Deterministic property changes in population models under random error perturbations

Abstract

It was found that essential properties (periodic and chaotic dynamics) of some simple ecological models need not be changed or destroyed by small random error perturbations although the perturbations can obscure orbits of periodic cycles of simple population models. It may not be generally realized that very small random error perturbations can significantly change the deterministic properties of some population models. We simulated 12 well-known population models. Among six of the 12 models, the deterministic properties are changed under random error perturbations. A host–parasitoid model was used as an example to illustrate our methodology. The observed property changes in that model included the following four aspects: (1) the periodic cycles changed to chaos; (2) chaos changed to periodic; (3) oscillation magnitude changed; and (4) complex dynamics became two blurred periodic orbits. Of the five other models in which deterministic property changes were observed, two of them were one-dimensional models and three were multi-dimensional models. The changes in properties are more striking in multi-dimensional models than that in one-dimensional models. All simulation results show that under random error perturbations, the equilibrium points and the two periodic cycles of some population models are stable. However, some complex dynamic behaviors of population models, such as n-periodic cycles ( n≥4), the regime of period-doubling, and the periodic windows, would be changed or blurred by random error perturbations. Thus these properties are very difficult to observe in the field or to verify in biological or ecological experiments.