COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

320 chapter 1212.19 Lotka–Volterra ModelWe extend the logistic map to the Lotka–Volterra Model (LVM) to describe twopopulations coexisting in the same geographical region. Letp(t)=prey density, P(t)=predator density. (12.53)In the absence of interactions between the species, we assume that the preypopulation p breeds at a per-capita rate of a, which would lead to exponentialgrowth:dpdt = ap, ⇒ p(t)=p(0)eat . (12.54)Yet exponential growth does not occur because the predators P eat more prey as theprey numbers increase. The interaction rate between predator and prey requiresboth to be present, with the simplest assumption being that it is proportional totheir joint probability:Interaction rate = bpP.This leads to a prey growth rate including both predation and breeding:dp= ap− bpP, (LVM-I for prey). (12.55)dtIf left to themselves, predators P will also breed and increase their population. Yetpredators need animals to eat, and if there are no other populations to prey upon,they will eat each other (or their young) at a per-capita mortality rate m:dPdt∣ = −mP, ⇒ P (t)=P (0)e −mt . (12.56)competitionHowever, once there are prey to interact with (read "eat") at the rate bpP , thepredator population will grow at the ratedPdt= ɛbpP − mP (LVM-I for predators), (12.57)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 320