## Bifurcation Diagram

Asked by pallav pal

### pallav pal (view profile)

on 4 Nov 2011
Latest activity Commented on by Walter Roberson

### Walter Roberson (view profile)

on 10 Jan 2017
Accepted Answer by Rick Rosson

### Rick Rosson (view profile)

I want to draw the bifurcation diagram fro the model.
dx/dt=rx(1-x/K)-mxy/(ax+by+c)
dy/dt=emxy/(ax+by+c)-dy-hy^2.
parameters are all +ve.
I have tryed to plot it but fails.
clear
r=0.806; a=15; b=16;c=17;e=0.333;d=0.3;h=0.01;K=200;
x (1)=0.7;
y (1)=0.11;
t (1)=0;
for m=6:1:22
for i=1:10000
t (i+1)=t (i)+.01;
x (i+1)= x (i)+0.01*[r*x (i)*(1-x (i)/K)-m*x (i)*y (i)/(a*x \
(i)+b*y (i)+c)];
y (i+1)=y (i)+.01*[e*m*x (i)*y (i)/(a*x (i)+b*y (i)+c)-d*y (i)-h*y \
(i)^2];
end
plot (m,x, 'b')
hold on;
end
xlabel ('m')
ylabel ('x')
figure (2)
for m=6:1:22
for i=1:10000
t (i+1)=t (i)+.01;
x (i+1)= x (i)+0.01*[r*x (i)*(1-x (i)/K)-m*x (i)*y (i)/(a*x \
(i)+b*y (i)+c)];
y (i+1)=y (i)+.01*[e*m*x (i)*y (i)/(a*x (i)+b*y (i)+c)-d*y (i)-h*y \
(i)^2];
end
plot (m,y, 'b')
hold on;
end
xlabel ('m')
ylabel ('y')
Please modify or help me to modify the matlab code to draw the following bifurcation diagram (parameter VS population):
1.Transcritical bifurcation (x vs m & y vs. m) around at m= 13.666
2. Saddle-node bifurcation (x vs m & y vs. m) around at m = 20.8.
3. Hopf-bifurcation (x vs m & y vs. m) at m=14.73, (d,h) = (0.02,0.001) and others are same.

Mirza

### Mirza (view profile)

on 4 Mar 2013
Plz try to read this. It has all relevant information. http://amath.colorado.edu/courses/2460/2009fall/Homeworks/bifurcation.pdf

### DEEPIHA PADMANATHAN (view profile)

on 10 Jan 2017
The requested page "/courses/2460/2009fall/Homeworks/bifurcation.pdf" could not be found. Page not found
Walter Roberson

on 10 Jan 2017

### Tags

Answer by Rick Rosson

### Rick Rosson (view profile)

on 6 Nov 2011

Hi Pallav,
There are several issues with the code that you posted. Some of these problems are syntax errors, some are non-standard coding style, and some are non-optimal formatting.
Syntax Errors
The biggest syntax error, and the main reason this code fails to run, is the improper use of square brackets [ and ] in places where parentheses (| and |) are required.
So, as a first step, please replace every instance of [ with (|, and likewise, every instance of |] with ).
MATLAB does use square brackets for certain purposes, but never as a way of grouping expressions within assignment statements to specify the order of evaluation. MATLAB always uses parentheses for that purpose, never square brackets.
Formatting
Please do not put a blank space between the name of an array variable and the open parenthesis that precedes the index of the array. Also, you should include at least one blank space before and after the = sign in any assignment statement. Likewise, you should also include at least one blank space on either side of any + or - sign in an expression consisting of more than one term.
So, for example, instead of writing
t (i+1)=t (i)+.01;
t(i+1) = t(i) + 0.01;
Although the first line is perfectly valid MATLAB syntax, it it horribly difficult to read and understand, whereas the second line is far easier to read and understand. The first line is also generally considered contrary to most MATLAB coding standards and conventions, whereas the second line is generally considered consistent with standard conventions.
Parameterization
It is often helpful to introduce one or more parameters into the code as a way of improving the readability and maintainability of the code. So, for example, instead of using the literal number 0.01 at several places throughout the code, it may make sense to define a parameter dt (meaning "delta-t" or "time increment") at the beginning of the script:
dt = 0.01;
and then replacing every instance of 0.01 with dt, such as:
t(i+1) = t(i) + dt;
and
x(i+1) = x(i) + dt*( ... );
and so on.
Pre-Allocation
It is very important to pre-allocate time t and the two state variables x and y to avoid having them grow during each iteration of the inner for loop. To do so, please insert the following four lines of code after the outer for loop and prior to the inner loop:
N = 10000;
x = zeros(N,1);
y = zeros(N,1);
t = zeros(N,1);
Although this code is not strictly necessary, it will reduce the run-time of your script quite substantially, and it will avoid unnecessarily fragmenting the memory during execution of the script.
Initial Conditions
Now that you have pre-allocated the time and state variables, you need to set the initial conditions for each of these three variables. So please cut the following three lines of code and paste them after the pre-allocation code and prior to the inner for loop:
x(1) = 0.7;
y(1) = 0.11;
t(1) = 0;
By moving these three lines inside the outer for loop but prior to the inner for loop, you will ensure that each instance of the parameter m will start with the correct initial conditions independent of the prior instance.
Modified Code
Here is what the code looks like after all of the changes mentioned above (plus a few others):
%%Initialize the environment
close all;
clear all;
clc;
%%Model parameters
r = 0.806;
a = 15;
b = 16;
c = 17;
e = 0.333;
d = 0.3;
h = 0.01;
K = 200;
%%Time parameters
dt = 0.01;
N = 10000;
%%Set-up figure and axes
figure;
ax(1) = subplot(2,1,1);
hold on
xlabel ('m');
ylabel ('x');
ax(2) = subplot(2,1,2);
hold on
xlabel ('m');
ylabel ('y');
%%Main loop
for m=6:1:22
x = zeros(N,1);
y = zeros(N,1);
t = zeros(N,1);
x(1) = 0.7;
y(1) = 0.11;
t(1) = 0;
for i=1:N
t(i+1) = t(i) + dt;
x(i+1) = x(i) + dt*( r*x(i)*(1-x(i)/K)-m*x(i)*y(i)/(a*x (i)+b*y(i)+c) );
y(i+1) = y(i) + dt*( e*m*x(i)*y(i)/(a*x(i)+b*y(i)+c)-d*y(i)-h*y(i)^2 );
end
plot(ax(1),m,x,'color','blue','marker','.');
plot(ax(2),m,y,'color','blue','marker','.');
end
HTH.
Rick

pallav pal

### pallav pal (view profile)

on 6 Nov 2011
Thank you. I have written dt = 0.01; at the beginning of the script and all 0.01 has been replaced by dt .
pallav pal

### pallav pal (view profile)

on 7 Nov 2011
Thank you very much for your detailed suggestions. It helps me to learn to make further codes. But, I am still unable to get the bifurcation diagrams. The fig that I am getting is not the bif diagram.
Rizwana Junaid

### Rizwana Junaid (view profile)

on 20 Dec 2011
ur system equations are differential equations not the discrete equations. so u should try to solve ur equations with solver ode45. so introduce solver command in the m loop.