Lotka-Volterra Equation The Equations Physical Meanings Of The

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

2.1 Prey

The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growthIn mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. Anything that grows by the same percentage every year (or every month, day, hour etc. is growing exponentially. For example, if the average number of 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.

2.2 Predators

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.

3 Solutions to the equations

The equations have periodic solutions which do not have a simple expression in terms of the usual trigonometric functionIn mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios ofs. However, an approximate linearisedLinearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point solution yields a simple harmonic motionSimple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic and can be described as that of a sine function (or equivalently a cosine function), with constant amplitude. It is ch with the population of predators leading that of prey by 90°.

4 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.

4.1 Population equilibrium

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

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

and

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.

4.2 Stability of the fixed points

The stability of the fixed points can be determined by performing a linearizationLinearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point using partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. They are useful in n-dimensional calculus and differential geometry. The partial derivative ofs.

The Jacobian matrix of the predator-prey model is

4.2.1 First fixed point

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

The eigenvalues of this matrix are

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 pointMultivariate calculus In mathematics, a saddle point is a point of a function of two variables which is a stationary point but not an local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves d.

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.

4.2.2 Second fixed point

Evaluating J at the second fixed point we get

The eigenvalues of this matrix are

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.

5 See also

Population dynamics

6 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.

Ecology

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Home > Lotka-Volterra equation

1 The equations

2 Physical meanings of the equations