In the LotkaVolterra equations, the interaction between

Chapter 9, Problem 13

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In the LotkaVolterra equations, the interaction between the two species is modeled byterms proportional to the product xy of the respective populations. If the prey population ismuch 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 thisproblem we consider an alternative model proposed by Rosenzweig and MacArthur.9(a) Consider the systemx= x1 0.2x 2yx + 6, y= y0.25 + xx + 6.Find all of the critical points of this system.(b) Determine the type and stability characteristics of each critical point.(c) Draw a direction field and phase portrait for this system.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.