Solved: The study of mathematical models for predicting the population dynamics of

Chapter 5, Problem 5

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The study of mathematical models for predicting the population dynamics of competing species has its origin in independent works published in the early part of the 20th century by A. J. Lotka and V. Volterra (see, for example, |Lol |, rLo21, and | Vo|.).Consider the problem of predicting the population of two species, one of which is a predator, whose population attime t is X2(t), feeding on the other, which is the prey, whose population is x\(t). We will assume that the prey always has an adequate food supply and that its birthrate at any time is proportional to the number of prey alive at thattime; that is, birthrate (prey) is k\X\(t). The death rate ofthe prey depends on both the number of prey and predators alive at that time. For simplicity, we assume death rate (prey) = {t)x2{t). The birthrate ofthe predator, on the other hand, depends on its food supply, x\{t) as well as on the number of predators available for reproduction purposes. For this reason, we assume that the birthrate (predator) is k-iX\(t)x2(t). The death rate ofthe predator will be taken as simply proportional to the number of predators alive at the time; that is, death rate (predator) = ^4X2(r). Since x[(t) and x^it) represent the change in the prey and predator populations, respectively, with respect to time, the problem is expressed by the system of nonlinear differential equations x\{t) = *1X1 (0 - k2X\(t)x2{t) and x'2{t) = 3*1 (0*2(0 - ^4*2(0- Solve this system for 0 < t <4, assuming thatthe initial population ofthe prey is 1000 and of the predators is 500 and that the constants are k\ 3, ki 0.002, 0.0006, and ^4 0.5. Sketch a graph ofthe solutions to this problem, plotting both populations with time, and describe the physical phenomena represented. Is there a stable solution to this population model? If so, for what values X| and X2 is the solution stable?