Chapter 7.4, Problem 15P

Harvesting in a Predator-Prey Relationship. In a predator-prey situation it may happen that one or perhaps both species are valuable sources of food (for example). Or, the prey may be regarded as a pest, leading to efforts to reduce their number. In a constant-effort model of harvesting, we introduce a term $-E_{1}x$ in the prey equation and a term $-E_{2}y$ in the predator equation. A constant-yield model ofharvesting is obtained by including the term $-H_1$ in the prey equation and the term $-H_2$ in the predator equation. The constants $E_1,E_2,H_1\text{and}{H}_2$ are always nonnegative. Problems14 and 15 deal with constant-effort harvesting, and Problem16 with constant-yield harvesting.

If we modify the Lotka-Volterra equation by including a self-limiting term $-\sigma x^2$ in the prey equation, and then assume constant-effort harvesting, we obtain the equations

$\begin{array}{l}x\text{'}=x(a-\sigma x-\alpha y-E_1)\\ y\text{'}=y(-c+\gamma x-E_2)\end{array}$

In the absence of harvesting, an equilibrium solution is $x=c/\gamma ,y=(a/\alpha )-(\sigma c)/(\alpha \gamma )$.

a) How does the equilibrium solution change if the prey isharvested, but not the predator $(E_1>0,E_2=0)$?

b) How does the equilibrium solution change if the predatoris harvested, but not the prey $(E_1=0,E_2>0)$?

c) How does the equilibrium solution change if both areharvested $(E_1>0,E_2>0)$?