In the Lotka--Volterra equations, the interaction between the two species is modeled by

Chapter 9, Problem 13

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In the Lotka--Volterra equations, the interaction between the two species is modeled by terms proportional to the product xy of the respective populations. If the prey population is much larger than the predator population, this may overstate the interaction; for example, a predator may hunt only when it is hungry and ignore the prey at other times. In this problem we consider an alternative model proposed by Rosenzweig and MacArthur.10 a. Consider the system x_ = x_1 x 5 2y x + 6 _, y_ = y_1 4 + x x + 6 _ . Find all of the critical points of this system. b. Determine the type and stability characteristics of each critical point. G c. Draw a direction field and phase portrait for this system. 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. Or, the prey species may be regarded as a pest, leading to efforts to reduce its numbers. In a constant-effort model of harvesting, we introduce a term E1x in the prey equation and a term E2 y in the predator equation, where E1 and E2 are measures of the effort invested in harvesting the respective species. A constant-yield model of harvesting is obtained by including the term H1 in the prey equation and the termH2 in the predator equation. The constants E1, E2, H1, and H2 are always nonnegative. 14 and 15 deal with constant-effort harvesting, and deals with constant-yield harvesting. 1