Predator-Prey Models

Pedro Ribeiro de AndradeGilberto Câmara

Acknowledgments and thanks

Many thanks to the following professors for making slides available on the internet that were reused by us

Abdessamad Tridane (ASU) Gleen Ledder (Univ of Nebraska) Roger Day (Illinois State University)

“nature red in tooth and claw”

One species uses another as a food resource: lynx and hare.

The Hudson’s Bay Company

hare and lynx populations (Canada)Note regular periodicity, and lag by lynx population peaks just after hare peaks

Predator-prey systems

The principal cause of death among the prey is being eaten by a predator.The birth and survival rates of the predators depend on their available food supply—namely, the prey.

Predator-prey systems

Two species encounter each other at a rate that is proportional to both populations

normal prey populationprey population

increasesprey population

increases

predator population increases

as more food

predator population decreases

as less foodprey population decreasesbecause of more predators

Predator-prey cycles

Generic Model

• f(x) prey growth term• g(y) predator mortality term• h(x,y) predation term• e - prey into predator biomass conversion coefficient

Lotka-Volterra Model

r - prey growth rate : Malthus lawm - predator mortality rate : natural mortalitya and b predation coefficients : b=eae prey into predator biomass conversion coefficient

Predator-prey population fluctuations in Lotka-Volterra model

Predator-prey systems

Suppose that populations of rabbits and wolves are described by the Lotka-Volterra equations with: k = 0.08, a = 0.001, r = 0.02, b = 0.00002

The time t is measured in months.

There are 40 wolfes and 1000 rabbits

Phase plane

Variation of one species in relation to the other

Phase trajectories: solution curve

A phase trajectory is a path traced out by solutions (R, W) as time goes by.

Equilibrium point

The point (1000, 80) is inside all the solution curves. It corresponds to the equilibrium solution R = 1000, W = 80.

Hare-lynx data

Hare-lynx data