Semilinear elliptic equations with generalized cubic nonlinearities
Junping Shi R. Shivaji
Conference Publications 2005, 2005(Special): 798-805 doi: 10.3934/proc.2005.2005.798
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
Jibin Li Kening Lu Junping Shi Chongchun Zeng
Discrete & Continuous Dynamical Systems - A 2009, 24(3): i-ii doi: 10.3934/dcds.2009.24.3i
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
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 245-258 doi: 10.3934/dcds.2008.21.245
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
Discrete & Continuous Dynamical Systems - B 2017, 22(3): 791-807 doi: 10.3934/dcdsb.2017039

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
Conference Publications 2005, 2005(Special): 510-517 doi: 10.3934/proc.2005.2005.510
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
Discrete & Continuous Dynamical Systems - A 2015, 35(4): 1743-1765 doi: 10.3934/dcds.2015.35.1743
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
Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay
Wenjie Zuo Junping Shi
Communications on Pure & Applied Analysis 2018, 17(3): 1179-1200 doi: 10.3934/cpaa.2018057

The existence of traveling wave solutions and wave train solutions of a diffusive ratio-dependent predator-prey system with distributed delay is proved. For the case without distributed delay, we first establish the existence of traveling wave solution by using the upper and lower solutions method. Second, we prove the existence of periodic traveling wave train by using the Hopf bifurcation theorem. For the case with distributed delay, we obtain the existence of traveling wave and traveling wave train solutions when the mean delay is sufficiently small via the geometric singular perturbation theory. Our results provide theoretical basis for biological invasion of predator species.

keywords: Holling-Tanner predator-prey system ratio-dependent distributed delay traveling wave solutions traveling wave train solutions
Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity
Junping Shi Ratnasingham Shivaji
Discrete & Continuous Dynamical Systems - A 2001, 7(3): 559-571 doi: 10.3934/dcds.2001.7.559
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
Communications on Pure & Applied Analysis 2016, 15(6): 2509-2526 doi: 10.3934/cpaa.2016047
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
Effect of rotational grazing on plant and animal production
Mayee Chen Junping Shi
Mathematical Biosciences & Engineering 2018, 15(2): 393-406 doi: 10.3934/mbe.2018017

It is a common understanding that rotational cattle grazing provides better yields than continuous grazing, but a quantitative analysis is lacking in agricultural literature. In rotational grazing, cattle periodically move among paddocks in contrast to continuous grazing, in which the cattle graze on a single plot for the entire grazing season. We construct a differential equation model of vegetation grazing on a fixed area to show that production yields and stockpiled forage are greater for rotational grazing than continuous grazing. Our results show that both the number of cattle per acre and stockpiled forage increase for many rotational configurations.

keywords: Rotational grazing differential equation model plant and animal production

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