The Influence of External Real and White Noise on the LOTKA‐VOLTERRA Model

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Abstract
The behaviour of a marginally stable predator‐prey system, the LOTKA‐VOLTERRA model is analyzed under the influence of parameter fluctuations. The cases of white and real noise are studied separately. It is shown that for white noise no stationary solution exists, but even for time tending to infinity explosion occurs only with zero probability. In the case of real noise the class of noise processes that permit a stationary solution is characterized by their spectral density. It turns out that this class consists of all stationary processes that do not contain the eigenfrequencies of the LOTKA‐VOLTERRA system.It is shown that for all other real noise processes a resonance phenomenon occurs and the solutions grow unboundedly.

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