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### In Project C of Chapter 4, it was shown that the …

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Problem 19

Predator-Prey Model. The Volterra-Lotka predatorprey model predicts some rather interesting behavior that is evident in certain biological systems. For example, suppose you fix the initial population of prey but increase the initial population of predators. Then the population cycle for the
prey becomes more severe in the sense that there is a long period of time with a reduced population of prey followed by a short period when the population of prey is very large.To demonstrate this behavior, use the vectorized Runge-Kutta algorithm for systems with $h=0.5$ 0.5 to approximate
the populations of prey $x$ and of predators $y$ over the period 30, 54 that satisfy the Volterra-Lotka system
\begin{aligned} x^{\prime} &=x(3-y) \\ y^{\prime} &=y(x-3) \end{aligned}
under each of the following initial conditions:
$$\begin{array}{ll}{\text { (a) } x(0)=2,} & {y(0)=4} \\ {\text { (b) } x(0)=2,} & {y(0)=5} \\ {\text { (c) } x(0)=2,} & {y(0)=7}\end{array}$$