Logistic equation is a simple stochastic carrying capacity

Abstract

The logistic model has long been used in ecological modelling for its simplicity and effectiveness. Variations on the logistic model are prolific but, to date, there are a limited number of models that incorporate the stochastic nature of the carrying capacity. This study proposes a modification to the logistic model to incorporate a second differential equation which describes the changes in the carrying capacity, thus treating the carrying capacity as a state variable. The carrying capacity is modelled via a stochastic differential equation that accounts for stochastic ('noisy') variations in the finite resources that the population relies on. The extinction probability distribution, expected solution paths, variance of the solution paths, and distribution of the population are computed using the Monte Carlo method. References

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