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.
Highlights
The analysis of predator-prey models considers different ecological phenomena affecting either one population or both populations. These phenomena can have strong consequences in the relationship between them and modifying the dynamical properties of a system describing it [1, 2]. Such is the case of an Allee effect affecting the prey population or else, the competition among predators (CAP) [3]; there are not enough studies to determine the real impact of these two phenomena in that dynamical relationship, when both act simultaneously in the interaction
We have analyzed a model considering competition among predators (CAP) and the prey population is affected by an Allee effect
By means of a diffeomorphism [30], we analyzed the topologically equivalent system (11) to the original one, depending only on four parameters; conditions for the existence of positive equilibrium points and their nature were established in some cases
Summary
The analysis of predator-prey models considers different ecological phenomena affecting either one population or both populations These phenomena can have strong consequences in the relationship between them and modifying the dynamical properties of a system describing it [1, 2]. In this work a modified Volterra model [4] is analyzed, in which the functional response is linear, assuming that (i) the prey growth rate is affected by a strong Allee effect [5] and (ii) there exists self-interference (interference) or competition among predators (CAP) [6]. We model the CAP with this last form presented in the Bazykin’s book [6]; we consider the linear functional response independent of predator density (i.e., only prey dependent), which means that any single predator affects the prey population growth rate independently of its conspecifics [15]. The obtained results will be compared with the predatorprey model in which competition among predators is not considered, partially studied in the book by Kot [24]; it will be compared with the model considering double Allee effect [5, 21], without self-interference among predators analyzed in [1, 2]
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