Hans H. Diebner's Research

Standard Lotka-Volterra Model

The predator-prey interaction model in the form presented here has been discussed in 1926 by V. Volterra in his essay "Variazione e fluttuazini del numero d'individui in specie animali conviventi." The prey species density x obeys in absence of a predator species (y = 0) a logistic growth dynamics The parameter b is reciprocally proportional to the environmental carrying capacity of the species. In other words, without interaction with a predator the prey species (resource) grows asymptotically until carrying capicity is reached. To the contrary, the predator is not able to survive without interaction with a resource and the population would exponentially decline: To conserve the population, the influx c'xy must - in the average - balance the exponential decline.

Occasionally, the notion "Lotka-Volterra model" is used for the dynamics for given b = 0, thus, without the saturation term. In the following, this case is discussed as a specific marginal case.

For b = 0 the carrying capacity tends towards infinity, enabling a potentially unlimited exponential growth for the prey species without impact of a predator (c = 0 or y = 0). Only the interaction with the predator is then able to limit the prey population size. However, in the boundary case of b = 0 both populations oscillate in a conservative mode. The amplitude is in this case determined by the initial values.

To the contrary, for b > 0 , i.e., for a finite carrying capacity, the system is dissipative and the population sizes obey a damped oscillation which eventually fades out to become stable at an invariant fixed point. One observes a transient oscillation only if the initial values are not set to the attracting fixed points from the beginning. After the transient phase, a transient oscillation can only be observed if the system receives an external perturbation.

The behaviour of the system can interactivelly be tested using the above slider to control the value of b.