Notes-Predator-Prey

Notes-Predator-Prey - The Predator-Prey Equations An...

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The Predator-Prey Equations An application of the nonlinear system of differential equations in mathematical biology / ecology: to model the predator-prey relationship of a simple eco-system. Suppose in a closed eco-system (i.e. no migration is allowed into or out of the system) there are only 2 types of animals: the predator and the prey. They form a simple food-chain where the predator species hunts the prey species, while the prey grazes vegetation. The size of the 2 populations can be described by a simple system of 2 nonlinear first order differential equations (a.k.a. the Lotka-Volterra equations , which originated in the study of fish populations of the Mediterranean during and immediately after WW I). Let x ( t ) denotes the population of the prey species, and y ( t ) denotes the population of the predator species. Then x ′ = a x ± α xy y ′ = ± c y + γ xy a , c , α , and γ are positive constants. Note that in the absence of the predators (when y = 0), the prey population would grow exponentially. If the preys are absence (when x = 0), the predator population would decay exponentially to zero due to starvation.
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This system has two critical points. One is the origin, and the other is in the first quadrant. 0 =
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Notes-Predator-Prey - The Predator-Prey Equations An...

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