Transcript Lotka-Volterra Predator

By: Alexandra Silva and Dani Hoover
Intro to Systems ESE 251
11/24/09
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
Phase Plot of Predator vs. Prey
 Set parameters
Phase plot
3
Y, predator
a=b=c=d=1
 Set initial conditions:
x=2 (prey), y=2
(predators)
 Equilibrium Point:
x=(c/d), y= (a/b)
 Counter-clockwise
motion
2.5
2
Equilibrium point (1,1)
1.5
1
0.5
0
0
0.5
1
1.5
X, prey
2
2.5
3
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
Phase Plot: Case 2
Phase plot
2
1.8
1.6
 When initial conditions
Y, predator
1.4
1.2
equal the equilibrium
point:
 Parameters:
a=b=c=d=1
Initial conditions:
x=1 (prey), y=1 (predators)
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
X, prey
1.2
1.4
1.6
1.8
2
Solution to L-V equations
2
Prey
Predator
1.8
1.6
population
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
time
30
35
40
45
50
Phase plot
2.5
Phase Plot: Case 3
changed:
 Parameters:
a=c=d=1, b=2
*Increase the death rate due to
predation
 Initial Conditions:
x=2 (prey), y=2 (predator)
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
X, prey
3
3.5
4
4.5
Solution to L-V equations
4.5
Prey
Predator
4
3.5
3
population
 When parameters are
Y, predator
2
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
time
30
35
40
45
50
Phase Plot: Case 4
 When a species dies out:
Phase plot
600
x=50 (prey), y=500
(predator)
500
400
Y, predator
 Parameters:
a=1, b=c=d=1
 Initial Conditions:
300
200
100
 Prey dies, therefore
predator dies too.
0
-10
0
10
20
X, prey
30
40
50
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
Phase Plot of Prey vs. Predator vs. Superpredator
Phase plot
Solution to L-V equations
5
Prey
Predator
Super Predator
4.5
5
4
3.5
population
Z, predator
4
3
2
3
2.5
2
1.5
1
2
4
1.5
1
3
1
Y, super predator
0.5
2
0.5
1
0
5
X, prey
Parameters:
a=b=c=d=1, e=0.5, f=0.01, g=0.02
Initial Condition:
X=1, Y=1, Z=1
10
15
20
25
time
30
35
40
45
50
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
Thank you
 Thank you to Anatoly for helping us with this
presentation and helping us to make programs in
MATLAB.
Questions?
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