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ESPM-EEP%202010%20Lecture%2014 - ESPM 104/EEP 115 Fall 2010...

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ESPM 104/EEP 115 Fall 2010: Lecture 14 1. Lotka-Volterra original prey-predator model: http://en.wikipedia.org/wiki/Lotka–Volterra_equation See various texts including Case 2000. Also see http://www.scholarpedia.org/article/Predator-prey_model x : density of prey species y : density of prey species a.) Absence of predators: prey grow exponentially at rate a >0 b.) Absence of prey: predators decay exponentially at rate c >0 c.) Mass action: predators encounter prey at a rate proportional to xy , and extract them with a factor of proportionality b >0 d.) Conversion: predators convert what they extract into their own biomass at a rate 0< k <1, with d = kb . In this case the equations are dx dt = ax bxy = x a by ( ) dy dt = cy + dxy = y ( dx c ) nullcline equations: x -nullclines : x =0 and y = a / b y -nullclines : y =0 and x=c / d Phase plane diagram and cyclic solutions 0 0 x y a/b c/d
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2. Prey-predator equations: replacing exponential with logistic
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