Predator Prey and the Lotka-Volterra Equations
dR/dt = a*R - b*R*F
dF/dt = e*b*R*F - c*F
eqn. 1
eqn. 2
where the parameters are defined by:




a is the natural net growth rate of rabbits in the absence of predation,
c is the natural death rate of foxes in the absence of food (rabbits),
b is the death rate per encounter of rabbits due to predation,
e is the efficiency of turning predated rabbits into foxes.
Equilibrium
dR/dt = 0
a*R=b*R*F
a=bF
a/b=Fequilibrium
and dF/dt = 0
e*b*R*F = c*F
e*b*R= c
Requilibrium=c/(e*b)
Figure 1. Equilibrium state of Fox / Rabbit Populations.
e*b = 0.00025 c=0.050
period = 28 yr a=0.1 b=0.0025
Question:
The equations controlling a predator prey interaction are determined to be
dR/dt = 0.6*R - 0.01*R*F
dF/dt =0 .5*R*F - .001*F
What are the predicted equilibrium values for R (Prey) and F (predator)?
Figure 2. Changing the period of oscillation without changing equilibria.
e*b = 0.0025 c=0.50
period = 28 yr a=0.1 b=0.0025 [1]
e*b = 0.00025 c=0.050
period = 90 yr a=0.1 b=0.0025 [2]
e*b = 0.025 c=5
period = 10 yr a=0.1 b=0.0025 [3]
There are two thing worth noting from these simulations. First the larger a and b are the
stiffer the system and the shorter the period. The same is true for larger values of e*b and
c. Also the amplitude of the oscillations increases with decreasing period. Of course the
initial conditions also control the amplitude but note the stiffness (or the period).
Question:
The equations controlling a predator prey interaction are determined to be:
dR/dt = a*R - b*R*F
dF/dt = e*b*R*F - c*F
Which set of conditions will yield the shortest period of oscillations for this system?
e*b = 0.0025 c=0.50
e*b = 0.00025 c=0.050
e*b = 0.025 c=5
e*b = 0.025 c=5
a=0.1 b=0.0025
a=0.1 b=0.0025
a=0.1 b=0.0025
a=1 b=0.025
[1]
[2]
[3]
[4]
For these scenarios how do the equilibrium values compare?
Figure 2. Changing the period of oscillation without changing equilibria.
eb = 0.00025 c=0.050
period = 90 yr a=0.1 b=0.0025 [1]
eb = 0.00025 c=0.050
period = 25 yr a=1 b=0.025 [2]
eb = 0.0025 c=.5
period = 8.5 yr a=1
b=0.025 [3]
Finally we draw an x-y scatter plot of the two populations. This is often referred to as a
phase plot.
Figure 3 . Phase plot of R vs F using eb = 0.00025
(period = 90 yr )
c=0.050
a=0.1
b=0.0025
Numerical Implementation of Competitive Lotka Volterra model
As both the predator and the prey compete for food and shelter in the forest, competition
sets in and the population of each species tends to control itself via a negative quadratic
effect , that is the population decreases with a rate directly proportional to the present
population of that species. Hence , higher the population , higher is the quadratic
decrease. This effect can be introduced in the model by incorporating a quadraic term in
each of equations (1) and (2) as follows :
dR/dt = a*R - b*R*F-alpha*R^2
dF/dt = e*b*R*F - c*F-beta*F^2
This effect was introduced in the model. The Parameters chosen for this simulation were
a = .1 , b = 0.0025 , e*b = 0.0025 and c = 0.5. The initial prey population was chosen to
be 150 and the Predator population was chosen to be 40 (equilibrium value) . This is the
predicted behavior to be expected when competition sets in Predator-Prey interactions.
eb = 0.0025 c=0.50
period = 30 yr
alpha = 0.00005 beta=0.0
a=0.1
b=0.0025
R v . F: 1 300
200
100
0
Page 2
40
F
80
5:04 PM Thu, Nov 03, 2011
Untitled
eb = 0.0025 c=0.50
period = 30 yr
alpha = 0.00005 beta=0.0
a=0.1
b=0.0025
Question: The equations controlling a predator prey interaction are determined to be:
dR/dt = a*R - b*R*F-alpha*R^2
dF/dt = e*b*R*F - c*F-beta*F^2
Which phase plot below corresponds to alpha=beta=0?
Question: The equations controlling a predator prey interaction are determined to be:
dR/dt = a*R - b*R*F-alpha*R^2
dF/dt = e*b*R*F - c*F-beta*F^2
Which phase plot below corresponds to the largest alpha assuming beta=0.0?
References
1.
http://www.stolaf.edu/people/mckelvey/envision.dir/lotka-volt.html
2.
http://www.personal.psu.edu/auk183/LotkaVolterra/LotkaVolterra1.html
Answers to selected questions
Question:
The equations controlling a predator prey interaction are determined to be
dR/dt = 0.6*R - 0.01*R*F
dF/dt =0 .5*R*F - .001*F
What are the predicted equilibrium values for R (Prey) and F (predator)?
Requilibrium=0.5/0.001=500 and Fequilibrium=.6/.01=60
Question:
The equations controlling a predator prey interaction are determined to be:
dR/dt = a*R - b*R*F
dF/dt = e*b*R*F - c*F
Which set of conditions will yield the shortest period of oscillations for this system?
e*b = 0.0025 c=0.50
e*b = 0.00025 c=0.050
e*b = 0.025 c=5
e*b = 0.025 c=5
a=0.1 b=0.0025
a=0.1 b=0.0025
a=0.1 b=0.0025
a=1 b=0.025
[1]
[2]
[3]
[4]
For these scenarios how do the equilibrium values compare?
All 4 conditions have the same equilibrium values.
Question: The equations controlling a predator prey interaction are determined to be:
dR/dt = a*R - b*R*F-alpha*R^2
dF/dt = e*b*R*F - c*F-beta*F^2
Which phase plot below corresponds to alpha=beta=0?
C
Question: The equations controlling a predator prey interaction are determined to be:
dR/dt = a*R - b*R*F-alpha*R^2
dF/dt = e*b*R*F - c*F-beta*F^2
Which phase plot below corresponds to the largest alpha assuming beta=0.0?
D