Jibin Li Kening Lu Junping Shi Chongchun Zeng
This special issue of Discrete and Continuous Dynamical Systems-A is dedicated to Peter W. Bates on the occasion of his 60th birthday, and in recognition of his outstanding contributions to infinite dimensional dynamical systems and the mathematical theory of phase transitions.
    Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.

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Dynamics of a reaction-diffusion system of autocatalytic chemical reaction
Jifa Jiang Junping Shi
The precise dynamics of a reaction-diffusion model of autocatalytic chemical reaction is described. It is shown that exactly either one, two, or three steady states exists, and the solution of dynamical problem always approaches to one of steady states in the long run. Moreover it is shown that a global codimension one manifold separates the basins of attraction of the two stable steady states. Analytic ingredients include exact multiplicity of semilinear elliptic equation, the theory of monotone dynamical systems and the theory of asymptotically autonomous dynamical systems.
keywords: autocatalytic chemical reaction asymptotic autonomous system convergence to equilibrium.
Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management
Renhao Cui Haomiao Li Linfeng Mei Junping Shi

A reaction-diffusion logistic population model with spatially nonhomogeneous harvesting is considered. It is shown that when the intrinsic growth rate is larger than the principal eigenvalue of the protection zone, then the population is always sustainable; while in the opposite case, there exists a maximum allowable catch to avoid the population extinction. The existence of steady state solutions is also studied for both cases. The existence of an optimal harvesting pattern is also shown, and theoretical results are complemented by some numerical simulations for one-dimensional domains.

keywords: Diffusive logistic equation harvesting protection zone bifurcation
On lane-emden type systems
Philip Korman Junping Shi
We consider a class of singular systems of Lane-Emden type \begin{equation} \nonumber \begin{cases} \Delta u + \la u^{p_1} v^{q_1}=0, & x\in D,\\ \Delta v + \la u^{p_2} v^{q_2}=0, & x\in D,\\ u=v=0, & x\in \partial D, \end{cases} \end{equation} with $p_1\le 0, \; p_2> 0, \; q_1> 0, \; q_2\le 0$, and $D$ a smooth domain in $\R^n$. In case the system is sublinear we prove existence of a positive solution. If $D$ is a ball in $\R^n$, we prove both existence and uniqueness of positive radially symmetric solution.
keywords: semilinear elliptic system uniqueness.
Traveling waves of a mutualistic model of mistletoes and birds
Chuncheng Wang Rongsong Liu Junping Shi Carlos Martinez del Rio
The existences of an asymptotic spreading speed and traveling wave solutions for a diffusive model which describes the interaction of mistletoe and bird populations with nonlocal diffusion and delay effect are proved by using monotone semiflow theory. The effects of different dispersal kernels on the asymptotic spreading speeds are investigated through concrete examples and simulations.
keywords: reaction-diffusion nonlocal traveling wave. asymptotic spreading speed Model of mistletoes and birds
Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity
Junping Shi Ratnasingham Shivaji
keywords: Semipositone Problems. Semilinear Elliptic Equation Exact Multiplicity Bifurcation
On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line
Fangfang Jiang Junping Shi Qing-guo Wang Jitao Sun
In this paper, we investigate the existence and uniqueness of crossing limit cycle for a planar nonlinear Liénard system which is discontinuous along a straight line (called a discontinuity line). By using the Poincaré mapping method and some analysis techniques, a criterion for the existence, uniqueness and stability of a crossing limit cycle in the discontinuous differential system is established. An application to Schnakenberg model of an autocatalytic chemical reaction is given to illustrate the effectiveness of our result. We also consider a class of discontinuous piecewise linear differential systems and give a necessary condition of the existence of crossing limit cycle, which can be used to prove the non-existence of crossing limit cycle.
keywords: Liénard system uniqueness. crossing limit cycle (non) existence Discontinuous system
The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere
Chjan C. Lim Junping Shi
The effects of a higher vorticity moment on a variational problem for barotropic vorticity on a rotating sphere is examined rigorously in the framework of the Direct Method. This variational model differs from previous work on the Barotropic Vorticity Equation (BVE) in relaxing the angular momentum constraint, which then allows us to state and prove theorems that give necessary and sufficient conditions for the existence and stability of constrained energy extremals in the form of super and sub-rotating solid-body steady flows. Relaxation of angular momentum is a necessary step in the modeling of the important tilt instability where the rotational axis of the barotropic atmosphere tilts away from the fixed north-south axis of planetary spin. These conditions on a minimal set of parameters consisting of the planetary spin, relative enstrophy and the fourth vorticity moment, extend the results of previous work and clarify the role of the higher vorticity moments in models of geophysical flows.
keywords: barotropic vorticity model higher vorticity moment variational method.
Relaxation oscillation profile of limit cycle in predator-prey system
Sze-Bi Hsu Junping Shi
It is known that some predator-prey system can possess a unique limit cycle which is globally asymptotically stable. For a prototypical predator-prey system, we show that the solution curve of the limit cycle exhibits temporal patterns of a relaxation oscillator, or a Heaviside function, when certain parameter is small.
keywords: predator-prey model. limit cycle Relaxation oscillator
Semilinear elliptic equations with generalized cubic nonlinearities
Junping Shi R. Shivaji
A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.
keywords: global bifurcation. semilinear reaction-diffusion equation

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