Lotka-Volterra Predator-Prey Model

Mr NicholsPHHS

HistoryAlfred Lotka Vito Volterra

-American biophysicist-Proposed the predator-prey model in 1925

-Italian mathematician-Proposed the predator-prey model in 1926

2-Species Models

Equations and VariablesX’ = ax – bxyY’ = -cy + dxy

X: the population of preyY: the population of predatorsa: natural growth rate of prey in the absence of

predationb: death rate due to predationc: natural death rate of predators in the

absence of preyd: growth rate due to predation

AssumptionsThe prey always has an unlimited supply of

food and reproduces exponentiallyThe food supply of the predators depend only

on the prey population (predators eat the prey only)

The rate of change of the population is proportional to the size of the population

The environment does not change in favor of one species

Steady-State Orbit explanation

A = Too many predators.B = Too few prey.C = Few predator and prey; prey can grow.D= Few predators, ample prey.

http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/2DS.html

3-Species Model (Super-predator)

Equations and Variables (for 3-species model)X’= ax-bxy (prey-- mouse)Y’= -cy+dxy-eyz (predator-- snake)Z’= -fz+gxz (super-predator-- owl)

a: natural growth rate of prey in the absence of predation

b: death rate due to predationc: natural death rate of predatord: growth rate due to predatione: death rate due to predation (by super-predator)f: natural death rate of super-predatorg: growth rate due to predation

Problems with Lotka-Volterra ModelsThe Lotka-Volterra model has infinite cycles

that do not settle down quickly. These cycles are not very common in nature.

Must have an ideal predator-prey system.In reality, predators may eat more than one

type of preyEnvironmental factors

Sources:http://www.cs.unm.edu/~forrest/classes/cs36

5/CS%20365/Lectures_files/lotka-volterra.pdfhttp://www.vanderbilt.edu/AnS/psychology/co

gsci/chaos/workshop/2DS.htmlhttp://en.wikipedia.org/wiki/Lotka%E2%80%9

3Volterra_equationhttp://isolatium.uhh.hawaii.edu/m206L/lab8/p

redator/predator.htmhttp://www4.ncsu.edu/eos/users/w/white/ww

w/white/ma302/less10.PDFhttp://www.cs.unm.edu/~forrest/classes/cs36

5/CS%20365/Lectures_files/lotka-volterra.pdfhttp://www.stolaf.edu/people/mckelvey/envisi

on.dir/lotka-volt.html