Predator-Prey Project (#1)

Introduction:

In this project we will see how the predator-prey model for a population of foxes and rabbits is interrelated. The basis for this model is derived from 4 different variables and 2 different differential equations.

Section 1:

To explain this model, one must first look at what causes the fox and rabbit populations to fluctuate. The primary causes are:

Foxes eat Rabbits.
Rabbits multiply at a fast rate.
The number of rabbits increases due to the reproduction rate of the rabbits. This causes the fox population to increase due to the abundance of food. The increase of foxes causes the rabbit population to decline because of consumption. The decline in the rabbit population then causes the fox population to decline due to lack of food supply. As the fox population dwindles, the rabbit population is allowed to rejuvinate, and the cycle begins again.

F₀ is the initial population of foxes, and R₀ is the initial population of rabbits.

F' is the rate of change over time of the population of foxes. It is based on the generic population model equation P'=kP. 'k' being the growth rate of the population, and 'P' being the population in terms of time (P(t)). In our model, 'k', for the foxes, is defined as -(df-R). This term states that the growth rate is equal to the birth rate of foxes (due to the rabbit population) minus the death rate of foxes. For R', 'k' is defined as (br-F), where br is equal to the birth rate of rabbits, and F is the death rate of rabbits due to the fox population. This model assumes no other ways for the population to be affected.

Section 2:

If F₀=br/

and R₀=df/

, then the time rate of change of the population will be equal to 0, as there will be just enough rabbits being born to sustain the fox population as both the rabbit and fox population are naturally decreasing. Therefore the changes in one population due to the other are at equilibrium.

The solution for this special case is the F(t) and R(t) will equal a constant value. This is because when you solve the equations, you find that 'k' is equal to 0, so in the final equation you have a constant (initial population) multiplied by e^k, which is equal to 1.

Section 3:

We chose the constants of proportionality as the following values:

Variable	Description	Value
df	Death rate of Foxes	0.150
	Birth rate of Foxes due to Rabbits	0.002
br	Birth rate of Rabbits	3.000
	Death rate of Rabbits due to Foxes	0.200

We chose these values due to the reasons mentioned in #1 above, and also so that the solution plot would be realistic (no negative populations, etc.) over our time period of 350 years. The solution plot of this system (see graph at the end) is a circle graphed with relation to the population of Rabbits and Foxes over time, with the center located at the equilibrium points (df/,br/). It is a circle because as the number of rabbits increases, the number of foxes increase, but then there are too many foxes, so the rabbit population decreases. As the rabbit population decreases, more foxes start to die, and both populations go down. After enough foxes have perished, the rabbits regenerates, and the cycle repeats.

Comments:

To see a more advanced version of this model with the ability to edit more variables, visit the WWW URL:

http://fisher.teorekol.lu.se/simulation_server.html

This project is also available off of the World Wide Web via the URL:

http://www.kluge.net/~felicity/proj/deq.html

Graph: