Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

Abstract

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.

Dynamical system analysis of quintom dark energy model

“…Thus at H c = 0, the system is structurally unstable as small perturbation at H c = 0, we get different characteristics of the vector fields. So, taking H c as a parameter, the bifurcation value is H c = 0 and bifurcation point is the origin [27]. At H c = 0, for n, m are even and v 0 > 0, the origin is a focus or center.…”

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

“…Here let the phase space C := C([−τ, 0], X C ). Our main focus is the stability of positive constant steady state E * (m * , a * ) with respect to the model (1.3), and the results of the boundary steady state E 0 (0, 1) can be seen in [22]. The linearization of system (1.3) at E * (m * , a * ) is given by…”

“…all other eigenvalues have non-zero real parts. In our previous paper [22], it has been proved that, the system (1.1) without diffusion can undergo Hopf bifurcation when parameters are chosen appropriately.…”

“…Motivated by their work, we study the spatiotemporal dynamics of system (1.3). Compared with the work of [22,24], we focus more on the common effects of delay and diffusion. Hence, a basic assumption is that the positive constant steady state is locally asymptotically stable under a homogeneous perturbation when time delay is equal to zero, and this assmption allows us to identify the importance of delay and diffusion in the process of pattern formation.…”

Dynamic Analysis on a Diffusive Two‐Enterprise Interaction Model with Two Delays

“…It is well known that diffusion factors are indispensable in the modeling of ecological and biological systems and often utilized to describe the spatial distributions of the densities of some certain substances, such as plants, animals, and other organisms. At present, there are a growing number of dynamic models described by reaction-diffusion equations with time delays in various application areas [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. For example, some scholars [19,23,[25][26][27] proposed several different diffusive predator-prey systems with delays to analyze the effect of diffusion on biological systems.…”