Pattern formation of a coupled two-cell Brusselator model

Abstract

We first study the stability of the unique positive constant solution (u, v, w, z) = (1, 1, 1, 1). Then, we give a priori estimate (positive upper and lower bounds) to the positive solution. At last, we obtain the non-existence and existence of positive nonconstant solutions as parameters d 1 , d 2 and b varied.

Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model

“…∂x (lπ, t) = 0, which is the system (4) of Zhou and Mu [17]. Applying the Corollary 3 of [17]), we have the following prior estimates.…”

“…By Theorem 1.2 of Zhou and Mu [17], we know that, there exists b 0 > 0, such that the system (3.22) has no non-constant solutions for b ∈ (0, b 0 ). Then for system (3.2), we have the following results.…”

“…Then, by employing the technique introduced in Zhou and Mu [17], we investigate the steady state bifurcation for the coupled Brusselator model (1.2).…”

“…Recently, Zhou and Mu [17] considered the existence and non-existence of positive non-trivial solutions of system (1.2) by utilizing the bifurcation theory and degree theory. Their theoretical analysis shows that the diffusion rate of this reaction and the size of the reactor play decisive roles in leading to the formation of stationary patterns.…”

Qualitative analysis of a strongly coupled predator–prey system with modified Holling–Tanner functional response

“…Finally, we remark that in the past decades, there has been much work on the existence and non-existence of non-constant positive steady states of ecological models with diffusion or cross-diffusion under the homogeneous Neumann boundary conditions. One can refer to [4,11,13,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The role of diffusion in modeling many physical, chemical and biological processes has been extensively studied.…”