Predator- Prey and ‘Evolon’- Behaviour Modelled by Random-Walk Processes

Abstract

In this contribution mathematical models for a global evolution and growth process (‘evolon’) and for the corresponding predator- prey processes are discussed. First by means of a general structure building principle a model for the evolon represented by the hyperlogistic growth law is transformed into a Lotka-Volterra system, where due to well known results on these systems qualitative analysis can be done. It turns out that within this three- dimensional Lotka- Volterra system a twodimensional predator- prey relation is responsible for all phenomena. Generalizing the special parameters of this predator- prey system results in a generalized deterministic evolon- model. In the following the question of the randomness of the process is dicussed. In order to treat also the ‘random basis’ of the process — the process is partially a random one, the deterministic continuous models are only approximations — a discrete model based on a random walk process for predator and prey is developed and investigated. Several rules in this random walk process are discussed. This discrete model based on random walk shows in the time domain the same qualitative behaviour as the deterministic model for the predator- prey process, the deterministic model has to be seen as approximation of the behaviour of the random model. Furthermore the randow walk model gives much more insight into the process because in each step interesting clusters representing the building of stable and unstable structures of the process can be observed.