Finding appropriate step size for set of ODE's numerically solved without an ODE solver

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So I'm trying to implement the following code to replicate this graph(parts a. and c. only)
Here is my code, the time step is increasing as we move from from 10^-4 to 10^5. So the step size, h, should be increasing in size as it goes along. I would think I would need an adaptive time step, but don't know how to implement unless I use the fancier Runge-Kutta Fehlberg 4/5 pair. I did post earlier but was left hanging with regards to this question, I'm trying to do this without using an ODE solver like ODE15 or ODE45
function cells
%constants
epsilon=0.0001;
M=30;
N=4;%double check if it plays role in code, mentioned in legend for graph
Ns1=20;
Ns2=10;
m=M;%constraint condition
%computing time variable
tstart=10^-4;
tend=10^5;
t=tstart;
n=500;%total time steps
logstart=log10(tstart);
logend=log10(tend);
lpm=[logstart:(logend-logstart)/n:logend];%log plot mesh
iplot=1;%counter
tarray=zeros(1,n+1);
tarray(iplot)=t;
%h=(10^-4);
%Numerical scheme for sigma<1/2
c=[Ns1,0,0,0,0];%initial conditions
carray=zeros(5,n+1);
carray(:,iplot)=c;%adding initial condition to array of values
while t<tend %loop for top graph implemented with Classical RK4
z=RK4(c,h,epsilon,m);
tm=10^(lpm(iplot));
if t>tm
iplot=iplot+1;
carray(:,iplot)=z;
tarray(iplot)=t;
end
c=z;
t=t+h;
end
logt=log10(tarray);
figure(1)
tiledlayout(2,1) %initiating display of multiple axes in one figure
nexttile%top plot
plot(logt,carray(2,:),'g')
hold on
plot(logt,carray(3,:),'y')
plot(logt,carray(4,:),'b')
plot(logt,carray(5,:),'k')
hold off
%%Numerical scheme for sigma>1/2
c=[Ns2,0,0,0,0];
carray=zeros(5,n+1);
carray(:,1)=c;
while t<tend%loop for bottom graph implemented with Classical RK4
z=RK4(c,h,epsilon,m);
tm=10^(lpm(iplot));
if t>tm
iplot=iplot+1;
carray(:,iplot)=z;
tarray(iplot)=t;
end
c=z;
t=t+h;
end
logt=log10(tarray);
nexttile%bottom plot
plot(logt,carray(2,:),'g');
hold on
plot(logt,carray(3,:),'y');
plot(logt,carray(4,:),'b');
plot(logt,carray(5,:),'k');
hold off
%Classical RK4
function W4=RK4(c,h,epsilon,m)
X1=c;
X2=c+h*(1/2)*rhs(X1,epsilon,m);
X3=c+h*(1/2)*rhs(X2,epsilon,m);
X4=c+h*rhs(X3,epsilon,m);
W4=c+h*((1/6)*rhs(X1,epsilon,m)+(1/3)*rhs(X2,epsilon,m)+(1/3)*rhs(X3,epsilon,m)+(1/6)*rhs(X4,epsilon,m));
%System of ODE's
function dcdt=rhs(c,epsilon,m)
dcdt(1)=-m*c(1)+epsilon*c(2);
dcdt(2)=-m*c(2)-epsilon*c(2)+m*c(1)+epsilon*c(3);
dcdt(3)=-m*c(3)-epsilon*c(3)+m*c(2)+epsilon*c(4);
dcdt(4)=-m*c(4)-epsilon*c(4)+m*c(3)+epsilon*c(5);
dcdt(5)=-epsilon*c(5)+m*c(4);

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