 | Lotka-Volterra equation: Encyclopedia II - Lotka-Volterra equation - Dynamics of the system
Lotka-Volterra equation - 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.
Lotka-Volterra equation - 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.
x(α − βy) = 0
− y(γ − δx) = 0
When solved for x and y the above system of equations yields
{y = 0,x = 0}
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.
Lotka-Volterra equation - 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
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 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. (In fact, this can only occur if the prey are artificially completely eradicated, causing the predators to die of starvation. If the predators are eradicated, the prey population grows without bound in this simple model).
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.
Other related archives1847, 1903, Alfred J. Lotka, Canada, Differential equations, Ecology, Fixed points, Hudson Bay company, Hudson's Bay Company, Jacobian matrix, Lotka-Volterra inter-specific competition equations, Population dynamics, V. Volterra, Vito Volterra, biological systems, differential equations, dingoes, exponential growth, hares, linearised, linearization, lynx, non-linear, parameters, partial derivatives, periodic, predator, prey, saddle point, simple harmonic motion, snowshoe hare, species, trigonometric functions, wallabies
 Adapted from the Wikipedia article "Dynamics of the system", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
