[Math] Difference equations and Matlab (some working)

Question

The difference equations below model the yearly populations of wolves and moose, measured in hundreds. The wolves kill the moose for food.

$$\begin{align}
x_n&=x_{n-1}-0.004x_{n-1}+0.002x_{n-1}y_{n-1}\\
y_n&=y_{n-1}+0.005y_{n-1}-0.010x_{n-1}y_{n-1}-0.001y_{n-1}^2
\end{align}$$

I think I've worked out what $x$ and $y$ are correctly.

$x_n$ is predator and $y_n$ is prey. I say this because the $x_n$ has a growth term

$$+ 0.002 x_{n-1}y_{n-1}$$

…which says that $x_n$ grows by interaction between $x$ and $y$. This is classic predation a.k.a. eating prey.

The $y$s have what looks like logistic growth:

$$y_{n-1} – 0.001y^2_{n-1}$$

which says this species has a population growth that is limited by the environment.

However, does anyone know what Matlab commands I would enter to answer this question?

If there are initially $50$ wolves and $300$ moose, use Matlab to obtain plot of $x$ and $y$ versus n on the same graph, for $0 \le n \le 10000$. Remember that $x$ and $y$ are measured in hundreds.

Answer 1

You can simply write it in this way:

 x(1) = 50;
 y(1) = 200;
 for n=2:10001
      x(n) = x(n-1) - 0.004*x(n-1) + 0.002*x(n-1)*y(n-1);
      y(n) = y(n-1) + 0.005*y(n-1) - 0.010*x(n-1)*y(n-1) - 0.001*(y(n-1))^2;
 end
 plot(1:10001,x);
 hold all;
 plot(1:10001,y);