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Volterra Lotka with Predator Satiation

# Volterra Lotka with Predator Satiation - Volterra-Lotka...

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Volterra-Lotka with Predator Satiation By: Benjamin Horstman & Caroline Lee

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ABSTRACT: The Volterra-Lotka equations are a first order non-linear differential system. They attempt to simulate the interaction of a predator species upon an exponentially reproducing prey species: predator: ( ) dx x y dt α β = - - prey: ( ) dy y x dt γ δ = - (1) However, these equations are very basic and many modifications exist, for example, making the prey growth logistic. Another addition is predator satiation . The concept behind predator satiation is predators can only eat so much before they become full. Thus, the rate at which the predators grow and the prey are preyed upon is not just c*x*y, where c is a growth constant. The modified Volterra-Lotka is as follows: ' by x ax x c ky = - + + ' ( ) fy y d ey y x c ky = - - + (2) We attempted to determine the bifurcations caused by varying parameter k over the range0 2 k x and holding all other variables constant. Matlab R2006a and module pplane7 were used for numeric analysis. We also analyzed equilibria a linearized version of the above equations analytically and predicted that the bifurcation values of k were 0.5, 1.2, & 1.7. Using pplane7 we verified the predictions of the linearized equations. We also saw the appearance of limit cycles, and an unpredicted bifurcation between k = 1.5 & 1.6. On the whole, the linearized equations accurately predicted the behavior of the nonlinear system. MAIN BODY: We used the modified Volterra-Lotka system from (2) above by setting the parameters as follows: .5, 1, 1, 1, .3, 1 a d e b c f = = = = = = ' .5 .3 y x x x ky = - + + ' (1 ) .3 y y y y x ky = - - + (3) Then, we varied parameter k over the range 0 2 k x . We wanted to determine two things with the modified system. (1) The bifurcation values of k. (2) Impact of bifurcations on predator & prey coexistence To attempt to analyze the system analytically, we computed the values of the equilibria by setting the x-nullclines equal to the y-nullclines (see appendix), producing three equilibria:
(0,0) (0,1) 2 (.255 .15 ) .15 , (1 .5 ) 1 .5 k k k - - - (4) Note that (0,0) and (0,1) are equilibria which do not depend on parameter k.

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Volterra Lotka with Predator Satiation - Volterra-Lotka...

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