Analysis of Predator – Prey
Models
By: Meredith Chiaro
April 29, 2009
Predator-Prey Model
• A model that studies the population growth
of two species in which one species is the
primary food source for the other
• Examples: wolves/rabbits, lions/gazelles,
birds/insects, etc.
Background
• Numerous Predator-Prey Models:
– Kolmogorov’s predator-prey model
• General predator-prey model
• Written in terms of two autonomous differential
equations
– Lotka-Volterra Model
• Differential equation model originally used for observing
predator fish in the Adriatic Sea
• Simplest model of predator-prey interactions
• Based on linear per capita growth rates
– Jacob – Monod Model
• Used for interactions with organisms (ie. Bacteria)
using nutrients
Importance
• The study of predator-prey interaction can
influence the evolution of that species.
• Predicting the outcome of species
interactions can help researchers analyze
how each of the species are structured
and sustained.
Discrete Implementation
• Discrete models describe micro-scale
events and short range interactions
• There is a time derivative
• For this project, a discrete model will be
implemented to show the interaction
between the predator and prey
Continuous Implementation
• Continuous models are good for extremely
large populations
• Describes long range effects and large
scale events
Model
• Analysis of Basic Model
– Limitations:
• Prey species has an unlimited supply of food
• No other threat to prey other than the one predator
• Predator is entirely dependent on the named prey
species as its only food supply
• If no predators were to exist, the prey would grow
exponentially
• The rate at which predators encounter prey is
jointly proportionally to the sizes of the two
populations
Model
• Thus, giving the equation for the prey
growth:
– dx/dt = ax – bxy
• Where a and b are both positive constants , x is
the prey population, and y is the predator
population
• Negative component of prey growth rate is
proportional to the product xy of the population
sizes
Model
• For the predator population:
– If there were no food supply, the population
would die out at a rate proportional to its size:
• dy/dt = -cy
– The growth of the predator population is
proportional to the deaths of its prey:
• dy/dt = -cy +pxy
Model
• Combining the prey growth and the
predator growth, the result is the LotkaVolterra Predator-Prey Model:
– dx/dt = ax-bxy
– dy/dt = -cy +pxy
• Where a,b,c,and p are positive constants
Model
• Focus for this problem will be on the
relationship between coyotes and
roadrunners in the desert southwest of the
United States.
Model
• Let’s assume:
– dC/dt = aC - bCR
– dR/dt = bCR – dR
• Where, C is the coyote population and R is the
roadrunner population alive at time t
• a is the natural growth rate of roadrunners in the
absence of predation
• b is the death rate of roadrunners due to predation
• d is the natural death rate of coyotes in the
absence of food (roadrunners)
Model
• By plugging the model into Excel the
following slides show results
Model
• Graph 1: t=10 years
Journal Publication
Questions ?