### Hans H. Diebner's Research

## Standard Lotka-Volterra Model

dx/dt =

dy/dt = -

*a*x -*b*x^{2}-*c*xydy/dt = -

*e*y -*c'*xy
Initial values: x

_{0}= y_{0}= 0.3. Integration using Euler algorithm with time step dt = 0.05. Trajectory or time series length, respectively: 8000 iterations.*x*obeys in absence of a predator species (

*y = 0*) a logistic growth dynamics

dx/dt =

The parameter *a*x -*b*x^{2}.*b*is reciprocally proportional to the environmental carrying capacity of the species. 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*.