Predator-prey Dynamics with Intraspecific Competition and an Allee Effect in the Predator Population

Abstract

The study of the Allee effect on the stability of equilibria of predatorprey systems is of recent interest to mathematicians, ecologists, and conservationists. Many theoretical models that include the Allee effect result in an unstable coexistence equilibrium. However, empirical evidence suggests that predator-prey systems exhibiting densitydependent growth at small population densities still can achieve coexistence in the long term. We review an often cited model that incorporates an Allee effect in the predator population resulting in an unstable coexistence equilibrium, and then present a novel extension to this model which includes a term modeling intraspecific competition within the predator population. The additional term penalizes predator population growth for large predator to prey density ratios. We use equilibrium analysis to define the regions in the parameter space where the coexistence equilibrium is stable, and show that there exist biologically reasonable parameter sets which produce a stable coexistence equilibrium for our model.

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

“…More than 50 years after the model has been proposed, the Rosenzweig-MacArthur predator-prey model [1] has been consistently developed by many scholars to approach the real world phenomena with more realistic mathematical models. The commonsensical modified Rosenzweig-MacArthur models are accomplishable by considering the biological perspectives of ecosystem conditions, for instance the stage structure [2,3], the refuge effect [4][5][6][7][8], the fear effect [9], the Allee effect [10,11], the intraspecific competition [12,13], the cannibalism [14], the infectious disease [15][16][17], and so forth.…”

On a Predator-Prey Model Involving Age and Spatial Structure

“…en, Λ is a contraction on the complete metric space L 2 + (Q 1 ) and using Banach's fixed point theorem, we conclude the existence of a unique fixed point u h nonnegative for Λ, so the unique couple (u h , v h ) is the unique solution of (9).…”

“…ey then prove the existence of a globally stable quasiperiodic solution of the associated non-autonomous model. For the mathematical and qualitative study of some prey-predator models, the reader can also consult the following articles [7][8][9][10][11][12][13][14].…”