find equilibrium points in matlab

Question

I need help with this material:

The dynamics growth of two populations is expressed by the system of equations: ($x=$ prey, $y=$predator, $0 \leq t \leq 30$)

$$\dot x=x(1-x)-\frac{2xy}{y+x}\qquad\dot y=-1.5y+\frac{2xy}{y+x}$$

Use Matlab to determine numerically the equilibrium points of the populations and their types (stable or unstable). Plot the graph of the dynamics of the two populations ($x$ and $y$ vs. $t$). Mark the equilibrium points on the graph.

I have no idea how to draw the graph and find the equilibrium points , please help me with this. thanks

Here are all the details

where : $a=2$, $b=2$, $m=1.5$, $0 \le t \le 30$

I got the following results when x=y=2 (Without critical points):

Is it possible?

Answer 1

Hints: This will guide you through the process and you can figure out how to do this in Matlab.

• To find the critical points, you want to simultaneously solve $x' = 0, y' = 0$. You will get two critical points at $$(x,y) = \left(\dfrac{1}{2}, \dfrac{1}{6}\right), (1, 0)$$

• You can then determine the types of critical points these are by finding the Jacobian, $J(x, y)$, and evaluating the eigenvalues of the $2x2$ Jacobian. In the phase portrait below, you can see we have a stable and an unstable critical point.

• The Phase portrait will show these two critical points and should look something like:

• For Matlab, you can see some examples here of how to draw that pahse protrait. Also, see MathWorks. Also see this book by S. Lynch.