These exercises explore the question "When one of two species in a colony is
desirable and the other is undesirable, is it better to use resources to
nurture the growth of the desirable species or to harvest the undesirable
one?"
Let \(x(t)\) and \(y(t)\) represent the populations of two competing species, with
\(x(t)\) the desirable species. Assume that if resources are invested in
promoting the growth of the desirable species, the population dynamics are
given by
$$
\begin{aligned}
&x^{\prime}=r(1-\alpha x-\beta y) x+\mu x \\
&y^{\prime}=r(1-\alpha y-\beta x) y
\end{aligned}
$$
If resources are invested in harvesting the undesirable species, the dynamics
are
$$
\begin{aligned}
&x^{\prime}=r(1-\alpha x-\beta y) x \\
&y^{\prime}=r(1-\alpha y-\beta x) y-\mu y
\end{aligned}
$$
In (10), \(r, \alpha, \beta\), and \(\mu\) are positive constants. For simplicity,
we assume the same parameter values for both species. For definiteness, assume
that \(\alpha>\beta>0\).
Consider system (9), which describes the strategy in which resources are
invested into nurturing the desirable species.
(a) Determine the four equilibrium points for the system.
(b) Show that it is possible, by investing sufficient resources (that is, by
making \(\mu\) large enough), to prevent equilibrium coexistence of the two
species.
(c) Assume that \(\mu\) is large enough to preclude equilibrium coexistence of
the two species. Compute the linearized system at each of the three physically
relevant equilibrium points. Determine the stability characteristics of the
linearized system at each of these equilibrium points.
(d) System (9) can be shown to be an almost linear system at each of the
equilibrium points. Use this fact and the results of part (c) to infer the
stability properties of system (9) at each of the three equilibrium points of
interest.
(e) Sketch the direction field. Will a sufficiently aggressive nurturing of
species \(x\) ultimately drive undesirable species \(y\) to extinction? If so,
what is the limiting population of species \(x\) ?
