Lotka-Volterra Prey and Predator Model

¡@

The Lotka-Volterra model is a model for an ecological predator-prey. If we assume that

1. the growth rate of prey A,

2. the rate at which predators destroy prey B ,

3. the death rate of predators C ,

4. the rate at which predators increase by consuming prey D,

are all constants, the following conditions hold.

1. A prey population x increases at a rate dx=Axdt (proportional to the number of prey) but is simultaneously destroyed by predators at a rate dx=-Bxydt (proportional to the product of the numbers of prey and predators).

2. A predator population y decreases at a rate dy=-Cydt (proportional to the number of predators), but increases at a rate dy=Dxydt (again proportional to the product of the numbers of prey and predators).

So, we can get two differential equations from the statements above:

dx/dt =  Ax  - Bxy

dy/dt  =  -Cy  + Dxy

¡@

Result:

In the model system, the predators thrive when there are plentiful preys but, ultimately, outstrip their food supply and decline. As the predator population is low the prey population will increase again. These dynamics continue in a cycle of growth and decline.

The result from the Lotka-Volterra model shows that the population of both preys and predators are oscillating in a periodic but non-trigonometric function. But up to first order approximation, we found that the the population of predators leading that of prey by 90¢X.

A close phase space diagram obtained from this model also proves that the populations are changing periodically.

¡@

¡@

¡@

Result in PDF form

¡@

java code for prey-predator model.

¡@

Parameter used in the code:

A = 0.07

B = 0.008

C = 0.2

D = 0.01

Initial prey population = 20

Initial predator population = 50

time interval for each step is 0.1, so that we will not miss the behavior in a small time region

and the total time is 2000, so that we can see the whole population behavior

¡@

¡@

¡@

Population Equilibrium

When and are both equal to 0, neither of the population levels is changing

So we have,

The solutions are {x = 0, y = 0} and {x = D/C, y = B/A }

The first solution means that there is no more living life for these two species, they are both extinct. The second solution means that there is a fixed point which both species can maintain their current population.

¡@

¡@

¡@