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The existence, direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady state of a reaction–diffusion–advection model with distributed delay and Dirichlet boundary condition are considered in this paper. By analyzing the characteristic equation and taking the bound of time delay as the bifurcation parameter, we show that Hopf bifurcation will occur when the shape parameter n ≥ 1 , and the positive steady state is always stable when n = 0. The properties of Hopf bifurcations are then obtained by center manifold reduction and norm form methods. The effects of the distributed delay, shape parameter and advection on Hopf bifurcation values are also investigated. Finally, numerical simulations illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive three-species food chain system with strong Allee effect subject to the Neumann boundary condition. The dynamics of the local system including stability of interior equilibria and Hopf bifurcation are discussed. For the diffusive system, we study the existence of non-negative solutions, stability of a positive homogeneous steady state, diffusion-driven Turing instability and the occurrence of Turing–Hopf bifurcation. Employing the multiple scale analysis, we derive the normal form of Turing–Hopf bifurcation, which helps us classify the dynamical behaviors of the diffusive system near the Turing–Hopf bifurcation point. [ABSTRACT FROM AUTHOR]

The spatiotemporal pattern induced by cross-diffusion of a toxic-phytoplankton-zooplankton model with nonmonotonic functional response is investigated in this paper. The linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes in the framework of a weakly nonlinear theory, and the stability analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, we illustrate the theoretical results via numerical simulations. It is shown that the spatiotemporal distribution of the plankton is homogeneous in the absence of cross-diffusion. However, when the cross-diffusivity is greater than the critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe patterns depending on the cross-diffusivity. Simultaneously, the impact of toxin-producing rate of toxic-phytoplankton (TPP) species and natural death rate of zooplankton species on pattern selection is also explored. [ABSTRACT FROM AUTHOR]

We propose a kind of delayed diffusive intraguild predation model with Holling II functional response in this paper. By analyzing the eigenvalue spectrum, it is found that the stability or instability of equilibria can be induced by delay. By utilizing the local bifurcation theory of partial functional differential equations, Hopf bifurcation of the proposed system with time delay as bifurcation parameter is investigated. It reveals that the time delay has a destabilizing effect in the intraguild predation model dynamics and a phenomenon of Hopf bifurcation occurs when the delay increases through a certain threshold. Then we give the explicit formulas to determine the direction, stability of Hopf bifurcation by utilizing the normal form method and center manifold reduction for PFDEs. Numerical simulations are performed to illustrate our theoretical results and show that delay and diffusion can induce the system into chaos and even trigger the emergence of different types of spatial patterns, including spiral wave pattern and chaotic wave pattern, which are induced by Hopf instability. [ABSTRACT FROM AUTHOR]