Lotka -- Volterra equations
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. (These are not to be confused with the Lotka-Volterra inter-specific competition equations, which describe populations that compete for the same resources.) They were proposed 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.
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2 Physical meanings of the equations 3 Dynamics of the system 4 Solutions to the equations 5 See also 6 Bibliography |
The usual form of the equations is:
When multiplied out, the equations take a form useful for physical interpretation.
The prey equation becomes:
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.
The predator equation becomes:
Hence the equation represents the change in the predator population as the growth of the predator population, minus natural death.
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 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.
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 at which this equilibrium is achieved depends on the chosen values of the parameters, α, β, γ, and δ.
The stability of the fixed points can be determined by performing a linear analysis using partial derivatives.
The Jacobian matrix of the predator-prey model is
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 in the majority of cases. However, as the fixed point at the origin is a saddle point, and hence unstable, we find that extinction is difficult in the model.
Evaluating J at the second fixed point we get
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°.
The equations
wherePhysical meanings of the equations
Prey
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. 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. Dynamics of the system
Population equilibrium
When solved for x and y the above system of equations yields
and
hence there are two equilibria.Stability of the fixed points
When evaluated at the steady state of (0,0) the Jacobian matrix J becomes
The eigenvalues of this equation 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 point.
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 it is in fact a centre. This means that the levels of the predator and prey populations cycle, and oscillate around this fixed point.Solutions to the equations
See also
Bibliography