Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

Abstract

The Kolmogorov model has been applied to many biological and environmental problems. We are particularly interested in one of its variants, that is, a Gauss-type predator-prey model that includes the Allee effect and Holling type-III functional response. Instead of using classic first order differential equations to formulate the model, fractional order differential equations are utilized. The existence and uniqueness of a nonnegative solution as well as the conditions for the existence of equilibrium points are provided. We then investigate the local stability of the three types of equilibrium points by using the linearization method. The conditions for the existence of a Hopf bifurcation at the positive equilibrium are also presented. To further affirm the theoretical results, numerical simulations for the coexistence equilibrium point are carried out.

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

“…The Hopf bifurcation is a local phenomenon when a stable equilibrium point loses its stability and all nearby solutions converge to a periodic solution namely limit-cycle if a parameter is varied [54,60,61]. It is shown that many fractional-order models involving the Caputo operator undergo a Hopf bifurcation which is driven by the order of the derivative (see [2,17,34,53,62]).…”

Dynamics for a fractional-order predator-prey model with group defense

“…Definition 3.2 25,26 We say that U * is an equilibrium point of system (3.1) if and only if h(t, U * ) = 0. Lemma 3.1 26,27 Let J(U * ) be the Jacobian matrix of model (3.1) near U * and λ…”

“…, then the equilibrium point U * is a saddle point. Lemma 3.2 26,27 Let U * , J(U * ) be equilibrium point, Jacobian matrix of system (3.1), respectively, and λ i be the eigenvalues of Jacobian matrix J(U * ) of system (2.1).…”

Dynamical analysis of an integrated pest management predator–prey model with weak Allee effect

“…We discuss existence,uniqueness and stability of the order-1 periodic orbit of system (3) based on the qualitative analysis of system (1).…”

“…The Allee effect is a phenomenon in biology characterized by a correlation between population size or density and the mean individual fitness (often measured as per capita population growth rate) of a population or species [11,15,17]. Many scholars investigated the rich dynamical behaviours of predator-prey model with Allee effect [1,7,16,18,21,24,34,40,44,45,48]. To consider that the predators are difficult to seek spouses when the species has a low population density, the Allee effect on predator is introduced to model phenomenon.…”