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# Lab5rev - Lab 5 Nonlinear Systems Goals In this lab you...

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Lab 5: Nonlinear Systems Goals In this lab you will use the pplane8 program to study two nonlinear systems by direct nu- merical simulation. The first model, from population biology, displays interesting nonlinear oscillations (so-called limit cycles ). The second is a system whose solutions depend on a parameter. Neither of these systems is described by exactly solvable systems of differen- tial equations. Although much may be learned from strictly theoretical analyses, we must ultimately rely on computational methods to extract quantitative predictions from these systems. Application 1: predator-prey species interactions In class we considered a model of predator-prey species interactions known as the Lotka- Volterra model (referred to in Section 6.3 as the predator-prey system ). If x describes the size of a population of rabbits and y describes a population of foxes (which feed on the population of rabbits) then the Lotka-Volterra model of their interactions says that there are positive constants a, b, c, d so that dx dt = x ( a - by ) , (1) dy dt = y ( - c + dx ) . (2) That is, the exponential growth rate of rabbits is decreased by the presence of foxes and the exponential death rate of foxes is decreased by the presence of rabbits. This model predicts some unlikely behavior. In the absence of foxes ( y = 0), equation (1) becomes dx/dt = ax . In other words, without any foxes the rabbits will always grow exponentially without bound. And even if the predator population is small, they will always eat the prey at a rate proportional to their product. In other words, 10 foxes surrounded by 100,000 rabbits would each have to eat ten times more than 10 foxes surrounded by 10,000 rabbits. If the rabbit population could be held at a fixed level x 0 > c/d , equation (2) becomes dy/dt = Cy where C = - c + dx 0 > 0. In other words, if the rabbit population is maintained at a given level, above some threshold, the fox population will always grow exponentially without bound. None of these predictions are ecologically reasonable. The following model addresses these problems. For positive values of r , a more reasonable model of the two populations is the system dx dt = x (1 - x ) - 5 xy 5 x + 1 , (3) 1

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dy dt = ry 1 - y x . (4) In the absence of predators, the prey satisfies the logistic equation with equilibrium popu- lation x = 1. In the presence of predators, prey is consumed at a rate 5 xy/ (5 x + 1). That is, if x is large compared to 1 / 5, then 5 xy/ (5 x + 1) y , and the predators consume prey at a rate proportional to the predator population. On the other hand, if x is small compared to 1 / 5 then 5 xy/ (5 x + 1) 5 xy and only then do the predators consume prey at a rate proportional to xy as in the Lotka-Volterra model. Furthermore, if the prey population x is held fixed somehow, we ignore the differential equation governing x ( t ) and replace x ( t ) by a constant, and we then see that the predator population satisfies a logistic growth equation
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