Solved: Harvesting in a PredatorPrey Relationship. In a predatorprey situation,it may

Chapter 9, Problem 16

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Harvesting in a PredatorPrey Relationship. In a predatorprey situation,it may happenthat one or perhaps both species are valuable sources of food. Or, the prey species maybe regarded as a pest, leading to efforts to reduce its numbers. In a constant-effort modelof harvesting, we introduce a term E1x in the prey equation and a term E2y in thepredator equation, where E1 and E2 are measures of the effort invested in harvestingthe respective species. A constant-yield model of harvesting is obtained by including theterm H1 in the prey equation and the term H2 in the predator equation. The constantsE1, E2, H1, and H2 are always nonnegative. 14 and 15 deal with constant-effortharvesting, and deals with constant-yield harvesting.In this problem we apply a constant-yield model of harvesting to the situation in Example 1.Consider the systemx = x(1 0.5y) H1, y = y(0.75 + 0.25x) H2,where H1 and H2 are nonnegative constants. Recall that if H1 = H2 = 0, then (3, 2) is anequilibrium solution for this system.(a) Before doing any mathematical analysis, think about the situation intuitively. How doyou think the populations will change if the prey alone is harvested? if the predator aloneis harvested? if both are harvested?(b) How does the equilibrium solution change if the prey is harvested, but not the predator(H1 > 0, H2 = 0)?(c) How does the equilibrium solution change if the predator is harvested, but not theprey (H1 = 0, H2 > 0)?(d) How does the equilibrium solution change if both are harvested (H1 > 0, H2 > 0)?