Spatiotemporal Dynamics of the Diffusive Mussel-Algae Model Near Turing-Hopf Bifurcation

Abstract

Intertidal mussels can self-organize into periodic spot, stripe, labyrinth, and gap patterns ranging from centimeter to meter scales. The leading mathematical explanations for these phenomena are the reaction-diffusion-advection model and the phase separation model. This paper continues the series studies on analytically understanding the existence of pattern solutions in the reactiondiffusion mussel-algae model. The stability of the positive constant steady state and the existence of Hopf and steady-state bifurcations are studied by analyzing the corresponding characteristic equation. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the explicit dynamical classification in its neighborhood by calculating and investigating the normal form on the center manifold. Using theoretical and numerical simulations, we demonstrates that this TH interaction would significantly enhance the diversity of spatial patterns and trigger the alternative paths for the pattern development.

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

“…Many of the previous researches for Turing-Hopf bifurcations are focused on reaction-diffusion systems (see previous studies 9,[13][14][15][16] ), but there are few related theoretical works for the time-delay systems to the best of our knowledge. However, the time delay really exists in real world and always exerts a profound influence on the dynamic behavior of the predator-prey system.…”

Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

“…Note that if (H1) holds, then D 0 = αr(r − 1)a * > 0. The transversality is proved by the recent work in [23], here we just state the following lemma without proof.…”

Spatiotemporal patterns induced by delay and cross‐fractional diffusion in a predator‐prey model describing intraguild predation

“…In the past decades, spatiotemporal dynamic patterns, especially in biology, mathematical ecology, and chemistry, have been investigated widely by many authors. [15][16][17][18][19][20][21][22][23][24][25][26][27] Currently, the spatiotemporal patterns of population distribution have been a hot topic because species abundance changes not only in time but also in space. Our first objective is to show that delay can induce spatially homogeneous and inhomogeneous periodic oscillatory patterns resulting from Hopf bifurcation.…”