Deterministic and Stochastic Research of Cubic Population Model with Harvesting

TL;DR

This paper investigates the impact of harvesting on populations with Allee effects using both deterministic and stochastic models, deriving equations to analyze population stability and probability distributions under different noise conditions.

Contribution

It extends the cubic population model by incorporating harvesting effects and stochastic noise, providing new insights into population dynamics under these influences.

Findings

Harvesting alters Allee thresholds and carrying capacity.

Stochastic noise influences population stability and distribution.

Different noise types affect population behavior distinctly.

Abstract

A detailed examination of the effect of harvesting on a population has been carried out by extending the standard cubic deterministic model by considering a population under Allee effect with a quadratic function representing harvesting. Weak and strong Allee effect transitions, carrying capacity, and Allee threshold change according to harvesting is first discussed in the deterministic model. A Fokker Planck equation has been obtained starting from a Langevin equation subject to correlated Gaussian white noise with zero mean, and an Approximate Fokker Planck Equation has been obtained from a Langevin equation subject to correlated Gaussian colored noise with zero mean. This allowed to calculate the stationary probability distributions of populations, and thus to discuss the effects of linear and nonlinear (Holling type-II) harvesting for populations under Allee effect and subject to white and colored noises respectively.