On an isolated island live a population of rabbits and a population of wolves. Let x(t) and y(t) be the numbers of rabbits and wolves on the island at time t, respectively. Our goal is to model x(t) and y(t). As is typical in modelling problems, we model quantities by describing their evolution over time, i.e., their rates of change. Assume that this evolution is governed by the following rules:
(1) The rabbits have an infinite food supply. The rabbit population grows at a rate
proportional to its size.
(2) The rabbits are eaten by the wolves at a rate proportional to the sizes of both rabbit
and wolf populations. Getting eaten by a wolf is the only way a rabbit dies.
(3) The population of rabbits is the only food source for the wolves. The wolf population
grows at a rate proportional to the sizes of both rabbit and wolf populations.
(4) The wolves are mortal; their population declines at a rate proportional to its size.
Thus, the populations are modeled by the following system of first-order differential equations: (this is the *) $$x'(t) = ax(t) - bx(t)y(t)$$ $$y'(t) = cx(t)y(t) - dy(t)$$
Exercise 3.
If x(t) $\ge$ 0 and $x'(t) = 0$, then $y(t) =$ _____. If y(t) $\ge$ 0 and $y'(t) = 0$, then $x(t) $ = ______. Your answers should be lines in the $xy$-plane. Plot them.
On the same plot, indicate the following regions:
$$R_{++} = {(x(t); y(t)) \in [0;1) * [0;1) : x_0(t) > 0 \,\text{and } y_0(t) > 0} $$ $$R_{+-} = {(x(t); y(t)) \in [0;1) * [0;1) : x_0(t) > 0 \,\text{and } y_0(t) < 0};$$ $$R_{-+} = {(x(t); y(t)) \in [0;1) * [0;1) : x_0(t) < 0 \,\text{and } y_0(t) > 0};$$ $$R_{--} = {(x(t); y(t)) \in [0;1) * [0;1) : x_0(t) < 0 \,\text{and } y_0(t) < 0}:$$
- On the same plot, roughly sketch some parametric "population trajectory" curves $(x(t), y(t))$ satisfying (* - equations above). Indicate direction along the curves with arrowheads.
So far I've found out that $$y(t) = a/b$$ $$x(t) = d/c$$ But I don't know how to plot the "population trajectory" curves.