Abstract
We investigate four predator–prey Rosenzweig–MacArthur models in which the prey exhibit herd behaviour and only the individuals on the edge of the herd are subjected to the predators’ attacks. The key concept is the herding index, i.e., the parameter defining the characteristic shape of the herd. We derive the population equations from the individual state transitions using the mechanistic approach and time scale separation method. We consider one predator and one prey species, linear and hyperbolic responses and the occurrence of predators’ intraspecific competition. For all models, we study the equilibria and their stability and we give the bifurcation analysis. We use standard numerical methods and the software Xppaut to obtain the one-parameter and two-parameter bifurcation diagrams.
Highlights
We study the dynamics of the model by Rosenzweig and MacArthur [16] with the α herd-Holling type II functional response f ( x ) = 1+axahxα derived in Section 2, conversion factor e of captured prey into new predators and per capita natural mortality rate m for the predators, logistic growth g( x ) = rx 1 −
The aim of this paper is to formalise, by means of illustrative examples, how prey herd behaviour can be modelled with ordinary differential equations from first principles
We use the mechanistic approach and time-scale separation method to derive from the individual mechanisms a Holling type II–like functional response for the predators, the herd-Holling type II functional response, which takes into account the prey herd shape
Summary
We investigate four predator–prey Rosenzweig–MacArthur models in which the prey exhibit herd behaviour and only the individuals on the edge of the herd are subjected to the predators’. The key concept is the herding index, i.e., the parameter defining the characteristic shape of the herd. We derive the population equations from the individual state transitions using the mechanistic approach and time scale separation method. We consider one predator and one prey species, linear and hyperbolic responses and the occurrence of predators’ intraspecific competition. We study the equilibria and their stability and we give the bifurcation analysis. We use standard numerical methods and the software Xppaut to obtain the one-parameter and two-parameter bifurcation diagrams
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