PROBLEM 2
Let x be the prey (rabbit) density and let y be the predator (fox) density. A simple growth relation
for the rabbits would be $\frac{dx}{dt} = ax$. Without the rabbits, foxes will perish, and a simple decay rate
would be $\frac{dy}{dt} = -by$. When both species re present, the rate at which rabbits are eaten may be
expected to be in proportion to the product xy, and the rate of growth of foxes may be expected to
increase in the same manner. Thus a feasible predator-prey model (the Volterra-Lotka model) is
$\frac{dx}{dt} = ax - pxy$, $\frac{dy}{dt} = -by + qxy$ where a, b, p, q are positive numbers.
Compute the solution for x(t) and y(t) with a = 2, b = 1, p = 0.01, q = 0.01 for $0 \le t \le 20$
with x(0) = 300, y(0) = 150.
Make plots, one for x and y as functions of t and one a phase plot with x on one axis and y on the
other.
