Abstract
In this article, a new predator-prey model having predator saturation is proposed. The model resembles a classical Rosenzweig-MacArthur type model, but comes with an added function, the population saturation function of the predator. This function of the predator population is a factor in the predator fertility term in the model. Consequently the model behaves better than the Rosenzweig-MacArthur model since all solutions are bounded within the population quadrant. An invariant region arises where the Poincare-Bendixon theorem can be applied. In most cases there is but a single critical value, either an attracting spiral point suggesting a stable population pair or an unstable node, resulting in a unique limit cycle. This model is fully described and an analysis of the stability of critical values is provided. The robustness of the model is demonstrated based on the classification of Gunawardena [8].
Highlights
There is a multitude of predator-prey models in the literature
The FGH model supports the feature that the solutions are bounded from the onset by the carrying capacity of the prey and the population saturation of the predator
There is a wide variation of all parameters values for which the FGH model is robust and results in a stable system having equilibrium values which lead to either limit cycles or attracting spiral points
Summary
There is a multitude of predator-prey models in the literature. These models originated with those of Lotka [12] (1925) and Volterra [15] (1931) and have been much refined, since . In this paper an original predator-prey model, referred to as the FGH model, is proposed and each term has a firm ecological basis for its inclusion and form. The model is, in essence, a classical Rosenzweig-MacArthur [14] type model with an added function which is called the population saturation of the predator. As this model has bounded solutions in the population quadrant, Poincaré-Bendixon theory [16] applies. It is shown that the FGH model, with specific parameter values, has equilibrium values which lead to either limit cycles or attracting spiral points
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