Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

Abstract

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

“…The innovation of this work lies in the complete analysis of Hopf bifurcation, Turing instability, Turing-Hopf bifurcation, and double Hopf bifurcation. In previous work, the authors often studied one type of codimension-two bifurcations, such as Turing-Hopf bifurcations or double Hopf bifurcations [2,42,36]. In our work, the time delay in prey growth and the diffusion of population could induce the emergence of both two types of codimension-two bifurcations.…”

“…Suppose K2a1 r1β < K 1 holds. If a 11 < 0 or one of the conditions (i-iii) in Theorem 2.6 hold, then the following statements are established for system (2). (i) In Case I, the positive equilibrium E * (u * , v * ) is locally asymptotically stable for all τ ≥ 0.…”

“…Studying the dynamic properties of Holling-Tanner model is informative in understanding the interaction between predator and prey, such as local and global stability of the unique positive equilibrium, periodic solution, and chaotic phenomenon. We want to explore the effect of the delay and diffusion terms on the dynamic characteristics of model (2). In previous literature, most of the work on Holling-Tanner or other types of models are related to the dynamic behavior, which is usually obtained by studying the properties of Hopf or steady-state bifurcations [28,27,8].…”

“…Hopf bifurcation, Turing bifurcation), we find that the predatorprey system can exhibit more abundant phenomena near codimension-two bifurcation points, such as spatial nonhomogeneous periodic or quasi-periodic solutions, multi-stable phenomenon, and chaotic attractors. The dynamical behaviors of the Turing-Hopf bifurcation and double Hopf bifurcation for positive equilibrium are investigated in view of [1,9,11,7,2]. For one thing, An and Jiang [1] extended the normal form methods given by Faria [15] and gave an explicit formula of the normal form for Turing-Hopf bifurcation.…”

“…Obviously, there is a trivial equilibrium (0, 0) and two boundary equilibria (K, 0), (0, K 2 /β) for system (2). Assume that E * (u * , v * ) is the coexisting equilibrium of the system (2), then u * ∈ (0, K) is a root of…”