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# Tutorial 15 - Lotka-Volterra model

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Overview

In this exercise, you will implement the famous predator-prey model called the Lotka-Volterra model using two alternative methods.

The model

The Lotka-Volterra model describes interactions between two species in an ecosystem, a predator and a prey. Since we are considering two species, the model will involve two differential equations, one which describes how the prey population changes and the second which describes how the predator population changes.

For concreteness, let us assume that the preys in our model are rabbits, and that the predators are foxes. If we let R(t) and F(t) represent the number of rabbits and foxes, respectively, that are alive at time t (year), then the Lotka-Volterra model is:

ΔR/Δt = aR - bRF
ΔF/Δt = ebRF - cF
where the parameters are defined by:

• a is the natural growth rate of rabbits (1/year) in the absence of predation,
• c is the natural death rate of foxes (1/year) in the absence of food (rabbits),
• b is the death rate per encounter of rabbits (1/(year F)) due to predation,
• e is the efficiency of turning predated rabbits into foxes (F/R).

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

Physical meanings of the equations

The Lotka-Volterra model makes a number of assumptions about the environment and evolution of the fox and rabbit populations:

1. The rabbit population finds ample food at all times.
2. The food supply of the fox population depends entirely on the rabbit population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and the genetic adaptation is sufficiently slow.

The rabbits are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this growth is represented in the equation above by the term aR. The rate of predation upon the rabbits is assumed to be proportional to the rate at which the foxes and the rabbits meet; this is represented above by bRF. If either the rabbits or the foxes are extinct, ( R or F is zero) then there can be no predation.

With these two terms ( aR and bRF), the differential equation for rabbits can be interpreted as: the change in the rabbit’s numbers is given by its own growth minus the rate at which it is preyed upon.

In the differential equation for the foxes, ebRF represents the growth of the fox population. (Note the similarity to the predation rate; however, a different constant is used ( eb), as the rate at which the fox population grows is not necessarily equal to the rate at which it consumes the rabbits). cF represents the loss rate of the foxes due to either natural death or emigration; it leads to an exponential decay in the absence of rabbits.

Hence, the equation for the foxes expresses the change in the fox population as growth fuelled by the food supply, minus natural death.

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References