Solved: In many fishery science models the rate at which a species is caught is assumed

Chapter 11, Problem 9

(choose chapter or problem)

In many fishery science models the rate at which a species is caught is assumed to be directly proportional to its abundance. If both predator and prey are being exploited in this manner, the Lotka-Volterra differential equations take the form

\(\begin{aligned}

&x^{\prime}=-a x+b x y-\varepsilon_{1} x \\

&y^{\prime}=-c x y+d y-\varepsilon_{2} y

\end{aligned}\)

where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are positive constants.

(a) When \(\varepsilon_{2}<d\), show that there is a new critical point in the first quadrant that is a center.

(b) Volterra's principle states that a moderate amount of exploitation increases the average number of prey and decreases the average number of predators. Is this fisheries model consistent with Volterra's principle?