Chapter 7.4, Problem 14P

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.

Applying a constant-effort model of harvesting to theLotka-Volterra equations (1), we obtain the system

$\begin{array}{cc}x\text{'}=x(a-\alpha y-E_1),& y\text{'}=y(-c+\gamma x-E_2).\end{array}$

When there is no harvesting, the equilibrium solution is $(c/\gamma ,a/\alpha )$

a) Before doing any mathematical analysis, think about thesituation intuitively. How do you think the populations willchange if the prey alone is harvested? if the predator alone isharvested? if both are harvested?

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

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

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