt.subplot(111)
+plt.plot(t,mass,linewidth=2)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$t$')
+ax1.set_ylabel(r'$\int_\Omega R + R_i \, dx$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+ax1.set_xlim(0,T)
+ax1.grid(linewidth=1.5)
+plt.tight_layout()
+plt.savefig('Fig3bmass.tiff')
+plt.show()
+
+
+#%%
+#animated plot
+import matplotlib.animation as animation
+fig = plt.figure(figsize=(4,3))
+ax = plt.subplot(111)
+ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax.set_xlabel(r'$x$')
+ax.set_ylabel('Activity')
+ax.spines["left"].set_linewidth(1.5)
+ax.spines["top"].set_linewidth(1.5)
+ax.spines["right"].set_linewidth(1.5)
+ax.spines["bottom"].set_linewidth(1.5)
+ax.set_xlim(0,1)
+ax.set_ylim(0,2)
+ax.grid(linewidth=1.5)
+title = plt.title(r'$b$=%1.2f, $\delta$=%1.2f' %(b, delta))
+line_u, = ax.plot(x,u[0,:],linewidth=4,color=(221/255,170/255,51/255),label=r'$R$')
+# line_v, = ax.plot(x,v[0,:],'--',linewidth=2,color=(0/255,68/255,136/255),label=r'$R_i$')
+plt.legend(loc=2)
+plt.tight_layout()
+
+bvals = np.zeros(np.shape(t))
+deltavals = np.zeros(np.shape(t))
+for i in range(len(t)):
+ ti = t[i]
+ if (ti>50) & (ti<=100):
+ # bvals[i] = b + 4/50*(ti-50)
+ bvals[i] = 4.5
+ deltavals[i] = delta
+ elif (ti>100) & (ti<=250):
+ bvals[i] = 4.5
+ deltavals[i] = delta+4.5/150*(ti-100)
+ elif ti>250:
+ bvals[i] = 4.5
+ deltavals[i] = delta+4.5/150*(250-100)
+ else:
+ bvals[i] = b
+ deltavals[i] = delta
+
+def animate(i):
+ title.set_text(r'$t$=%1.2f, $b$=%1.2f, $\delta$=%1.2f' %(t[i],bvals[i], deltavals[i]))
+ line_u.set_ydata(u[i,:])
+ # line_v.set_ydata(v[i,:])
+ return line_u, title #line_v, title
+
+ani = animation.FuncAnimation(fig,animate,frames=M)
+ani.save("Movie 1.mp4",fps=30,dpi=300)
diff --git a/Fig4b.py b/Fig4b.py
new file mode 100644
index 0000000..afb4829
--- /dev/null
+++ b/Fig4b.py
@@ -0,0 +1,255 @@
+import matplotlib
+# matplotlib.use('Qt5Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+from scipy.integrate import odeint
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+import scipy.io
+
+plt.rc("text", usetex=False)
+plt.rc("font", **{'family':'sans-serif', 'sans-serif':['Helvetica'], 'size':10})
+
+def f(u, v, b, gamma, n, RT, delta):
+ return (b+gamma*u**n/(1+u**n))*v - delta*u
+
+def g(u, v, b, gamma, n, RT, delta):
+ return -f(u,v,b, gamma, n, RT, delta)
+
+def F(R):
+ sharp = 10
+ switch = 1
+ magnitude = 0.001
+ return magnitude/(1+np.exp(-2*sharp*(R-switch)))
+
+def xmdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (spring*(xp-xm-1) - F(u[0]))/viscosity
+
+def xpdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (-spring*(xp-xm-1) + F(u[-1]))/viscosity
+
+def rdPDE(y, t, b, gamma, n, RT, delta, Du, Dv, dx):
+ """
+ The ODEs are derived using the method of lines.
+ https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#mol
+ """
+ # The vectors u and v are interleaved in y. We define
+ # views of u and v by slicing y.
+ #the ode for L is stored at the end of y
+ u = y[0:-2:2]
+ v = y[1:-2:2]
+ xm = y[-2]
+ xp = y[-1]
+
+ l = xp-xm
+
+ if (t>50):
+ b = 4
+
+ delta = 3 + 1*(l-1)
+
+ # dydt is the return value of this function.
+ dydt = np.empty_like(y)
+
+ dudt = dydt[0:-2:2]
+ dvdt = dydt[1:-2:2]
+ dxmdt = dydt[-2]
+ dxpdt = dydt[-1]
+
+ dudt[0] = f(u[0], v[0], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[0] + 2.0*u[1]) / dx**2 - u[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[1:-1] = f(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * np.diff(u,2) / dx**2 - u[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[-1] = f(u[-1], v[-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[-1] + 2.0*u[-2]) / dx**2 - u[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dvdt[0] = g(u[0], v[0], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[0] + 2.0*v[1]) / dx**2 - v[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[1:-1] = g(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * np.diff(v,2) / dx**2 - v[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[-1] = g(u[-1], v[-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[-1] + 2.0*v[-2]) / dx**2 - v[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dxmdt = xmdot(u,v,xm,xp,t)
+ dxpdt = xpdot(u,v,xm,xp,t)
+
+ dydt[0:-2:2] = dudt
+ dydt[1:-2:2] = dvdt
+ dydt[-2] = dxmdt
+ dydt[-1] = dxpdt
+
+ return dydt
+
+# %%
+# %%time
+L = 1
+N = 1000
+x = np.linspace(0, L, N)
+dx = 1/N
+T = 1000
+M = 2500
+t = np.linspace(0, T, M)
+
+b, gamma, n, RT, delta = 0.1, 5, 6, 2, 3
+Du = 0.01
+Dv = 10
+
+#b = 0.1 steadystate
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+
+y0 = np.zeros(2*N+2)
+xm0 = 0
+xp0 = 1
+y0[0:-2:2] = u0
+y0[1:-2:2] = v0
+y0[-2] = xm0
+y0[-1] = xp0
+
+
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx))
+
+# %%
+
+u = sol[:,0:-2:2]
+v = sol[:,1:-2:2]
+xm = sol[:,-2]
+xp = sol[:,-1]
+l = xp - xm
+
+scipy.io.savemat('data.mat',dict(t=t,l=l,u=u))
+
+# %%
+fig = plt.figure("fig1",figsize=(5.2,2.6))
+ax1 = plt.subplot(121)
+pmesh = plt.pcolormesh(x,t,u,cmap=cm.hot)
+cbar = fig.colorbar(pmesh,ax=ax1)
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$\bar x$')
+ax1.set_ylabel(r'$t$')
+ax1.set_title(r'$R(\bar x,t)$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+
+ax2 = plt.subplot(122)
+pmesh =plt.pcolormesh(x,t,v,cmap=cm.hot)
+ax2.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+cbar = fig.colorbar(pmesh,ax=ax2)
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax2.set_xlabel(r'$\bar x$')
+ax2.set_ylabel(r'$t$')
+ax2.set_title(r'$R_i(\bar x,t)$')
+ax2.spines["left"].set_linewidth(1.5)
+ax2.spines["top"].set_linewidth(1.5)
+ax2.spines["right"].set_linewidth(1.5)
+ax2.spines["bottom"].set_linewidth(1.5)
+
+plt.tight_layout()
+plt.savefig('Fig14_fixed.png',dpi=1200)
+plt.show()
+
+# %%
+Xgrid, Tgrid = np.meshgrid(x,t)
+Xgrid = np.empty_like(Xgrid)
+for i in range(len(t)):
+ Xgrid[i,:] = x*l[i]+xm[i]
+# %%
+
+fig = plt.figure("fig1",figsize=(2.6,2.6))
+ax1 = plt.subplot(111)
+pmesh = plt.pcolormesh(Xgrid,Tgrid,u,cmap=cm.hot,vmin=0,vmax=2)
+cbar = fig.colorbar(pmesh,ax=ax1)
+xmline = plt.plot(xm,t,linewidth=1.5,color='k')
+xpline = plt.plot(xp,t,linewidth=1.5,color='k')
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+# ax1.set_xlabel(r'$x$')
+# ax1.set_ylabel(r'$t$')
+# ax1.set_title(r'$R(x,t)$')
+ax1.set_facecolor((0.7,0.7,0.7))
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+
+plt.tight_layout()
+plt.savefig('Polar.tiff',dpi=1200)
+# plt.savefig('Fig14_moving.eps')
+plt.show()
+#%%
+#
+fig = plt.figure(figsize=(4,3))
+plt.plot(t,l)
+plt.savefig('length_3b.tif')
+# plt.plot(t,delta)
+
+# %%
+# check mass conservation
+print(np.sum(dx*l[0]*(u[0,:] + v[0,:])))
+print(np.sum(dx*l[-1]*(u[-1,:] + v[-1,:])))
+mass = []
+for i in range(len(t)):
+ mass.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+
+fig = plt.figure("fig2",figsize=(4,3))
+ax1 = plt.subplot(111)
+plt.plot(t,mass,linewidth=2)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$t$')
+ax1.set_ylabel(r'$\int_\Omega R + R_i \, dx$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+ax1.set_xlim(0,T)
+ax1.grid(linewidth=1.5)
+plt.tight_layout()
+# plt.savefig('Fig14mass.png',dpi=1200)
+plt.show()
+
+# #%%
+# # animated plot
+
+movieon = 1
+if movieon == 1:
+ import matplotlib.animation as animation
+ fig = plt.figure(figsize=(4,3))
+ ax = plt.subplot(111)
+ ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ ax.set_xlabel(r'$x$')
+ ax.set_ylabel('Activity')
+ ax.spines["left"].set_linewidth(1.5)
+ ax.spines["top"].set_linewidth(1.5)
+ ax.spines["right"].set_linewidth(1.5)
+ ax.spines["bottom"].set_linewidth(1.5)
+ ax.set_xlim(np.min(Xgrid),np.max(Xgrid))
+ ax.set_ylim(0,np.max(u)+0.5)
+ ax.grid(linewidth=1.5)
+ # title = plt.title(r'$b$=%1.2f, $\delta$=%1.2f' %(b, delta))
+ line_u, = ax.plot(Xgrid[0,:],u[0,:],linewidth=4,color=(221/255,170/255,51/255),label=r'$R$')
+ # line_v, = ax.plot(Xgrid[0,:],v[0,:],'--',linewidth=2,color=(0/255,68/255,136/255),label=r'$R_i$')
+ plt.legend(loc=2)
+ vertline = ax.axvline(Xgrid[0,-1],ls='-',color='k')
+ vertline2 = ax.axvline(Xgrid[0,0],ls='-',color='k')
+ plt.tight_layout()
+
+ def animate(i):
+ # title.set_text(r'$b$=%1.2f, $\delta$=%1.2f' %(bvals[i], deltavals[i]))
+ line_u.set_xdata(Xgrid[i,:])
+ # line_v.set_xdata(Xgrid[i,:])
+ line_u.set_ydata(u[i,:])
+ # line_v.set_ydata(v[i,:])
+ vertline.set_xdata(Xgrid[i,-1])
+ vertline2.set_xdata(Xgrid[i,0])
+ return line_u, vertline, vertline2
+
+ ani = animation.FuncAnimation(fig,animate,frames=M)
+ ani.save("Movie 2.mp4",fps=30,dpi=300)
diff --git a/Fig4c.py b/Fig4c.py
new file mode 100644
index 0000000..2c46aa3
--- /dev/null
+++ b/Fig4c.py
@@ -0,0 +1,269 @@
+import matplotlib
+# matplotlib.use('Qt5Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+from scipy.integrate import odeint
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+import scipy.io
+
+
+plt.rc("text", usetex=False)
+plt.rc("font", **{'family':'sans-serif', 'sans-serif':['Helvetica'], 'size':10})
+
+def f(u, v, b, gamma, n, RT, delta):
+ return (b+gamma*u**n/(1+u**n))*v - delta*u
+
+def g(u, v, b, gamma, n, RT, delta):
+ return -f(u,v,b, gamma, n, RT, delta)
+
+def F(R):
+ sharp = 10
+ switch = 1
+ magnitude = 0.001
+ return magnitude/(1+np.exp(-2*sharp*(R-switch)))
+
+def xmdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (spring*(xp-xm-1) - F(u[0]))/viscosity
+
+def xpdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (-spring*(xp-xm-1) + F(u[-1]))/viscosity
+
+def rdPDE(y, t, b, gamma, n, RT, delta, Du, Dv, dx):
+ """
+ The ODEs are derived using the method of lines.
+ https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#mol
+ """
+ # The vectors u and v are interleaved in y. We define
+ # views of u and v by slicing y.
+ #the ode for L is stored at the end of y
+ u = y[0:-2:2]
+ v = y[1:-2:2]
+ xm = y[-2]
+ xp = y[-1]
+
+ l = xp-xm
+
+ if (t>50):
+ b = 1.5
+
+ delta = 3 + 1*(l-1)
+
+ # dydt is the return value of this function.
+ dydt = np.empty_like(y)
+
+ dudt = dydt[0:-2:2]
+ dvdt = dydt[1:-2:2]
+ dxmdt = dydt[-2]
+ dxpdt = dydt[-1]
+
+ dudt[0] = f(u[0], v[0], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[0] + 2.0*u[1]) / dx**2 - u[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[1:-1] = f(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * np.diff(u,2) / dx**2 - u[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[-1] = f(u[-1], v[-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[-1] + 2.0*u[-2]) / dx**2 - u[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dvdt[0] = g(u[0], v[0], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[0] + 2.0*v[1]) / dx**2 - v[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[1:-1] = g(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * np.diff(v,2) / dx**2 - v[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[-1] = g(u[-1], v[-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[-1] + 2.0*v[-2]) / dx**2 - v[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dxmdt = xmdot(u,v,xm,xp,t)
+ dxpdt = xpdot(u,v,xm,xp,t)
+
+ dydt[0:-2:2] = dudt
+ dydt[1:-2:2] = dvdt
+ dydt[-2] = dxmdt
+ dydt[-1] = dxpdt
+
+ return dydt
+
+# %%
+# %%time
+L = 1
+N = 1000
+x = np.linspace(0, L, N)
+dx = 1/N
+T = 5000
+M = 2500
+t = np.linspace(0, T, M)
+
+b, gamma, n, RT, delta = 0.1, 5, 6, 2, 3
+
+Du = 0.01
+Dv = 10
+
+#polarized ic
+# width = 0.1
+# height = 2
+# u0 = np.zeros(N)
+# u0[x > 1-width] = height
+# v0 = (RT-width*height)*np.ones(np.size(x))
+
+#b = 0.1 steadystate
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+
+# u0ic = RT
+# u0 = u0ic*np.ones(np.size(x))
+# v0 = (RT-u0ic)*np.ones(np.size(x))
+
+
+y0 = np.zeros(2*N+2)
+xm0 = 0
+xp0 = 1
+y0[0:-2:2] = u0
+y0[1:-2:2] = v0
+y0[-2] = xm0
+y0[-1] = xp0
+
+
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx))
+
+# %%
+
+u = sol[:,0:-2:2]
+v = sol[:,1:-2:2]
+xm = sol[:,-2]
+xp = sol[:,-1]
+l = xp - xm
+
+scipy.io.savemat('data.mat',dict(t=t,l=l,u=u))
+# %%
+
+fig = plt.figure("fig1",figsize=(5.2,2.6))
+ax1 = plt.subplot(121)
+pmesh = plt.pcolormesh(x,t,u,cmap=cm.hot)
+cbar = fig.colorbar(pmesh,ax=ax1)
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$\bar x$')
+ax1.set_ylabel(r'$t$')
+ax1.set_title(r'$R(\bar x,t)$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+
+ax2 = plt.subplot(122)
+pmesh =plt.pcolormesh(x,t,v,cmap=cm.hot)
+ax2.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+cbar = fig.colorbar(pmesh,ax=ax2)
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax2.set_xlabel(r'$\bar x$')
+ax2.set_ylabel(r'$t$')
+ax2.set_title(r'$R_i(\bar x,t)$')
+ax2.spines["left"].set_linewidth(1.5)
+ax2.spines["top"].set_linewidth(1.5)
+ax2.spines["right"].set_linewidth(1.5)
+ax2.spines["bottom"].set_linewidth(1.5)
+
+plt.tight_layout()
+# plt.savefig('Fig13_fixed.png',dpi=1200)
+plt.show()
+
+# %%
+fig = plt.figure(figsize=(4,3))
+plt.plot(t,l)
+plt.savefig('length_4c.tif')
+
+# %%
+Xgrid, Tgrid = np.meshgrid(x,t)
+Xgrid = np.empty_like(Xgrid)
+for i in range(len(t)):
+ Xgrid[i,:] = x*l[i]+xm[i]
+
+fig = plt.figure("fig1",figsize=(2.6,2.6))
+ax1 = plt.subplot(111)
+pmesh = plt.pcolormesh(Xgrid,Tgrid,u,cmap=cm.hot,vmin=0,vmax=2)
+cbar = fig.colorbar(pmesh,ax=ax1)
+xmline = plt.plot(xm,t,linewidth=1.5,color='k')
+xpline = plt.plot(xp,t,linewidth=1.5,color='k')
+
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+# ax1.set_xlabel(r'$x$')
+# ax1.set_ylabel(r'$t$')
+# ax1.set_title(r'$R(x,t)$')
+ax1.set_facecolor((0.7,0.7,0.7))
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+#
+
+plt.tight_layout()
+plt.savefig('oscillation.tiff',dpi=1200)
+# plt.savefig('Fig14_moving.eps')
+plt.show()
+
+# %%
+# check mass conservation
+print(np.sum(dx*l[0]*(u[0,:] + v[0,:])))
+print(np.sum(dx*l[-1]*(u[-1,:] + v[-1,:])))
+mass = []
+for i in range(len(t)):
+ mass.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+
+fig = plt.figure("fig2",figsize=(4,3))
+ax1 = plt.subplot(111)
+plt.plot(t,mass,linewidth=2)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$t$')
+ax1.set_ylabel(r'$\int_\Omega R + R_i \, dx$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+ax1.set_xlim(0,T)
+ax1.grid(linewidth=1.5)
+plt.tight_layout()
+plt.savefig('mass_4c.png',dpi=1200)
+plt.show()
+#
+# #%%
+# # animated plot
+movieon = 1
+if movieon == 1:
+ import matplotlib.animation as animation
+ fig = plt.figure(figsize=(4,3))
+ ax = plt.subplot(111)
+ ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ ax.set_xlabel(r'$x$')
+ ax.set_ylabel('Activity')
+ ax.spines["left"].set_linewidth(1.5)
+ ax.spines["top"].set_linewidth(1.5)
+ ax.spines["right"].set_linewidth(1.5)
+ ax.spines["bottom"].set_linewidth(1.5)
+ ax.set_xlim(np.min(Xgrid),np.max(Xgrid))
+ ax.set_ylim(0,np.max(u)+0.5)
+ ax.grid(linewidth=1.5)
+ # title = plt.title(r'$b$=%1.2f, $\delta$=%1.2f' %(b, delta))
+ line_u, = ax.plot(Xgrid[0,:],u[0,:],linewidth=4,color=(221/255,170/255,51/255),label=r'$R$')
+ # line_v, = ax.plot(Xgrid[0,:],v[0,:],'--',linewidth=2,color=(0/255,68/255,136/255),label=r'$R_i$')
+ plt.legend(loc=2)
+ vertline = ax.axvline(Xgrid[0,-1],ls='-',color='k')
+ vertline2 = ax.axvline(Xgrid[0,0],ls='-',color='k')
+ plt.tight_layout()
+
+ def animate(i):
+ # title.set_text(r'$b$=%1.2f, $\delta$=%1.2f' %(bvals[i], deltavals[i]))
+ line_u.set_xdata(Xgrid[i,:])
+ # line_v.set_xdata(Xgrid[i,:])
+ line_u.set_ydata(u[i,:])
+ # line_v.set_ydata(v[i,:])
+ vertline.set_xdata(Xgrid[i,-1])
+ vertline2.set_xdata(Xgrid[i,0])
+ return line_u, vertline, vertline2
+
+ ani = animation.FuncAnimation(fig,animate,frames=M)
+ani.save("Movie 3.mp4",fps=30,dpi=300)
diff --git a/Fig5/0.5_data.mat b/Fig5/0.5_data.mat
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diff --git a/Fig5/1.0_data.mat b/Fig5/1.0_data.mat
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diff --git a/Fig5/1.5_data.mat b/Fig5/1.5_data.mat
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diff --git a/Fig5/2.0_data.mat b/Fig5/2.0_data.mat
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index 0000000..63089fb
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diff --git a/Fig5/2.5_data.mat b/Fig5/2.5_data.mat
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index 0000000..e025f5c
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diff --git a/Fig5/3.0_data.mat b/Fig5/3.0_data.mat
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index 0000000..f205f2c
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diff --git a/Fig5/3.5_data.mat b/Fig5/3.5_data.mat
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diff --git a/Fig5/4.0_data.mat b/Fig5/4.0_data.mat
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diff --git a/Fig5/4.5_data.mat b/Fig5/4.5_data.mat
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diff --git a/Fig5/5.0_data.mat b/Fig5/5.0_data.mat
new file mode 100644
index 0000000..0ee183b
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diff --git a/Fig5/Fig5a.m b/Fig5/Fig5a.m
new file mode 100644
index 0000000..43540e4
--- /dev/null
+++ b/Fig5/Fig5a.m
@@ -0,0 +1,56 @@
+bettercolors
+uiopen('/Users/zmurchc/Dropbox/Apps/Overleaf/neutrophil-tension-paper/code/Fig5/RT_2.fig',1)
+% figure;
+% hold on
+% %
+% width=5.2;
+% height=5.2/2;
+% x0 = 5;
+% y0 = 5;
+% fontsize = 12;
+% figure('Units','inches','Position',[x0 y0 width/2 height],'PaperPositionMode','auto');
+% Fig1b = subplot(1,1,1);
+% % xlabel(Fig1b,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% % ylabel(Fig1b,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+%
+% set(Fig1b,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% hold on
+% grid
+% Fig1b.XLim = [0 5.5];
+% Fig1b.YLim = [0 1.6];
+% Fig1b.Box = 'on';
+% ylabel({'$L$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% xlabel({''},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+
+% set(gca,'LineWidth',1.5)
+
+
+bvals = linspace(0.5,5,10);
+for j = 1:length(bvals)
+ bvalue = bvals(j)
+ filename = [num2str(bvalue,'%.1f'),'_data.mat'];
+ load(filename)
+
+ is_polar = zeros(size(u,1),1);
+ color = zeros(size(u,1),3);
+ for i = 1:size(u,1)
+ is_polar(i) = is_polarized(u(i,:));
+ color(i,:) = (is_polar(i)>0)*bright(6,:)+(is_polar(i)==0)*bright(2,:);
+ end
+
+ active_u = zeros(size(u,1),1);
+ for i = 1:size(u,1)
+ active_u(i) = mean(u(i,:));
+ end
+
+ % color = color.*(bvalue/3);
+ scatter(bvalue*ones(size(l)),l,16,color,'filled')
+ % scatter(bvalue+bvalue*t/35000,l,16,color,'filled')
+ % scatter(active_u,l,8,color,'filled')
+ % scatter(t,l,16,color,'filled')
+ % ylim([0.98 1.37])
+ % xlim([0 1.4])
+end
+ylim([0 1.85])
+
+print(1,'Fig5a','-depsc','-painters')
diff --git a/Fig5/Fig5b.m b/Fig5/Fig5b.m
new file mode 100644
index 0000000..321687a
--- /dev/null
+++ b/Fig5/Fig5b.m
@@ -0,0 +1,137 @@
+bettercolors
+
+% figure;
+% hold on
+% %
+width=3.4;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 10;
+figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+
+Fig1a = subplot(2,2,1);
+% xlabel(Fig1a,{'$t$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+%
+set(Fig1a,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+ylabel(Fig1a,'Length','FontUnits','points','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+
+hold on
+grid
+% Fig1b.XLim = [0 5.5];
+% Fig1b.YLim = [0 1.6];
+Fig1a.Box = 'on';
+Fig1a.XColor = 'k';
+Fig1a.YColor = 'k';
+
+set(gca,'LineWidth',1.5)
+
+bvals = linspace(0.5,5,10);
+
+bI = bvals(1:2);
+bII = bvals(3);
+bIII = bvals(8);
+bIV = bvals(end);
+
+for j = 1:length(bIII)
+ bvalue = bIII(j)
+ filename = [num2str(bvalue,'%.1f'),'_data.mat'];
+ load(filename)
+
+ is_polar = zeros(size(u,1),1);
+ differences = zeros(size(u,1),1);
+ color = zeros(size(u,1),3);
+ for i = 1:size(u,1)
+ is_polar(i) = is_polarized(u(i,:));
+ differences(i) = max(u(i,:))-min(u(i,:));
+ color(i,:) = (is_polar(i)>0)*bright(6,:)+(is_polar(i)==0)*bright(2,:);
+ end
+
+ active_u = zeros(size(u,1),1);
+ for i = 1:size(u,1)
+ active_u(i) = mean(u(i,:));
+ end
+
+ % color = color.*(bvalue/3);
+ % scatter(bvalue*ones(size(l)),l,16,color,'filled')
+ % scatter(bvalue+bvalue*t/35000,l,16,color,'filled')
+ % scatter(active_u,l,8,color,'filled')
+ scatter(Fig1a,t,l,10,color,'filled')
+ xlim([0 2000])
+
+
+ Fig1b = subplot(2,2,3);
+ Fig1b.Box = 'on';
+ Fig1b.XColor = 'k';
+ Fig1b.YColor = 'k';
+ set(gca,'LineWidth',1.5)
+ set(Fig1b,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+ hold on
+ ylabel(Fig1b,{'max - min Rac'},'FontUnits','points','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+ xlabel(Fig1b,{'Time'},'FontUnits','points','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+ grid
+
+ scatter(Fig1b,t,differences,10,color,'filled')
+ xlim([0 2000])
+
+end
+
+Fig1c = subplot(2,2,2);
+% xlabel(Fig1a,{'$t$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+% ylabel(Fig1a,{'$L$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+%
+set(Fig1c,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+hold on
+grid
+% Fig1b.XLim = [0 5.5];
+% Fig1b.YLim = [0 1.6];
+Fig1c.Box = 'on';
+Fig1c.XColor = 'k';
+Fig1c.YColor = 'k';
+set(gca,'LineWidth',1.5)
+
+for j = 1:length(bII)
+ bvalue = bII(j)
+ filename = [num2str(bvalue,'%.1f'),'_data.mat'];
+ load(filename)
+
+ is_polar = zeros(size(u,1),1);
+ differences = zeros(size(u,1),1);
+ color = zeros(size(u,1),3);
+ for i = 1:size(u,1)
+ is_polar(i) = is_polarized(u(i,:));
+ differences(i) = max(u(i,:))-min(u(i,:));
+ color(i,:) = (is_polar(i)>0)*bright(6,:)+(is_polar(i)==0)*bright(2,:);
+ end
+
+ active_u = zeros(size(u,1),1);
+ for i = 1:size(u,1)
+ active_u(i) = mean(u(i,:));
+ end
+
+ % color = color.*(bvalue/3);
+ % scatter(bvalue*ones(size(l)),l,16,color,'filled')
+ % scatter(bvalue+bvalue*t/35000,l,16,color,'filled')
+ % scatter(active_u,l,8,color,'filled')
+ scatter(Fig1c,t,l,10,color,'filled')
+ xlim([0 2000])
+
+
+
+ Fig1d = subplot(2,2,4);
+ Fig1d.Box = 'on';
+ Fig1d.XColor = 'k';
+ Fig1d.YColor = 'k';
+ set(gca,'LineWidth',1.5)
+ set(Fig1d,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+
+ hold on
+ % xlabel(Fig1d,{'Time'},'FontUnits','points','FontWeight','normal','FontSize',10,'FontName','Helvetica')
+ grid
+
+ scatter(Fig1d,t,differences,10,color,'filled')
+ xlim([0 2000])
+
+end
+
+print(1,'Fig5b','-depsc','-painters')
diff --git a/Fig5/RT_2.fig b/Fig5/RT_2.fig
new file mode 100644
index 0000000..eb49cd9
Binary files /dev/null and b/Fig5/RT_2.fig differ
diff --git a/Fig5/RT_2.m b/Fig5/RT_2.m
new file mode 100644
index 0000000..408123b
--- /dev/null
+++ b/Fig5/RT_2.m
@@ -0,0 +1,384 @@
+bettercolors;
+
+global cds sys
+
+sys.gui.pausespecial=0; %Pause at special points
+sys.gui.pausenever=1; %Pause never
+sys.gui.pauseeachpoint=0; %Pause at each point
+
+syshandle=@funcs; %Specify system file
+
+SubFunHandles=feval(syshandle); %Get function handles from system file
+RHShandle=SubFunHandles{2}; %Get function handle for ODE
+
+b = 0.1;
+L= 0.9;
+RT=2;
+delta0=3;
+delta1=1;
+l0=1;
+gma=5;
+n=6;
+V=0;
+
+local = 1;
+codim = 0;
+
+xinit=[0;0]; %Set ODE initial condition
+
+%Specify ODE function with ODE parameters set
+RHS_no_param=@(t,x)RHShandle(t,x,b,L,gma,n,RT,delta0,delta1,l0,V);
+
+%Set ODE integrator parameters.
+options=odeset;
+options=odeset(options,'RelTol',1e-12);
+options=odeset(options,'AbsTol',1e-12);
+
+%Integrate until a steady state is found.
+[tout xout]=ode45(RHS_no_param,[0,200],xinit,options);
+
+xinit=xout(size(xout,1),:);
+
+pvec=[b,L,gma,n,RT,delta0,delta1,l0,V]'; % Initialize parameter vector
+
+ap=1;
+
+[x0,v0]=init_EP_EP(syshandle, xinit', pvec, ap); %Initialize equilibrium
+
+
+opt=contset;
+opt=contset(opt,'MaxNumPoints',500); %Set numeber of continuation steps
+opt=contset(opt,'MaxStepsize',0.1); %Set max step size
+opt=contset(opt,'Singularities',1); %Monitor singularities
+opt=contset(opt,'Eigenvalues',1); %Output eigenvalues
+opt=contset(opt,'InitStepsize',0.01); %Set Initial stepsize
+
+[x1,v1,s1,h1,f1]=cont(@equilibrium,x0,v0,opt);
+
+opt=contset(opt,'backward',1);
+[x1b,v1b,s1b,h1b,f1b]=cont(@equilibrium,x0,v0,opt);
+
+
+if local == 1
+%%%% LOCAL BRANCHES
+
+ind = 2;
+xBP=x1(1:2,s1(ind).index); %Extract branch point
+pvec(ap)=x1(3,s1(ind).index); %Extract branch point
+[x0,v0]=init_BP_EP(syshandle, xBP, pvec, s1(ind), 0.05);
+% opt=contset(opt,'backward',1);
+% [x3b,v3b,s3b,h3b,f3b]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+opt = contset(opt,'backward',0);
+[x3,v3,s3,h3,f3]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+
+% ind = 3;
+% xBP=x1(1:2,s1(ind).index); %Extract branch point
+% pvec(ap)=x1(3,s1(ind).index); %Extract branch point
+% [x0,v0]=init_BP_EP(syshandle, xBP, pvec, s1(ind), 0.001);
+% % opt = contset(opt,'backward',1);
+% % [x4b,v4b,s4b,h4b,f4b]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+% opt = contset(opt,'backward',0);
+% [x4,v4,s4,h4,f4]=cont(@equilibrium,x0,v0,opt); %Switch branches and continue.
+
+end
+% CODIM 1 plots
+
+
+width=5.2;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 10;
+f = figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+Fig2a = subplot(1,2,1);
+xlabel(Fig2a,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+ylabel(Fig2a,{'$R^\ell$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+Fig2aTitle = title(Fig2a,{'(a)'},'FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica');
+set(Fig2a,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+grid
+Fig2a.XLim = [0 6];
+Fig2a.YLim = [0 2];
+Fig2a.Box = 'on';
+set(gca,'LineWidth',1.5)
+hold on
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%GLOBAL curves
+curves = {'1','1b'};
+
+for i = 1:length(curves)
+ xeqcurve=eval(['x' curves{i}]);
+ minevaleq=eval(['f' curves{i} '(2,:)']); %This is the last eigenvalue. That is the one that determines stability
+
+ L=length(xeqcurve(1,:));
+
+ curveind=1;
+ lengthind=0;
+ maxlengthind=0;
+ evalstart=floor(heaviside(minevaleq(1)));
+ datamateq=zeros(4,L);
+
+ for i=1:L
+ evalind=floor(heaviside(minevaleq(i)));
+ if evalstart~=evalind
+ curveind=curveind+1;
+ i;
+ evalstart=evalind;
+ maxlengthind=max(lengthind,maxlengthind);
+ lengthind=0;
+ end
+ datamateq(1,i)=xeqcurve(3,i); % This is the parameter that is varied.
+ datamateq(2,i)=xeqcurve(1,i); % This is the dependent axis of the bifurcation plot. The one you wish to plot
+ datamateq(3,i)=evalind;
+ datamateq(4,i)=curveind;
+
+ lengthind=lengthind+1;
+ end
+
+ maxlengthind=max(maxlengthind,lengthind);
+
+ curveindeq=curveind;
+
+ for i=1:curveindeq
+ index=find(datamateq(4,:)==i);
+ eval(['curve' num2str(i) 'eq' '=datamateq(1:3,index);']);
+ end
+
+ for i=1:curveindeq
+ stability=eval(['curve' num2str(i) 'eq(3,1)']);
+ if stability==0
+ plotsty='-';
+ else
+ plotsty=':';
+ end
+
+ plotcolor='k';
+
+ plotstr=strcat(plotcolor,plotsty);
+
+ plot(eval(['curve' num2str(i) 'eq(1,:)']),eval(['curve' num2str(i) 'eq(2,:)']),plotstr,'Linewidth',3)
+ hold on
+ end
+end
+
+if local == 1
+%%%LOCAL
+curves = {'3'};
+
+for i = 1:length(curves)
+ xeqcurve=eval(['x' curves{i}]);
+ minevaleq=eval(['f' curves{i} '(2,:)']); %This is the last eigenvalue. That is the one that determines stability
+
+ L=length(xeqcurve(1,:));
+
+ curveind=1;
+ lengthind=0;
+ maxlengthind=0;
+ evalstart=floor(heaviside(minevaleq(1)));
+ datamateq=zeros(4,L);
+
+ for i=1:L
+ evalind=floor(heaviside(minevaleq(i)));
+ if evalstart~=evalind
+ curveind=curveind+1;
+ i;
+ evalstart=evalind;
+ maxlengthind=max(lengthind,maxlengthind);
+ lengthind=0;
+ end
+ datamateq(1,i)=xeqcurve(3,i); % This is the parameter that is varied.
+ datamateq(2,i)=xeqcurve(2,i); % This is the dependent axis of the bifurcation plot. The one you wish to plot
+ datamateq(3,i)=evalind;
+ datamateq(4,i)=curveind;
+
+ lengthind=lengthind+1;
+ end
+
+ maxlengthind=max(maxlengthind,lengthind);
+
+ curveindeq=curveind;
+
+ for i=1:curveindeq
+ index=find(datamateq(4,:)==i);
+ eval(['curve' num2str(i) 'eq' '=datamateq(1:3,index);']);
+ end
+
+ for i=1:curveindeq
+ stability=eval(['curve' num2str(i) 'eq(3,1)']);
+ if stability==0
+ plotsty='-';
+ else
+ plotsty=':';
+ end
+
+ mycolor = highcontrast(2,:);
+ plotstr=plotsty;
+
+ plot(eval(['curve' num2str(i) 'eq(1,:)']),eval(['curve' num2str(i) 'eq(2,:)']),plotstr,'Color',mycolor,'Linewidth',2)
+ hold on
+ end
+end
+
+end
+
+opt=contset;
+opt=contset(opt,'MaxNumPoints',5500); %Set numeber of continuation steps
+opt=contset(opt,'MaxStepsize',1); %Set max step size
+opt=contset(opt,'Singularities',1); %Monitor singularities
+opt=contset(opt,'Eigenvalues',1); %Output eigenvalues
+opt=contset(opt,'InitStepsize',0.5); %Set Initial stepsize
+aps = [1,2];
+
+%
+% ind = 2
+% xtmp = x3(1:end-1,s3(ind).index);
+% pvec(ap) = x3(end,s3(ind).index);
+% [x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+% opt = contset(opt,'backward',0);
+% [x5,v5,s5,h5,f5] = cont(@limitpoint,x0,v0,opt);
+% opt = contset(opt,'backward',1);
+% [x5b,v5b,s5b,h5b,f5b] = cont(@limitpoint,x0,v0,opt);
+
+% ind = 3
+% xtmp = x3(1:end-1,s3(ind).index);
+% pvec(ap) = x3(end,s3(ind).index);
+% [x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+% opt = contset(opt,'backward',0);
+% [x6,v6,s6,h6,f6] = cont(@limitpoint,x0,v0,opt);
+% opt = contset(opt,'backward',1);
+% [x6b,v6b,s6b,h6b,f6b] = cont(@limitpoint,x0,v0,opt);
+
+ind = 4
+xtmp = x3(1:end-1,s3(ind).index);
+pvec(ap) = x3(end,s3(ind).index);
+[x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x7,v7,s7,h7,f7] = cont(@limitpoint,x0,v0,opt);
+ opt = contset(opt,'backward',1);
+[x7b,v7b,s7b,h7b,f7b] = cont(@limitpoint,x0,v0,opt);
+
+% ind = 5;
+% xtmp = x3(1:end-1,s3(ind).index);
+% pvec(ap) = x3(end,s3(ind).index);
+% [x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+% opt = contset(opt,'backward',0);
+% [x8,v8,s8,h8,f8] = cont(@limitpoint,x0,v0,opt);
+% opt = contset(opt,'backward',1);
+% [x8b,v8b,s8b,h8b,f8b] = cont(@limitpoint,x0,v0,opt);
+%
+% ind = 6;
+% xtmp = x3(1:end-1,s3(ind).index);
+% pvec(ap) = x3(end,s3(ind).index);
+% [x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+% opt = contset(opt,'backward',0);
+% [x9,v9,s9,h9,f9] = cont(@limitpoint,x0,v0,opt);
+% opt = contset(opt,'backward',1);
+% [x9b,v9b,s9b,h9b,f9b] = cont(@limitpoint,x0,v0,opt);
+%
+% ind = 7;
+% xtmp = x3(1:end-1,s3(ind).index);
+% pvec(ap) = x3(end,s3(ind).index);
+% [x0,v0] = init_LP_LP(syshandle,xtmp,pvec,aps);
+% opt = contset(opt,'backward',0);
+% [x10,v10,s10,h10,f10] = cont(@limitpoint,x0,v0,opt);
+% opt = contset(opt,'backward',1);
+% [x10b,v10b,s10b,h10b,f10b] = cont(@limitpoint,x0,v0,opt);
+
+
+%BPs
+ind = 2
+xtmp = x1(1:end-1,s1(ind).index);
+pvec(ap) = x1(end,s1(ind).index);
+[x0,v0] = init_BP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x11,v11,s11,h11,f11] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x11b,v11b,s11b,h11b,f11b] = cont(@limitpoint,x0,v0,opt);
+
+ind = 7
+xtmp = x1(1:end-1,s1(ind).index);
+pvec(ap) = x1(end,s1(ind).index);
+[x0,v0] = init_BP_LP(syshandle,xtmp,pvec,aps);
+opt = contset(opt,'backward',0);
+[x12,v12,s12,h12,f12] = cont(@limitpoint,x0,v0,opt);
+opt = contset(opt,'backward',1);
+[x12b,v12b,s12b,h12b,f12b] = cont(@limitpoint,x0,v0,opt);
+
+% ind = 5;
+% xtmp = x3(1:end-1,s3(ind).index);
+% pvec(ap) = x3(end,s3(ind).index);
+% [x0,v0] = init_BP_LP(syshandle,xtmp,pvec,aps);
+% opt = contset(opt,'backward',0);
+% [x13,v13,s13,h13,f13] = cont(@limitpoint,x0,v0,opt);
+% opt = contset(opt,'backward',1);
+% [x13b,v13b,s13b,h13b,f13b] = cont(@limitpoint,x0,v0,opt);
+
+%%%
+close all
+
+width=5.2;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 10;
+figure('Units','inches','Position',[x0 y0 width/2 height],'PaperPositionMode','auto');
+Fig5a = subplot(1,1,1);
+xlabel(Fig5a,{'$b$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+
+set(Fig5a,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+hold on
+grid
+Fig5a.XLim = [0 5.5];
+Fig5a.YLim = [0 2.2];
+Fig5a.Box = 'on';
+Fig5a.XColor = 'k';
+Fig5a.YColor = 'k';
+ylabel({'$L$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+set(gca,'LineWidth',1.5)
+curves = {'5','6','7','8'};
+
+linecolor(5) = {highcontrast(2,:)};
+linecolor(6) = {highcontrast(2,:)};
+linecolor(7) = {highcontrast(2,:)};
+linecolor(8) = {highcontrast(2,:)};
+linecolor(9) = {highcontrast(2,:)};
+linecolor(10) = {highcontrast(2,:)};
+linecolor(11) = {highcontrast(3,:)};
+linecolor(12) = {highcontrast(3,:)};
+linecolor(13) = {highcontrast(3,:)};
+
+linestyle(5) = {'-'};
+linestyle(6) = {'-'};
+linestyle(7) = {'-'};
+linestyle(8) = {'-'};
+linestyle(9) = {'-'};
+linestyle(10) = {'-'};
+linestyle(11) = {'--'};
+linestyle(12) = {'--'};
+linestyle(13) = {'--'};
+
+for i = [7,11,12]
+ i
+ xeqcurve=eval(['x' num2str(i)]);
+ plot(xeqcurve(end-1,:),xeqcurve(end,:),'LineStyle',linestyle{i},'Color',linecolor{i},'LineWidth',4)
+ xeqcurve = eval(['x' num2str(i) 'b']);
+ plot(xeqcurve(end-1,:),xeqcurve(end,:),'LineStyle',linestyle{i},'Color',linecolor{i},'LineWidth',4)
+end
+
+% scatter(0.1,7,'ko','filled')
+% text(0.1,7,' (b)')
+%
+% scatter(0.1,3,'ko','filled')
+% text(0.1,3,' (c)')
+%
+% scatter(4.5,6,'ko','filled')
+% text(4.5,6,' (d)')
+%
+% scatter(4.5,3,'ko','filled')
+% text(4.5,3,' (e)')
+
+
+
+% print(1,'1a','-depsc')
+% print(1,'Fig5a','-depsc','-painters')
+savefig(1,'RT_2.fig')
diff --git a/Fig5/funcs.m b/Fig5/funcs.m
new file mode 100644
index 0000000..1dfc1e8
--- /dev/null
+++ b/Fig5/funcs.m
@@ -0,0 +1,46 @@
+function out = Fig1_Functions
+out{1} = @init;
+out{2} = @fun_eval;
+out{3} = [];
+out{4} = [];
+out{5} = [];
+out{6} = [];
+out{7} = [];
+out{8} = [];
+out{9} = [];
+
+function y = A(R,b,gma,n)
+ y = b+gma*R^n/(1+R^n);
+
+function y = delta(L,delta0,delta1,l0)
+ y = delta0 + delta1 * (L-l0);
+
+function y = f(L,V)
+ y = 1; %L/(L+V);
+
+
+% --------------------------------------------------------------------------
+function dydt = fun_eval(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
+dydt=[A(y(1),b,gma,n)*f(L,V)*(RT/L-y(1)) - delta(L,delta0,delta1,l0)*y(1);A(y(2),b,gma,n)*f(L,V)*(RT/L-y(1)) - delta(L,delta0,delta1,l0)*y(2);];
+
+% --------------------------------------------------------------------------
+function [tspan,y0,options] = init
+handles = feval(PosFeedMutInhib);
+y0=[0];
+options = odeset('Jacobian',[],'JacobianP',[],'Hessians',[],'HessiansP',[]);
+tspan = [0 10];
+
+% --------------------------------------------------------------------------
+function jac = jacobian(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
+% --------------------------------------------------------------------------
+function jacp = jacobianp(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
+% --------------------------------------------------------------------------
+function hess = hessians(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
+% --------------------------------------------------------------------------
+function hessp = hessiansp(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
+%---------------------------------------------------------------------------
+function Tension3 = der3(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
+%---------------------------------------------------------------------------
+function Tension4 = der4(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
+%---------------------------------------------------------------------------
+function Tension5 = der5(t,y,b,L,gma,n,RT,delta0,delta1,l0,V)
diff --git a/Fig5/is_polarized.m b/Fig5/is_polarized.m
new file mode 100644
index 0000000..6ed321c
--- /dev/null
+++ b/Fig5/is_polarized.m
@@ -0,0 +1,12 @@
+function y = is_polarized(rac)
+ % #takes rac and rho, two vectors which #describe concentration over space.
+ % #Determine if polarized pattern exists...
+
+ tol = 0.1;
+
+ if abs(max(rac) - min(rac)) > tol
+ y = 1;
+ else
+ y = 0;
+ end
+end
diff --git a/Fig5/pdeSimulation.py b/Fig5/pdeSimulation.py
new file mode 100644
index 0000000..fec81ae
--- /dev/null
+++ b/Fig5/pdeSimulation.py
@@ -0,0 +1,287 @@
+import matplotlib
+# matplotlib.use('Qt5Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+from scipy.integrate import odeint
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+import scipy.io
+
+plt.rc("text", usetex=False)
+plt.rc("font", family="sans-serif", size=10)
+
+def f(u, v, b, gamma, n, RT, delta):
+ return (b+gamma*u**n/(1+u**n))*v - delta*u
+
+def g(u, v, b, gamma, n, RT, delta):
+ return -f(u,v,b, gamma, n, RT, delta)
+
+def F(R):
+ sharp = 10
+ switch = 1
+ magnitude = 0.001
+ return magnitude/(1+np.exp(-2*sharp*(R-switch)))
+
+def xmdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (spring*(xp-xm-1) - F(u[0]))/viscosity
+
+def xpdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (-spring*(xp-xm-1) + F(u[-1]))/viscosity
+
+def rdPDE(y, t, b, gamma, n, RT, delta, Du, Dv, dx, bjump):
+ """
+ The ODEs are derived using the method of lines.
+ https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#mol
+ """
+ # The vectors u and v are interleaved in y. We define
+ # views of u and v by slicing y.
+ #the ode for L is stored at the end of y
+ u = y[0:-2:2]
+ v = y[1:-2:2]
+ xm = y[-2]
+ xp = y[-1]
+
+ l = xp-xm
+
+ if (t>50):
+ b = bjump
+
+ delta = 3 + 1*(l-1)
+
+ # dydt is the return value of this function.
+ dydt = np.empty_like(y)
+
+ dudt = dydt[0:-2:2]
+ dvdt = dydt[1:-2:2]
+ dxmdt = dydt[-2]
+ dxpdt = dydt[-1]
+
+ dudt[0] = f(u[0], v[0], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[0] + 2.0*u[1]) / dx**2 - u[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[1:-1] = f(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * np.diff(u,2) / dx**2 - u[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[-1] = f(u[-1], v[-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[-1] + 2.0*u[-2]) / dx**2 - u[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dvdt[0] = g(u[0], v[0], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[0] + 2.0*v[1]) / dx**2 - v[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[1:-1] = g(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * np.diff(v,2) / dx**2 - v[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[-1] = g(u[-1], v[-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[-1] + 2.0*v[-2]) / dx**2 - v[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dxmdt = xmdot(u,v,xm,xp,t)
+ dxpdt = xpdot(u,v,xm,xp,t)
+
+ dydt[0:-2:2] = dudt
+ dydt[1:-2:2] = dvdt
+ dydt[-2] = dxmdt
+ dydt[-1] = dxpdt
+
+ return dydt
+
+# %%
+
+def pdesimulation(bjump):
+ L = 1
+ N = 1000
+ x = np.linspace(0, L, N)
+ dx = 1/N
+ T = 5000
+ M = 2500
+ t = np.linspace(0, T, M)
+
+ b, gamma, n, RT, delta = 0.1, 5, 6, 2, 3
+ # b, gamma, n, RT, delta = 0.1, 5, 6, 2.5, 3
+ Du = 0.01
+ Dv = 10
+
+ #polarized ic
+ # width = 0.1
+ # height = 2
+ # u0 = np.zeros(N)
+ # u0[x > 1-width] = height
+ # v0 = (RT-width*height)*np.ones(np.size(x))
+
+ #b = 0.1 steadystate
+ u0ic = 0.05645
+ u0 = u0ic*np.ones(np.size(x))
+ v0 = (RT-u0ic)*np.ones(np.size(x))
+
+ # u0ic = RT
+ # u0 = u0ic*np.ones(np.size(x))
+ # v0 = (RT-u0ic)*np.ones(np.size(x))
+
+
+ y0 = np.zeros(2*N+2)
+ xm0 = 0
+ xp0 = 1
+ y0[0:-2:2] = u0
+ y0[1:-2:2] = v0
+ y0[-2] = xm0
+ y0[-1] = xp0
+
+ sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx, bjump))
+
+ u = sol[:,0:-2:2]
+ v = sol[:,1:-2:2]
+ xm = sol[:,-2]
+ xp = sol[:,-1]
+ l = xp - xm
+
+ filename = str(bjump)+'_data.mat'
+ scipy.io.savemat(filename,dict(t=t,l=l,u=u,v=v,xm=xm,xp=xp))
+ return filename
+
+bvals = np.linspace(0.5,5,num=10)
+for bvalue in bvals:
+ print(bvalue)
+ pdesimulation(bvalue)
+
+# # %%
+# fig = plt.figure("fig1",figsize=(5.2,2.6))
+# ax1 = plt.subplot(121)
+# pmesh = plt.pcolormesh(x,t,u,cmap=cm.viridis)
+# cbar = fig.colorbar(pmesh,ax=ax1)
+# # plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+# cbar.outline.set_linewidth(1.5)
+# cbar.ax.tick_params(width=1.5)
+# ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+# ax1.set_xlabel(r'$\bar x$')
+# ax1.set_ylabel(r'$t$')
+# ax1.set_title(r'$R(\bar x,t)$')
+# ax1.spines["left"].set_linewidth(1.5)
+# ax1.spines["top"].set_linewidth(1.5)
+# ax1.spines["right"].set_linewidth(1.5)
+# ax1.spines["bottom"].set_linewidth(1.5)
+#
+# ax2 = plt.subplot(122)
+# pmesh =plt.pcolormesh(x,t,v,cmap=cm.viridis)
+# ax2.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+# cbar = fig.colorbar(pmesh,ax=ax2)
+# # plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+# cbar.outline.set_linewidth(1.5)
+# cbar.ax.tick_params(width=1.5)
+# ax2.set_xlabel(r'$\bar x$')
+# ax2.set_ylabel(r'$t$')
+# ax2.set_title(r'$R_i(\bar x,t)$')
+# ax2.spines["left"].set_linewidth(1.5)
+# ax2.spines["top"].set_linewidth(1.5)
+# ax2.spines["right"].set_linewidth(1.5)
+# ax2.spines["bottom"].set_linewidth(1.5)
+#
+# plt.tight_layout()
+# plt.savefig('Fig14_fixed.png',dpi=600)
+# plt.show()
+#
+# # %%
+# Xgrid, Tgrid = np.meshgrid(x,t)
+# Xgrid = np.empty_like(Xgrid)
+# for i in range(len(t)):
+# Xgrid[i,:] = x*l[i]+xm[i]
+# # %%
+#
+# fig = plt.figure("fig1",figsize=(2.6,2.6))
+# ax1 = plt.subplot(111)
+# pmesh = plt.pcolormesh(Xgrid,Tgrid,u,cmap=cm.viridis)
+# cbar = fig.colorbar(pmesh,ax=ax1)
+# # plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+# cbar.outline.set_linewidth(1.5)
+# cbar.ax.tick_params(width=1.5)
+# ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+# # ax1.set_xlabel(r'$x$')
+# # ax1.set_ylabel(r'$t$')
+# # ax1.set_title(r'$R(x,t)$')
+# ax1.set_facecolor((0.7,0.7,0.7))
+# ax1.spines["left"].set_linewidth(1.5)
+# ax1.spines["top"].set_linewidth(1.5)
+# ax1.spines["right"].set_linewidth(1.5)
+# ax1.spines["bottom"].set_linewidth(1.5)
+# #
+# # ax2 = plt.subplot(122)
+# # pmesh =plt.pcolormesh(Xgrid,Tgrid,v,cmap=cm.viridis)
+# # ax2.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+# # cbar = fig.colorbar(pmesh,ax=ax2)
+# # # plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+# # cbar.outline.set_linewidth(1.5)
+# # cbar.ax.tick_params(width=1.5)
+# # ax2.set_xlabel(r'$x$')
+# # ax2.set_ylabel(r'$t$')
+# # ax2.set_title(r'$R_i(x,t)$')
+# # ax2.set_facecolor((0.7,0.7,0.7))
+# # ax2.spines["left"].set_linewidth(1.5)
+# # ax2.spines["top"].set_linewidth(1.5)
+# # ax2.spines["right"].set_linewidth(1.5)
+# # ax2.spines["bottom"].set_linewidth(1.5)
+#
+# plt.tight_layout()
+# plt.savefig('Polar.tiff',dpi=600)
+# # plt.savefig('Fig14_moving.eps')
+# plt.show()
+# #%%
+# #x
+# fig = plt.figure(figsize=(4,3))
+# plt.plot(t,l)
+# plt.savefig('length_3b.tif')
+# # plt.plot(t,delta)
+#
+# # %%
+# # check mass conservation
+# print(np.sum(dx*l[0]*(u[0,:] + v[0,:])))
+# print(np.sum(dx*l[-1]*(u[-1,:] + v[-1,:])))
+# mass = []
+# for i in range(len(t)):
+# mass.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+#
+# fig = plt.figure("fig2",figsize=(4,3))
+# ax1 = plt.subplot(111)
+# plt.plot(t,mass,linewidth=2)
+# ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+# ax1.set_xlabel(r'$t$')
+# ax1.set_ylabel(r'$\int_\Omega R + R_i \, dx$')
+# ax1.spines["left"].set_linewidth(1.5)
+# ax1.spines["top"].set_linewidth(1.5)
+# ax1.spines["right"].set_linewidth(1.5)
+# ax1.spines["bottom"].set_linewidth(1.5)
+# ax1.set_xlim(0,T)
+# ax1.grid(linewidth=1.5)
+# plt.tight_layout()
+# # plt.savefig('Fig14mass.png',dpi=600)
+# plt.show()
+#
+# # #%%
+# # # animated plot
+#
+# movieon = 0
+# if movieon == 1:
+# import matplotlib.animation as animation
+# fig = plt.figure(figsize=(4,3))
+# ax = plt.subplot(111)
+# ax.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+# ax.set_xlabel(r'$x$')
+# ax.set_ylabel('Activity')
+# ax.spines["left"].set_linewidth(1.5)
+# ax.spines["top"].set_linewidth(1.5)
+# ax.spines["right"].set_linewidth(1.5)
+# ax.spines["bottom"].set_linewidth(1.5)
+# ax.set_xlim(np.min(Xgrid),np.max(Xgrid))
+# ax.set_ylim(0,np.max(u)+0.5)
+# ax.grid(linewidth=1.5)
+# # title = plt.title(r'$b$=%1.2f, $\delta$=%1.2f' %(b, delta))
+# line_u, = ax.plot(Xgrid[0,:],u[0,:],linewidth=4,color=(221/255,170/255,51/255),label=r'$R$')
+# # line_v, = ax.plot(Xgrid[0,:],v[0,:],'--',linewidth=2,color=(0/255,68/255,136/255),label=r'$R_i$')
+# plt.legend(loc=2)
+# vertline = ax.axvline(Xgrid[0,-1],ls='-',color='k')
+# vertline2 = ax.axvline(Xgrid[0,0],ls='-',color='k')
+# plt.tight_layout()
+#
+# def animate(i):
+# # title.set_text(r'$b$=%1.2f, $\delta$=%1.2f' %(bvals[i], deltavals[i]))
+# line_u.set_xdata(Xgrid[i,:])
+# # line_v.set_xdata(Xgrid[i,:])
+# line_u.set_ydata(u[i,:])
+# # line_v.set_ydata(v[i,:])
+# vertline.set_xdata(Xgrid[i,-1])
+# vertline2.set_xdata(Xgrid[i,0])
+# return line_u, vertline, vertline2
+#
+# ani = animation.FuncAnimation(fig,animate,frames=M)
+# ani.save("Movie 2.mp4",fps=30,dpi=300)
diff --git a/Fig6/Fig6b.m b/Fig6/Fig6b.m
new file mode 100644
index 0000000..0e77c54
--- /dev/null
+++ b/Fig6/Fig6b.m
@@ -0,0 +1,66 @@
+bettercolors
+load('dataFig6b.mat')
+
+% figure;
+% hold on
+% %
+width=3.4/2;
+height=5.2/2;
+x0 = 5;
+y0 = 5;
+fontsize = 10;
+figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+
+Fig1a = subplot(2,1,1);
+% xlabel(Fig1a,{'$t$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+%
+set(Fig1a,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+ylabel(Fig1a,'Area','FontUnits','points','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+
+hold on
+grid
+% Fig1b.XLim = [0 5.5];
+% Fig1b.YLim = [0 1.6];
+Fig1a.Box = 'on';
+Fig1a.XColor = 'k';
+Fig1a.YColor = 'k';
+
+set(gca,'LineWidth',1.5)
+idx = 1:1600;
+
+t = timestep(idx)*0.01;
+is_polar = zeros(length(idx),1);
+color = zeros(length(idx),3);
+for i = 1:length(idx)
+ color(i,:) = (Rac_Max_Min(i)>0.15)*bright(6,:)+(Rac_Max_Min(i)<=0.15)*bright(2,:);
+end
+%
+scatter(Fig1a,t,Area(idx),10,color,'filled')
+xlim([0 10])
+xticks([0 5 10])
+ylim([1 1.3])
+yticks([1 1.1 1.2 1.3])
+yticklabels({'1', '', '', '1.3'})
+xticklabels({'0', '', '10'})
+
+Fig1b = subplot(2,1,2);
+Fig1b.Box = 'on';
+Fig1b.XColor = 'k';
+Fig1b.YColor = 'k';
+set(gca,'LineWidth',1.5)
+set(Fig1b,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Helvetica')
+hold on
+ylabel(Fig1b,{'max - min Rac'},'FontUnits','points','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+xlabel(Fig1b,{'Time'},'FontUnits','points','FontWeight','normal','FontSize',12,'FontName','Helvetica')
+grid
+
+scatter(Fig1b,t,Rac_Max_Min(idx),10,color,'filled')
+xlim([0 10])
+ylim([0 0.8])
+yticks([0 0.4 0.8])
+xticks([0 5 10])
+yticklabels({'0', '', '0.8'})
+xticklabels({'0', '', '10'})
+
+
+print(1,'Fig6b.eps','-depsc','-painters')
diff --git a/Fig6/Fig6data.csv b/Fig6/Fig6data.csv
new file mode 100644
index 0000000..8655a32
--- /dev/null
+++ b/Fig6/Fig6data.csv
@@ -0,0 +1,2001 @@
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diff --git a/Fig6/RD_0.tif b/Fig6/RD_0.tif
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index 0000000..fa61c28
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diff --git a/Fig6/RD_0_crop.tif b/Fig6/RD_0_crop.tif
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index 0000000..023ed3c
Binary files /dev/null and b/Fig6/RD_0_crop.tif differ
diff --git a/Fig6/RD_120.tif b/Fig6/RD_120.tif
new file mode 100644
index 0000000..4c2fee0
Binary files /dev/null and b/Fig6/RD_120.tif differ
diff --git a/Fig6/RD_120_crop.tif b/Fig6/RD_120_crop.tif
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index 0000000..8aece29
Binary files /dev/null and b/Fig6/RD_120_crop.tif differ
diff --git a/Fig6/RD_160.tif b/Fig6/RD_160.tif
new file mode 100644
index 0000000..44dc9f4
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diff --git a/Fig6/RD_160_crop.tif b/Fig6/RD_160_crop.tif
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index 0000000..10d3379
Binary files /dev/null and b/Fig6/RD_160_crop.tif differ
diff --git a/Fig6/RD_20.tif b/Fig6/RD_20.tif
new file mode 100644
index 0000000..70f5d2c
Binary files /dev/null and b/Fig6/RD_20.tif differ
diff --git a/Fig6/RD_20_crop.tif b/Fig6/RD_20_crop.tif
new file mode 100644
index 0000000..c40085e
Binary files /dev/null and b/Fig6/RD_20_crop.tif differ
diff --git a/Fig6/RD_40.tif b/Fig6/RD_40.tif
new file mode 100644
index 0000000..a2df3c5
Binary files /dev/null and b/Fig6/RD_40.tif differ
diff --git a/Fig6/RD_40_crop.tif b/Fig6/RD_40_crop.tif
new file mode 100644
index 0000000..561696b
Binary files /dev/null and b/Fig6/RD_40_crop.tif differ
diff --git a/Fig6/RD_80.tif b/Fig6/RD_80.tif
new file mode 100644
index 0000000..12c4b5a
Binary files /dev/null and b/Fig6/RD_80.tif differ
diff --git a/Fig6/RD_80_crop.tif b/Fig6/RD_80_crop.tif
new file mode 100644
index 0000000..058a9e4
Binary files /dev/null and b/Fig6/RD_80_crop.tif differ
diff --git a/Fig6/crop.m b/Fig6/crop.m
new file mode 100644
index 0000000..645ec88
--- /dev/null
+++ b/Fig6/crop.m
@@ -0,0 +1,11 @@
+files = {'RD_0.tif','RD_20.tif','RD_40.tif','RD_80.tif','RD_120.tif','RD_160.tif'}
+names = {'RD_0_crop.tif','RD_20_crop.tif','RD_40_crop.tif','RD_80_crop.tif','RD_120_crop.tif','RD_160_crop.tif'}
+
+for i=1:length(files)
+ f = files{i};
+ g = names{i};
+ I = imread(f);
+ rect = [150,500,1300,1050];
+ J = imcrop(I(:,:,1:3),rect);
+ imwrite(J,g);
+end
diff --git a/Fig6/dataFig6b.mat b/Fig6/dataFig6b.mat
new file mode 100644
index 0000000..eea3a31
Binary files /dev/null and b/Fig6/dataFig6b.mat differ
diff --git a/LICENSE.txt b/LICENSE.txt
new file mode 100644
index 0000000..c3f5b52
--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,674 @@
+GNU GENERAL PUBLIC LICENSE
+ Version 3, 29 June 2007
+
+Copyright (C) 2007 Free Software Foundation, Inc.
+Everyone is permitted to copy and distribute verbatim copies
+of this license document, but changing it is not allowed.
+
+ Preamble
+
+The GNU General Public License is a free, copyleft license for
+software and other kinds of works.
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+The licenses for most software and other practical works are designed
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+share and change all versions of a program--to make sure it remains free
+software for all its users. We, the Free Software Foundation, use the
+GNU General Public License for most of our software; it applies also to
+any other work released this way by its authors. You can apply it to
+your programs, too.
+
+When we speak of free software, we are referring to freedom, not
+price. Our General Public Licenses are designed to make sure that you
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diff --git a/README.md b/README.md
new file mode 100644
index 0000000..1d77647
--- /dev/null
+++ b/README.md
@@ -0,0 +1,31 @@
+#Source Code
+
+This repository contains code used to generate figures for our paper on mechanosensing as an adaptation for neutrophil polarity.
+
+Written by Cole Zmurchok, https://zmurchok.github.io.
+
+## Contents
+
+Each file generates a part of a figure:
+
+- bettercolors.m is a Matlab script that changes the default Matlab colors.
+- Figure2a_driver.m generates the LPA bifurcation diagram in Figure 2A. It requires the file Fig2_Functions.m (that has the ODE system) and the software Matcont
+- Fig2b-e.py is a Python file used to solve the reaction-diffusion PDE to produce the remaining panels in Figure 2. Scipy, Matplotlib and Numpy are needed. This code was based off of the tutorial at https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#mol
+- Fig3a.m generates Figure 3A, and requires the file Fig2_Functions.m
+- Fig3b.py generates Figure 3B by solving the RD-PDE as before.
+- Fig4b.py and Fig4c.py by solving the moving-boundary RD-PDE.
+- The directory Fig5/ contains several files:
+ - 'X_data.mat' is a Matlab data file containing data from Python simulations of the reaction-diffusion PDE system for different stimulus levels X.
+ - Each data file is produced by the Python script pdeSimulation.py
+ - The File RT_2.m generates the bifurcation diagram shown in Figure 5a using Matcont and the function file funcs.m, and saves it as RT_2.fig.
+ - Fig5a.m and Fig5b.m generate panels A and B respectively. Fig4a requires the file RT_2.fig. You will need to update the path to RT_2.fig in the Fig5a.m file.
+ - is_polarized.m is a function used to determine if a spatial distribution of Rac is polarized or not at a single time step.
+- The directory Fig6/ contains data from the 2D simulations.
+ - Fig6data.csv is the simulation data (time, area, mass, max - min Rac activity), and dataFig6b.mat is a Matlab version of this csv file that contains column vectors of each of these quantities.
+ - crop.m crops the simulation output images to be a consistent size to make the subpanels of Figure 6A.
+ - Fig6b.m generates the plots used in Figure 6B from the data in dataFig6b.mat.
+- SIFig1.py solves the RD PDE as in Figure 4B with successively increasing numbers of grid points.
+
+## License
+
+GNU GPL v3 (see License.txt)
diff --git a/SIFig1.py b/SIFig1.py
new file mode 100644
index 0000000..a932ff8
--- /dev/null
+++ b/SIFig1.py
@@ -0,0 +1,349 @@
+import matplotlib
+# matplotlib.use('Qt5Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+from scipy.integrate import odeint
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+import scipy.io
+
+plt.rc("text", usetex=False)
+plt.rc("font", **{'family':'sans-serif', 'sans-serif':['Helvetica'], 'size':10})
+
+def f(u, v, b, gamma, n, RT, delta):
+ return (b+gamma*u**n/(1+u**n))*v - delta*u
+
+def g(u, v, b, gamma, n, RT, delta):
+ return -f(u,v,b, gamma, n, RT, delta)
+
+def F(R):
+ sharp = 10
+ switch = 1
+ magnitude = 0.001
+ return magnitude/(1+np.exp(-2*sharp*(R-switch)))
+
+def xmdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (spring*(xp-xm-1) - F(u[0]))/viscosity
+
+def xpdot(u,v,xm,xp,t):
+ viscosity = 1
+ spring = 0.001
+ return (-spring*(xp-xm-1) + F(u[-1]))/viscosity
+
+def rdPDE(y, t, b, gamma, n, RT, delta, Du, Dv, dx):
+ """
+ The ODEs are derived using the method of lines.
+ https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#mol
+ """
+ # The vectors u and v are interleaved in y. We define
+ # views of u and v by slicing y.
+ #the ode for L is stored at the end of y
+ u = y[0:-2:2]
+ v = y[1:-2:2]
+ xm = y[-2]
+ xp = y[-1]
+
+ l = xp-xm
+
+ if (t>50):
+ b = 4
+
+ delta = 3 + 1*(l-1)
+
+ # dydt is the return value of this function.
+ dydt = np.empty_like(y)
+
+ dudt = dydt[0:-2:2]
+ dvdt = dydt[1:-2:2]
+ dxmdt = dydt[-2]
+ dxpdt = dydt[-1]
+
+ dudt[0] = f(u[0], v[0], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[0] + 2.0*u[1]) / dx**2 - u[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[1:-1] = f(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * np.diff(u,2) / dx**2 - u[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dudt[-1] = f(u[-1], v[-1], b, gamma, n, RT, delta) + ( Du / l**2 ) * (-2.0*u[-1] + 2.0*u[-2]) / dx**2 - u[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dvdt[0] = g(u[0], v[0], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[0] + 2.0*v[1]) / dx**2 - v[0] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[1:-1] = g(u[1:-1], v[1:-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * np.diff(v,2) / dx**2 - v[1:-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+ dvdt[-1] = g(u[-1], v[-1], b, gamma, n, RT, delta) + ( Dv / l**2 ) * (-2.0*v[-1] + 2.0*v[-2]) / dx**2 - v[-1] * (xpdot(u,v,xm,xp,t) - xmdot(u,v,xm,xp,t)) / l
+
+ dxmdt = xmdot(u,v,xm,xp,t)
+ dxpdt = xpdot(u,v,xm,xp,t)
+
+ dydt[0:-2:2] = dudt
+ dydt[1:-2:2] = dvdt
+ dydt[-2] = dxmdt
+ dydt[-1] = dxpdt
+
+ return dydt
+
+# %%
+# %%time
+L = 1
+N = 100
+x = np.linspace(0, L, N)
+dx = 1/N
+T = 5000
+M = 2500
+t = np.linspace(0, T, M)
+
+b, gamma, n, RT, delta = 0.1, 5, 6, 2, 3
+Du = 0.01
+Dv = 10
+
+#polarized ic
+# width = 0.1
+# height = 2
+# u0 = np.zeros(N)
+# u0[x > 1-width] = height
+# v0 = (RT-width*height)*np.ones(np.size(x))
+
+#b = 0.1 steadystate
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+
+# u0ic = RT
+# u0 = u0ic*np.ones(np.size(x))
+# v0 = (RT-u0ic)*np.ones(np.size(x))
+
+
+y0 = np.zeros(2*N+2)
+xm0 = 0
+xp0 = 1
+y0[0:-2:2] = u0
+y0[1:-2:2] = v0
+y0[-2] = xm0
+y0[-1] = xp0
+
+
+sol = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx))
+
+# %%
+
+u = sol[:,0:-2:2]
+v = sol[:,1:-2:2]
+xm = sol[:,-2]
+xp = sol[:,-1]
+l = xp - xm
+
+# scipy.io.savemat('data.mat',dict(t=t,l=l,u=u))
+
+# %%
+fig = plt.figure("fig1",figsize=(5.2,2.6))
+ax1 = plt.subplot(121)
+pmesh = plt.pcolormesh(x,t,u,cmap=cm.inferno)
+cbar = fig.colorbar(pmesh,ax=ax1)
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$\bar x$')
+ax1.set_ylabel(r'$t$')
+ax1.set_title(r'$R(\bar x,t)$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+
+ax2 = plt.subplot(122)
+pmesh =plt.pcolormesh(x,t,v,cmap=cm.inferno)
+ax2.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+cbar = fig.colorbar(pmesh,ax=ax2)
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax2.set_xlabel(r'$\bar x$')
+ax2.set_ylabel(r'$t$')
+ax2.set_title(r'$R_i(\bar x,t)$')
+ax2.spines["left"].set_linewidth(1.5)
+ax2.spines["top"].set_linewidth(1.5)
+ax2.spines["right"].set_linewidth(1.5)
+ax2.spines["bottom"].set_linewidth(1.5)
+
+plt.tight_layout()
+plt.savefig('test3_fixed.tiff',dpi=600)
+plt.show()
+
+# %%
+Xgrid, Tgrid = np.meshgrid(x,t)
+Xgrid = np.empty_like(Xgrid)
+for i in range(len(t)):
+ Xgrid[i,:] = x*l[i]+xm[i]
+# %%
+
+fig = plt.figure("fig1",figsize=(2.6,2.6))
+ax1 = plt.subplot(111)
+pmesh = plt.pcolormesh(Xgrid,Tgrid,u,cmap=cm.inferno,vmin=0,vmax=2)
+cbar = fig.colorbar(pmesh,ax=ax1)
+xmline = plt.plot(xm,t,linewidth=1.5,color='k')
+xpline = plt.plot(xp,t,linewidth=1.5,color='k')
+# plt.axhline(y=20,linestyle='--',linewidth=2,color='w')
+cbar.outline.set_linewidth(1.5)
+cbar.ax.tick_params(width=1.5)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+# ax1.set_xlabel(r'$x$')
+# ax1.set_ylabel(r'$t$')
+# ax1.set_title(r'$R(x,t)$')
+ax1.set_facecolor((0.7,0.7,0.7))
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+
+
+plt.tight_layout()
+plt.savefig('test3.tiff',dpi=600)
+plt.show()
+#%%
+fig = plt.figure(figsize=(2.6,2.6))
+ax1 = plt.subplot(111)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+plt.plot(t,l)
+ax1.set_xlabel(r'$t$')
+ax1.set_ylabel(r'$l$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+ax1.set_xlim(0,T)
+ax1.grid(linewidth=1.5)
+plt.tight_layout()
+plt.savefig('test3_length.tiff',dpi=600)
+plt.show()
+# %%
+# check mass conservation
+print(np.sum(dx*l[0]*(u[0,:] + v[0,:])))
+print(np.sum(dx*l[-1]*(u[-1,:] + v[-1,:])))
+mass = []
+for i in range(len(t)):
+ mass.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+
+#%%
+
+N = 200
+x = np.linspace(0, L, N)
+dx = 1/N
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+y0 = np.zeros(2*N+2)
+xm0 = 0
+xp0 = 1
+y0[0:-2:2] = u0
+y0[1:-2:2] = v0
+y0[-2] = xm0
+y0[-1] = xp0
+sol2 = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx))
+u = sol2[:,0:-2:2]
+v = sol2[:,1:-2:2]
+xm = sol2[:,-2]
+xp = sol2[:,-1]
+l = xp - xm
+mass2 = []
+for i in range(len(t)):
+ mass2.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+
+
+N = 400
+x = np.linspace(0, L, N)
+dx = 1/N
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+y0 = np.zeros(2*N+2)
+xm0 = 0
+xp0 = 1
+y0[0:-2:2] = u0
+y0[1:-2:2] = v0
+y0[-2] = xm0
+y0[-1] = xp0
+sol3 = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx))
+u = sol3[:,0:-2:2]
+v = sol3[:,1:-2:2]
+xm = sol3[:,-2]
+xp = sol3[:,-1]
+l = xp - xm
+mass3 = []
+for i in range(len(t)):
+ mass3.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+
+
+N = 800
+x = np.linspace(0, L, N)
+dx = 1/N
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+y0 = np.zeros(2*N+2)
+xm0 = 0
+xp0 = 1
+y0[0:-2:2] = u0
+y0[1:-2:2] = v0
+y0[-2] = xm0
+y0[-1] = xp0
+sol4 = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx))
+u = sol4[:,0:-2:2]
+v = sol4[:,1:-2:2]
+xm = sol4[:,-2]
+xp = sol4[:,-1]
+l = xp - xm
+mass4 = []
+for i in range(len(t)):
+ mass4.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+
+
+N = 1600
+x = np.linspace(0, L, N)
+dx = 1/N
+u0ic = 0.05645
+u0 = u0ic*np.ones(np.size(x))
+v0 = (RT-u0ic)*np.ones(np.size(x))
+y0 = np.zeros(2*N+2)
+xm0 = 0
+xp0 = 1
+y0[0:-2:2] = u0
+y0[1:-2:2] = v0
+y0[-2] = xm0
+y0[-1] = xp0
+sol5 = odeint(rdPDE, y0, t, args=(b, gamma, n, RT, delta, Du, Dv, dx))
+u = sol5[:,0:-2:2]
+v = sol5[:,1:-2:2]
+xm = sol5[:,-2]
+xp = sol5[:,-1]
+l = xp - xm
+mass5 = []
+for i in range(len(t)):
+ mass5.append(np.sum(dx*l[i]*(u[i,:] + v[i,:])))
+
+
+
+#%%
+from mpltools import annotation
+
+N = [100,200,400,800,1600]
+
+def error(mass):
+ return np.max(np.abs(np.array(mass)-RT))
+
+errors = [error(mass),error(mass2),error(mass3),error(mass4),error(mass5)]
+
+fig = plt.figure("fig3",figsize=(3,3))
+ax1 = plt.subplot(111)
+plt.loglog(N,errors,'o-')
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10 , width=1.5)
+ax1.set_xlabel(r'$N$')
+ax1.set_ylabel(r'$|| e ||_\infty$')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+# ax1.set_xlim(0,T)
+ax1.grid(linewidth=1.5)
+annotation.slope_marker((350, 0.0007), (-1,1))
+# plt.legend(location='center right')
+plt.tight_layout()
+plt.savefig('test3_mass_error.tiff',dpi=1200)
+# plt.savefig('test3_mass_error.eps')
+plt.show()
diff --git a/SIFig2.csv b/SIFig2.csv
new file mode 100644
index 0000000..8655a32
--- /dev/null
+++ b/SIFig2.csv
@@ -0,0 +1,2001 @@
+timestep,Area,Mass,Rac_Max_Min
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diff --git a/SIFig2.py b/SIFig2.py
new file mode 100644
index 0000000..aeaee2b
--- /dev/null
+++ b/SIFig2.py
@@ -0,0 +1,65 @@
+import matplotlib.pyplot as plt
+import numpy as np
+from matplotlib import cm
+
+plt.rc("text", usetex=False)
+plt.rc("font", **{'family':'sans-serif', 'sans-serif':['Helvetica'], 'size':10})
+
+t = []
+A = []
+M = []
+P = []
+
+t,A,M,P = np.genfromtxt('SIFig2.csv',skip_header=1,unpack=True,delimiter=',')
+
+
+# fig = plt.figure(figsize=(2.6,2.6))
+# ax1 = plt.subplot(111)
+# ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+# plt.plot(t,A)
+# ax1.set_xlabel(r'$t$')
+# ax1.set_ylabel(r'$A$')
+# ax1.spines["left"].set_linewidth(1.5)
+# ax1.spines["top"].set_linewidth(1.5)
+# ax1.spines["right"].set_linewidth(1.5)
+# ax1.spines["bottom"].set_linewidth(1.5)
+# ax1.grid(linewidth=1.5)
+# plt.tight_layout()
+# plt.savefig('Fig6_Area.tiff',dpi=600)
+# plt.savefig('Fig6_Area.eps',dpi=600)
+
+#plt.show()
+
+fig = plt.figure(figsize=(3,3))
+ax1 = plt.subplot(111)
+ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+plt.plot(t,M)
+ax1.set_xlabel(r'Time Step')
+ax1.set_ylabel(r'Total Rac')
+ax1.spines["left"].set_linewidth(1.5)
+ax1.spines["top"].set_linewidth(1.5)
+ax1.spines["right"].set_linewidth(1.5)
+ax1.spines["bottom"].set_linewidth(1.5)
+ax1.grid(linewidth=1.5)
+plt.tight_layout()
+plt.savefig('SIFig2.tiff',dpi=1200)
+# plt.savefig('Fig6_Mass.eps',dpi=600)
+
+#plt.show()
+#
+# fig = plt.figure(figsize=(2.6,2.6))
+# ax1 = plt.subplot(111)
+# ax1.tick_params(axis="both", direction="in", which="both", right=True, top=True, labelsize=10, width=1.5)
+# plt.plot(t,P)
+# ax1.set_xlabel(r'$t$')
+# ax1.set_ylabel(r'$R_{Max}-R_{Min}$')
+# ax1.spines["left"].set_linewidth(1.5)
+# ax1.spines["top"].set_linewidth(1.5)
+# ax1.spines["right"].set_linewidth(1.5)
+# ax1.spines["bottom"].set_linewidth(1.5)
+# ax1.grid(linewidth=1.5)
+# plt.tight_layout()
+# plt.savefig('Fig6_Polarity.tiff',dpi=600)
+# plt.savefig('Fig6_Polarity.eps',dpi=600)
+#
+# #plt.show()
diff --git a/bettercolors.m b/bettercolors.m
new file mode 100644
index 0000000..bdf5b00
--- /dev/null
+++ b/bettercolors.m
@@ -0,0 +1,42 @@
+%https://personal.sron.nl/~pault/#sec:colour_blindness
+bright = [68,119,170; 102,204,238; 34,136,51; 204,187,68; 238,102,119; 170,51,119]/255;
+highcontrast = [0,0,0; 0,68,136; 187,85,102; 221,170,51]/255;
+muted = [51,24,136;136,204,238;68,170,153;17,119,51;153,153,51;221,204,119;204,102,119;136,34,85;170,68,153]/255;
+iridescent = [254,251,233;252,247,213;245,243,193;234,240,181;221,236,191;208,231,202;194,227,210;181,221,216;168,216,220;155,210,225;141,203,228;129,196,231;123,188,231;126,178,228;136,165,221;147,152,210;155,138,196;157,125,178;154,112,158;144,99,136;128,87,112;104,73,87;70,53,58]/255;
+
+set(groot,'defaultAxesColorOrder',bright)
+
+% x = linspace(0,15,200);
+%
+% width=4;
+% height=3;
+% x0 = 5;
+% y0 = 5;
+% fontsize = 12;
+% figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+% hold on
+% for i=1:10
+% plot(x,sin(x+i/2)*(10-i),'LineWidth',2)%'Color',iridescent(i,:),'LineWidth',2);
+% end
+%
+% xlabel({'$x$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% ylabel({'$y$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% set(gca,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% set(gca,'Box','on')
+% set(gca,'LineWidth',1.5)
+% grid
+%
+% figure('Units','inches','Position',[x0 y0 width height],'PaperPositionMode','auto');
+% [X,Y] = meshgrid(-8:.5:8);
+% R = sqrt(X.^2 + Y.^2) + eps;
+% Z = sin(R)./R;
+% surf(X,Y,Z)
+% xlabel({'$x$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% ylabel({'$y$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% zlabel({'$z$'},'FontUnits','points','Interpreter','latex','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% set(gca,'Units','normalized','FontUnits','points','FontWeight','normal','FontSize',fontsize,'FontName','Times')
+% set(gca,'Box','on')
+% set(gca,'LineWidth',1.5)
+% c = colorbar;
+% c.LineWidth=1.5;
+% colormap(iridescent)