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Lotka-Volterra equation

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The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. They were proposed independently by Vito Volterra and Alfred J. Lotka in the 1920s. A classic model using the equations is of the population dynamics of the lynx and the snowshoe hare, popularised due to the extensive data collected on the relative populations of the two species by the Hudson Bay company during the 19th century.

Contents

The equations

The usual form of the equations is:

<math>\frac{dx}{dt} = x(\alpha - \beta y)<math>
<math>\frac{dy}{dt} = -y(\gamma - \delta x)<math>

where

  • y is the number of some predator (for example, foxes);
  • x is the number of its prey (for example, rabbits);
  • t represents the development of the two populations against time; and
  • α, β, γ and δ are parameters representing the interaction of the two species.

Physical meanings of the equations

When multiplied out, the equations take a form useful for physical interpretation.

Prey

The prey equation becomes:

<math>\frac{dx}{dt} = \alpha x - \beta xy<math>

The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

Predators

The predator equation becomes:

<math>\frac{dy}{dt} = \delta xy - \gamma y<math>

In this equation, δxy represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). γy represents the natural death of the predators; it is an exponential decay.

Hence the equation represents the change in the predator population as the growth of the predator population, minus natural death.

Solutions to the equations

The equations have periodic solutions which do not have a simple expression in terms of the usual trigonometric functions. However, an approximate linearised solution yields a simple harmonic motion with the population of predators leading that of prey by 90°.

Missing image
Volterra_lotka_dynamics.PNG
Image:Volterra_lotka_dynamics.PNG

Dynamics of the system

In the model system, the predators thrive when there are plentiful prey 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.

Population equilibrium

Population equilibrium occurs in the model when neither of the population levels are changing, i.e. when both of the differential equations are equal to 0.

<math>x(\alpha - \beta y) = 0<math>
<math>-y(\gamma - \delta x) = 0<math>

When solved for x and y the above system of equations yields

<math>\{y = 0 , x = 0\}<math>

and

<math>\{y = \frac{\alpha}{\beta}, x = \frac{\gamma}{\delta}\}<math>,

hence there are two equilibria.

The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depends on the chosen values of the parameters, α, β, γ, and δ.

Stability of the fixed points

The stability of the fixed points can be determined by performing a linearization using partial derivatives.

The Jacobian matrix of the predator-prey model is

<math>J(x,y) = \begin{bmatrix}

\alpha - \beta y & -\beta x \\ \delta y & \delta x - \gamma \\ \end{bmatrix}<math>

First fixed point

When evaluated at the steady state of (0,0) the Jacobian matrix J becomes

<math>J(0,0) = \begin{bmatrix}

\alpha & 0 \\ 0 & -\gamma \\ \end{bmatrix}<math>

The eigenvalues of this matrix are

<math>\lambda_1 = \alpha , \lambda_2 = -\gamma<math>

In the model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point.

The stability of this fixed point is of importance. If it was stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, we find that the extinction of both species is difficult in the model.

Second fixed point

Evaluating J at the second fixed point we get

<math>J(\frac{\gamma}{\delta},\frac{\alpha}{\beta}) = \begin{bmatrix}

0 & -\frac{\beta \gamma}{\delta} \\ \frac{\alpha \delta}{\beta} & 0 \\ \end{bmatrix}<math>

The eigenvalues of this matrix are

<math>\lambda_1 = i \sqrt{\alpha \gamma} , \lambda_2 = -i \sqrt{\alpha \gamma}

<math>

As the eigenvalues are both complex, this fixed point is a focus. The real part is zero in both cases so more specifically it is a centre. This means that the levels of the predator and prey populations cycle, and oscillate around this fixed point.

See also

Bibliography

  • E. R. Leigh (1968) The ecological role of Volterra's equations, in Some Mathematical Problems in Biology - a modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903.
  • Understanding Nonlinear Dynamics. Daniel Kaplan and Leon Glass.
  • V. Volterra. Variations and fluctuations of the number of individuals in animal species living together. In Animal Ecology. McGraw-Hill, 1931. Translated from 1928 edition by R. N. Chapman.de:Volterra-Gesetze
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