Traveling waves and their stability in a coupled reaction diffusion system
Xiaojie Hou Wei Feng
Communications on Pure & Applied Analysis 2011, 10(1): 141-160 doi: 10.3934/cpaa.2011.10.141
We study the traveling wave solutions to a reaction diffusion system modeling the public goods game with altruistic behaviors. The existence of the waves is derived through monotone iteration of a pair of classical upper- and lower solutions. The waves are shown to be unique and strictly monotonic. A similar KPP wave like asymptotic behaviors are obtained by comparison principle and exponential dichotomy. The stability of the traveling waves with non-critical speed is investigated by spectral analysis in the weighted Banach spaces.
keywords: uniqueness stability. spectrum existence asymptotic rates Traveling wave
Zhaosheng Feng Wei Feng
Discrete & Continuous Dynamical Systems - S 2014, 7(6): i-i doi: 10.3934/dcdss.2014.7.6i
As we all know, many biological and physical systems, such as neuronal systems and disease systems, are featured by certain nonlinear and complex patterns in their elements and networks. These phenomena carry significant biological and physical information and regulate down-stream mechanism in many instances. This issue of Discrete and Continuous Dynamical Systems, Series S, comprises a collection of recent works in the general area of nonlinear differential equations and dynamical systems, and related applications in mathematical biology and engineering. The common themes of this issue include theoretical analysis, mathematical models, computational and statistical methods on dynamical systems and differential equations, as well as applications in fields of neurodynamics, biology, and engineering etc.
    Research articles contributed to this issue explore a large variety of topics and present many of the advances in the field of differential equations, dynamical systems and mathematical modeling, with emphasis on newly developed theory and techniques on analysis of nonlinear systems, as well as applications in natural science and engineering. These contributions not only present valuable new results, ideas and techniques in nonlinear systems, but also formulate a few open questions which may stimulate further study in this area. We would like to thank the authors for their excellent contributions, the referees for their tireless efforts in reviewing the manuscripts and making suggestions, and the chief editors of DCDS-S for making this issue possible. We hope that these works will help the readers and researchers to understand and make future progress in the field of nonlinear analysis and mathematical modeling.
Dynamics in 30species preadtor-prey models with time delays
Wei Feng
Conference Publications 2007, 2007(Special): 364-372 doi: 10.3934/proc.2007.2007.364
We study a differential equation system with diffusion and time delays which models the dynamics of predator-prey interactions within three biological species. Our main focus is on the persistence (non-extinction) of u-species which is at the bottom of the nutrient hierarchy, and the permanence effect (long-term survival of all the predators and prey) in this model. When u-species persists in the absence of its predators, we generate a condition on the interaction rates to ensure that it does not go extinction under the predation of the v- and w-species. With certain additional conditions, we can further obtain the permanence effect (long-term survival of all three species) in the ecological system. Our proven results also explicitly present the effects of all the environmental data (growth rates and interaction rates) on the ultimate bounds of the three biological species. Numerical simulations of the model are also given to demonstrate the pattern of dynamics (extinction, persistence, and permanence)in the ecological model.
keywords: Long-term survival and permanence. Differential equation models Time delays Mathematical Ecology Predator-prey systems
Global periodicity in a class of reaction-diffusion systems with time delays
Wei Feng Xin Lu
Discrete & Continuous Dynamical Systems - B 2003, 3(1): 69-78 doi: 10.3934/dcdsb.2003.3.69
In this paper we study a class of reaction-diffusion systems modelling the dynamics of "food-limited" populations with periodic environmental data and time delays. The existence of a global attracting positive periodic solution is first established in the model without time delay. It is further shown that as long as the magnitude of the instantaneous self-limitation effects is larger than that of the time-delay effects, the positive periodic solution is also the global attractor in the time-delay system. Numerical simulations for both cases (with or without time delays) demonstrate the same asymptotic behavior (extinction or converging to the positive $T$-periodic solution, depending on the growth rate of the species).
keywords: Reaction-diffusion systems time-delay system.
Zhaosheng Feng Wei Feng
Communications on Pure & Applied Analysis 2011, 10(5): i-ii doi: 10.3934/cpaa.2011.10.5i
This issue of Communications on Pure and Applied Analysis, comprises a collection in the general area of nonlinear systems and analysis, and related applications in mathematical biology and engineering. During the past few decades people have seen an enormous growth of the applicability of dynamical systems and the new developments of related dynamical concepts. This has been driven by modern computer power as well as by the discovery of advanced mathematical techniques. Scientists in all disciplines have come to realize the power and beauty of the geometric and qualitative techniques developed during this period. More importantly, they have been able to apply these techniques to a various nonlinear problems ranging from physics and engineering to biology and ecology, from the smallest scales of theoretical particle physics up to the largest scales of cosmic structure. The results have been truly exciting: systems which once seemed completely intractable from an analytical point of view can now be studied geometrically and qualitatively. Chaotic and random behavior of solutions of various systems is now understood to be an inherent feature of many nonlinear systems, and the geometric and numerical methods developed over the past few decades contributed significantly in those areas.
Global stability in a class of reaction-diffusion systems with time-varying delays
Wei Feng Xin Lu
Conference Publications 1998, 1998(Special): 253-261 doi: 10.3934/proc.1998.1998.253
Please refer to Full Text.
keywords: stability and asymptotic behavior. time-varying delay effects Diffusive logistic equations
Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain
Wei Feng Nicole Rocco Michael Freeze Xin Lu
Discrete & Continuous Dynamical Systems - S 2014, 7(6): 1215-1230 doi: 10.3934/dcdss.2014.7.1215
In this paper, we study a new model as an extension of the Rosenzweig-MacArthur tritrophic food chain model in which the super-predator consumes both the predator and the prey. We first obtain the ultimate bounds and conditions for exponential convergence for these populations. We also find all possible equilibria and investigate their stability or instability in relation with all the ecological parameters. Our main focus is on the conditions for the existence, uniqueness and stability of a coexistence equilibrium. The complexity of the dynamics in this model is theoretically discussed and graphically demonstrated through various examples and numerical simulations.
keywords: stability and asymptotic behavior extinction or coexistence numerical simulations. limiting nutrient response Tri-trophic food-chain models
Stability and pattern in two-patch predator-prey population dynamics
Wei Feng Jody Hinson
Conference Publications 2005, 2005(Special): 268-279 doi: 10.3934/proc.2005.2005.268
In this paper we explore the dynamics of predator-prey interactions in two patches which are coupled by the diffusion of the predator. The purpose of this exploration to find upper and lower bounds for the populations, and to discuss the complexity and stability of the equilibriums. Some of our results relax the conditions, which are given in earlier papers, necessary for stability of the equilibrium solutions associated with this model. Numerical simulations are also provided to graphically demonstrate the population dynamics of this model utilizing some of the relaxed conditions for stability and new conditions for instability.
keywords: Reaction-diffusion systems Food chain models Stability and asymptotic behavior. time delays
On existence of wavefront solutions in mixed monotone reaction-diffusion systems
Wei Feng Weihua Ruan Xin Lu
Discrete & Continuous Dynamical Systems - B 2016, 21(3): 815-836 doi: 10.3934/dcdsb.2016.21.815
In this article, we give an existence-comparison theorem for wavefront solutions in a general class of reaction-diffusion systems. With mixed quasi-monotonicity and Lipschitz condition on the set bounded by coupled upper-lower solutions, the existence of wavefront solution is proven by applying the Schauder Fixed Point Theorem on a compact invariant set. Our main result is then applied to well-known examples: a ratio-dependent predator-prey model, a three-species food chain model of Lotka-Volterra type and a three-species competition model of Lotka-Volterra type. For each model, we establish conditions on the ecological parameters for the presence of wavefront solutions flowing towards the coexistent states through suitably constructed upper and lower solutions. Numerical simulations on those models are also demonstrated to illustrate our theoretical results.
keywords: mixed monotone functions numerical simulations. coexistence in ecological models Reaction-diffusion systems existence of wavefront solutions
Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective
Wei Feng Shuhua Hu Xin Lu
Conference Publications 2003, 2003(Special): 544-553 doi: 10.3934/proc.2003.2003.544
keywords: Optimal control cancer chemotherapy. three compartment model field of extremals

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