Transcript Lotka-Volterra Predator

Mr Nichols
PHHS
History
 Alfred Lotka
-American biophysicist
-Proposed the predatorprey model in 1925
 Vito Volterra
-Italian mathematician
-Proposed the predatorprey model in 1926
2-Species Models
Equations and Variables
 X’ = ax – bxy
 Y’ = -cy + dxy
 X: the population of prey
 Y: the population of predators
 a: natural growth rate of prey in the absence of
predation
 b: death rate due to predation
 c: natural death rate of predators in the absence of prey
 d: growth rate due to predation
Assumptions
 The prey always has an unlimited supply of food and
reproduces exponentially
 The 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/works
hop/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 predation
 c: natural death rate of predator
 d: growth rate due to predation
 e: death rate due to predation (by super-predator)
 f: natural death rate of super-predator
 g: growth rate due to predation
Problems with Lotka-Volterra Models
 The 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 prey
 Environmental factors
Sources:
 http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectu






res_files/lotka-volterra.pdf
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/work
shop/2DS.html
http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equati
on
http://isolatium.uhh.hawaii.edu/m206L/lab8/predator/predator
.htm
http://www4.ncsu.edu/eos/users/w/white/www/white/ma302/l
ess10.PDF
http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectu
res_files/lotka-volterra.pdf
http://www.stolaf.edu/people/mckelvey/envision.dir/lotkavolt.html