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