Stochastic spatiotemporal diffusive predator-prey systems
Guanqi Liu Yuwen Wang

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
A double saddle-node bifurcation theorem
Ping Liu Junping Shi Yuwen Wang
In this paper, we consider an abstract equation $F(\lambda,u)=0$ with one parameter $\lambda$, where $F\in C^p(\mathbb{R} \times X, Y)$, $p\geq 2$, is a nonlinear differentiable mapping, and $X,Y$ are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a "double" saddle-node bifurcation theorem with a two-dimensional kernel. Applications include a perturbed problem and a semilinear elliptic equation.
keywords: two-mimensional kernel Saddle-node bifurcation Morse lemma.
Pattern formation of a coupled two-cell Schnakenberg model
Guanqi Liu Yuwen Wang
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

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