Pattern formation of a coupled two-cell Schnakenberg model
Guanqi Liu Yuwen Wang
Discrete & Continuous Dynamical Systems - S 2017, 10(5): 1051-1062 doi: 10.3934/dcdss.2017056
In this paper, we study a coupled two-cell Schnakenberg model with homogenous Neumann boundary condition, i.e.,
$\left\{ \begin{gathered} -d_1Δ u=a-u+u^2v+c(w-u),&\text{ in } Ω, \\-d_2Δ v=b-u^2v,&\text{ in } Ω , \\-d_1Δ w=a-w+w^2z+c(u-w),&\text{ in } Ω, \\-d_2Δ z=b-w^2z,&\text{ in } Ω, \\\dfrac{\partial u}{\partial ν}=\dfrac{\partial v}{\partial ν}=\dfrac{\partial w}{\partial ν}=\dfrac{\partial z}{\partial ν}=0, &\text{ on } \partialΩ.\end{gathered} \right.$
We give a priori estimate to the positive solution. Meanwhile, we obtain the non-existence and existence of positive non-constant solution as parameters
$ d_1, d_2, a$
and b changes.
keywords: Schnakenberg model two-cell pattern formation a priori estimate steady state non-existence and existence
Stochastic spatiotemporal diffusive predator-prey systems
Guanqi Liu Yuwen Wang
Communications on Pure & Applied Analysis 2018, 17(1): 67-84 doi: 10.3934/cpaa.2018005

In this paper, a spatiotemporal diffusive predator-prey system with Holling type-Ⅲ is considered. By using a Lyapunov-like function, it is proved that the unique local solution of the system must be a a global one if the interaction intensity is small enough. A comparison theorem is used to show that the system can be extinction or stability in mean square under some additional conditions. Finally, an unique invariant measure for the system is obtained.

keywords: Global solution spatiotemporal stochastic Holling type Ⅲ extinction stability in mean square invariant measure

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