arXiv subjects

Zuolin Shen

Publications and source records attributed to Zuolin Shen.

4 recordsLinked to original sources

arXiv2020

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of the center manifold reduction technique and normal form theory, we present a new algorithm for computing normal forms associated with the codimension-two double Hopf bifurcation of nonlocal reaction-diffusion equations. The theoretical results are applied to a predator-prey model, and complex dynamic behaviors such as spatially nonhomogeneous periodic oscillations and spatially nonhomogeneous quasi-periodic oscillations could occur.

math.DS↗

arXiv2019

Hopf bifurcation of a delayed single population model with patch structure

In this paper, we show the existence of Hopf bifurcation of a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the "local" Hopf bifurcation values over all patches. On the other hand, when the dispersal rate tends to infinity, the Hopf bifurcation value tends to that of the "average" model.

math.DS↗

arXiv2019

Bifurcation Analysis in A Diffusive Mussel-Algae Model with Delay

In this paper, we consider the dynamics of a delayed reaction-diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

math.DS↗

arXiv2018

Spatiotemporal patterns near the Turing-Hopf bifurcation in a delay-diffusion mussel-algae model

The spatiotemporal patterns of a reaction diffusion mussel-algae system with a delay subject to Neumann boundary conditions is considered. The paper is a continuation of our previous studies on delay-diffusion mussel-algae model. The global existence and positivity of solutions are obtained. The stability of the positive constant steady state and existence of Hopf bifurcation and Turing bifurcation are discussed by analyzing the distribution of eigenvalues. Furthermore, the dynamic classifications near the Turing-Hopf bifurcation point are obtained in the dimensionless parameter space by calculating the normal form on the center manifold, and the spatiotemporal patterns consisting of spatially homogeneous periodic solutions, spatially inhomogeneous steady states, and spatially inhomogeneous periodic solutions are identified in this parameter space through some numerical simulations. Both theoretical and numerical results reveal that the Turing-Hopf bifurcation can enrich the diversity of spatial distribution of populations.

math.DS↗