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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 detailsenter image description here

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):

enter image description here

Is it possible?

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  • $\begingroup$ You haven't told us what z and w are. $\endgroup$ – Emily Jan 10 '14 at 19:46
  • $\begingroup$ Hi new user! $$\color{red}{\Large\text{Welcome to Math.SE!}}$$ Don't worry about it now (since you're new) but you might like to know that we prefer to use MathJax here :) $\endgroup$ – Shaun Jan 10 '14 at 19:54
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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:

enter image description here

  • 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.
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  • $\begingroup$ thanks a lot, Which point is a stable ? $\endgroup$ – user3183171 Jan 10 '14 at 22:50
  • $\begingroup$ $(1/2, 1/6)$. You can actually see these points on the phase portrait. $(1,0)$ is an unstable saddle point. Please recall to upvote and/or accepts answers that are helpful. $\endgroup$ – Amzoti Jan 10 '14 at 23:13
  • $\begingroup$ I know it's not customary, but no idea how to write the code to this exercise and I have to submit it in a few hours, please help me. Exercise: 1. Use Matlab to determine numerically the equilibrium points of the populations and their types (stable or unstable). 2. Plot the graph of the dynamics of the two populations (x and y, t). Mark the equilibrium points on the graph. 3. Plot the graph (phase plane plot) (x, y). Mark the equilibrium points on the graph. Thanks again $\endgroup$ – user3183171 Jan 10 '14 at 23:29
  • $\begingroup$ Ok, thank you. If someone can help me with the code, I'd very much appreciate it. $\endgroup$ – user3183171 Jan 10 '14 at 23:42
  • $\begingroup$ There is a stack exchange for it, but they expect the same as this site - that you do some work first. See: stackoverflow.com/questions/tagged/matlab $\endgroup$ – Amzoti Jan 10 '14 at 23:43

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