Competition among Predators and Allee Effect on Prey, Their Influence on a Gause‐Type Predation Model

Abstract

Interference or competition among predators (CAP) has often been ruled out in depredation models, although there are varied mathematical forms to describe and incorporate it into this interaction. In this work, we present the most known of these descriptions and one of them will be used in a modified Volterra model. Moreover, of this ecological phenomenon, a simple and strong Allee effect affecting the prey population will be considered in the relationship. An important feature of the new model is to have until two positive equilibrium points, to the difference with the Volterra model (without Allee effect); hence different and interesting dynamic situations appear in the system. Conditions for the existence and local stability of equilibria are determined. The boundedness of solutions, the existence of a limit cycle and a separatrix curve are also proven. Besides, the main properties of the model are examined from an ecological point of view. To make a comparative discussion of our results, an Appendix is added with the main properties of models, in which neither the Allee effect nor the competition among predators is considered. Some simulations are shown to endorse our results.

Collaboration between predators and weak Allee effect in prey. Consequences on the dynamics of a predation model

“…Hemos verificado que el sistema de ecuaciones diferenciales no lineales tiene una dinámica más variada que la del modelo Leslie original [30], y también es lo suficientemente diferente al sistema que considera competencia (o interferencia) entre individuos de la población de depredadores [23].…”

“…Estos fenómenos no sólo tienen consecuencias en la relación entre las especies, sino qur también, modifican las propiedades dinámicas del sistema que describe la interacción [23,25,26].…”

“…Existe un amplio conjunto de estudios que confirman el efecto estabilizador que tiene la competencia entre depredadores sobre el sistema que describe la interacción [23]. Esto se corrobora por la existencia de una gran región del espacio de parámetros donde el sistema posee un único punto de equilibrio en el plano de fase, el cual es globalmente asintóticamente estable (gas) [23].…”

Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge

“…In the study of ecology, understanding the impact of Allee effect plays an important role because Allee effects can greatly increase the likelihood of local and global extinction. Accordingly, the most significant consequences of the Allee effect [9,10,27,33,34,36,40,44,45] for saving the endangered species and revealing the evolution process that has given improvement to extant biodiversity. Pal et al [25] have considered a delayed predator-prey model with strong Allee effect in prey population growth where the growth function is governed by multiplicative Allee effect on the prey population.…”

Coexistence in seasonally varying predator–prey systems with Allee effect

“…More than 90 years after their introduction [50,51], Lotka-Volterra predator-prey models are a topic which attracts the attention of many researchers from a large range of points of view. Concerning recent works, we mention examples of results on predator-prey models dealing with autonomous ODEs [49,19], with delay [29,8] and stochastic [28,46] systems of equations, time-discrete [53,22], network [54,47] and fractional [35] models. Cross diffusion and pattern formation are also topics with recent results [9,13,45] as well as the derivation of classical functional responses [33,18,12] and the impact of seasonality in ODEs models [27,30].…”