# 8 Predator Prey

Published on January 1, 2008

Author: Laurie

Source: authorstream.com

Predator-Prey Models:  Predator-Prey Models Dr. Tim Taylor Lecture 8 Lotka-Volterra:  Lotka-Volterra A simple example of a predator-prey model is two interacting populations for which the level of one population depends on the level of the other population. Lotka (1925) and Volterra (1926) first studied this as a way of proposing models for interaction species. They proposed: N(t) is the population of prey and P(t) is the population of the predator at time t. The model they proposed: Lotka-Volterra:  Lotka-Volterra The problem with the original equation ca be seen if we look at the boundary conditions, as well as other conditions. Boundary condition 1: P=0 P=0 so, N t N0 There is no limit on N population Lotka-Volterra:  Lotka-Volterra Condition 2: This says that an N increase the number of N eaten by each P increases to infinity. Improvements:  Improvements To improve the Lotka-Volterra model, we can start with the population growth model developed in class. This model gives limited growth for the prey. We need a term that represents the amount of predation that will occur as a function of the population of N Possible models of predator response R(N)::  Possible models of predator response R(N): Lokta-Volterra model R(N)=A Possible models of predator response R(N)::  Possible models of predator response R(N): b. R(N)= if N>>>B then as N→0 Possible models of predator response R(N)::  Possible models of predator response R(N): c. If N >>> B, and as as Possible models of predator response R(N)::  Possible models of predator response R(N): d. This is similar to part (b) which was described by the equation R(N): Real World:  Real World Which model is representative of what happens in the real world? When N is small, it is hard for the predator to find any prey to eat. As N increases, the predator can eat some because there are more, and thus are easier to find. At some population N, the predator cannot eat any more so A is the maximum amount one predator can eat in a given time period. Rewriting the prey function:  Rewriting the prey function Now to describe the predator model, Kp must be a function of the population of N. Think of it as the carrying capacity of P per unit N. Let

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