8 Predator Prey

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