Dynamics of A Single Population Model with Memory Effect and Spatial Heterogeneity

In this paper, a single population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method of upper and lower solutions, which is asymptotically stable for relatively small memorized diffusion. However, after the memorized diffusion rate exceeding a critical value, spatial inhomogeneous periodic solution can be generated through Hopf bifurcation, if the integral of intrinsic growth rate over the domain is negative. Such phenomenon will never happen, if only memorized diffusion or spatially heterogeneity is presented, and therefore must be induced by their joint effects. This indicates that the memorized diffusion will bring about spatial-temporal patterns in the overall hostile environment. When the integral of intrinsic growth rate over the domain is positive, it turns out that the steady state is still asymptotically stable. Finally, the possible dynamics of the model is also discussed, if the boundary condition is replaced by Dirichlet condition

Turing-Hopf bifurcation and spatiotemporal patterns in a ratio-dependent diffusive Holling-Tanner system with time delay

The Turing-Hopf type spatiotemporal patterns in a diffusive Holling-Tanner model with discrete time delay is considered. A global Turing bifurcation theorem for $\tau=0$ and a local Turing bifurcation theorem for $\tau>0$ are given by the method of eigenvalue analysis and prior estimation. Further considering the degenerated situation, the existence of Bogdanov-Takens bifurcation and Turing-Hopf bifurcation are obtained. The normal form method is used to study the explicit dynamics near the Turing-Hopf singularity, and we establish the existence of various self-organized spatiotemporal patterns, such as two non-constant steady states (stripe patterns) coexist and two spatially inhomogeneous periodic solutions (spot patterns) coexist. 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 theoretically explained, when there is another Turing bifurcation curve which is relatively closed to the studied one

Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay

The normal forms up to the third order for a Hopf-steady state bifurcation of a general system of partial functional differential equations (PFDEs) is derived based on the center manifold and normal form theory of PFDEs. This is a codimension-two degenerate bifurcation with the characteristic equation having a pair of simple purely imaginary roots and a simple zero root, and the corresponding eigenfunctions may be spatially inhomogeneous. The PFDEs are reduced to a three-dimensional system of ordinary differential equations and precise dynamics near bifurcation point can be revealed by two unfolding parameters. The normal forms are explicitly written as functions of the Fr\'echet derivatives up to the third orders and characteristic functions of the original PFDEs, and they are presented in a concise matrix notation, which greatly eases the applications to the original PFDEs and is convenient for computer implementation. This provides a user-friendly approach of showing the existence and stability of patterned stationary and time-periodic solutions with spatial heterogeneity when the parameters are near a Turing-Hopf bifurcation point, and it can also be applied to reaction-diffusion systems without delay and the retarded functional differential equations without diffusion