Two species competition with an inhibitor involved
Georg Hetzer Wenxian Shen
Discrete & Continuous Dynamical Systems - A 2005, 12(1): 39-57 doi: 10.3934/dcds.2005.12.39
The dynamics of the solution flow of a two-species Lotka-Volterra competition model with an extra equation for simple inhibitor dynamics is investigated. The model fits into the abstract framework of two-species competition systems (or $K$-monotone systems), but the equilibrium representing the extinction of both species is not a repeller. This feature distinguishes our problem from the case of classical two-species competition without inhibitor (classical case for short), where a basic assumption requires that equilibrium to be a repeller. Nevertheless, several results similar to those in the classical case, such as competitive exclusion and the existence of a "thin" separatrix, are obtained, but differently from the classical case, coexistence of the two species or extinction of one of them may depend on the initial conditions. As in almost all two species competition models, the strong monotonicity of the flow (with respect to a certain order on $\mathbb R^3$) is a key ingredient for establishing the main results of the paper.
keywords: convergence results competitive exclusion strong monotonicity Lotka-Volterra two-species competition inhibitor Poincaré-Bendixson Theorem.
Global existence for a functional reaction-diffusion problem from climate modeling
Georg Hetzer
Conference Publications 2011, 2011(Special): 660-671 doi: 10.3934/proc.2011.2011.660
Motivated by coupling an energy balance climate model and a two-species competition model for the bio-sphere, one is led to study the existence of non-negative mild solutions for set-valued functional reaction-diffusion equations involving a memory term and a nonlocal Volterra-type operator. A global existence and boundedness result is established in an m-accretive setting.
keywords: m-accretive upper semi-continuous multi slow diff usion global bounded mild solution
Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems
Georg Hetzer Wenxian Shen
Discrete & Continuous Dynamical Systems - A 2015, 35(4): i-iii doi: 10.3934/dcds.2015.35.4i
The strong interest in infinite dimensional dissipative systems originated from the observation that the dynamics of large classes of partial differential equations and systems resembles the behavior known from the modern theory of finite-dimensional dynamical systems. Reaction-diffusion problems are typical examples in this context. In biological applications linear diffusion represents random dispersal of a species, but in many cases other dispersal strategies occur, which has led to models with cross diffusion and nonlocal dispersal.

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Characterization of turing diffusion-driven instability on evolving domains
Georg Hetzer Anotida Madzvamuse Wenxian Shen
Discrete & Continuous Dynamical Systems - A 2012, 32(11): 3975-4000 doi: 10.3934/dcds.2012.32.3975
In this paper we establish a general theoretical framework for Turing diffusion-driven instability for reaction-diffusion systems on time-dependent evolving domains. The main result is that Turing diffusion-driven instability for reaction-diffusion systems on evolving domains is characterised by Lyapunov exponents of the evolution family associated with the linearised system (obtained by linearising the original system along a spatially independent solution). This framework allows for the inclusion of the analysis of the long-time behavior of the solutions of reaction-diffusion systems. Applications to two special types of evolving domains are considered: (i) time-dependent domains which evolve to a final limiting fixed domain and (ii) time-dependent domains which are eventually time periodic. Reaction-diffusion systems have been widely proposed as plausible mechanisms for pattern formation in morphogenesis.
keywords: evolving domains Turing diffusion-driven instability exponential dichotomy. Reaction-diffusion systems Lyapunov exponents evolution semigroup theory growing domains nonautonomous systems
Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal
Georg Hetzer Tung Nguyen Wenxian Shen
Communications on Pure & Applied Analysis 2012, 11(5): 1699-1722 doi: 10.3934/cpaa.2012.11.1699
Coexistence and extinction for two species Volterra-Lotka competition systems with nonlocal dispersal are investigated in this paper. Sufficient conditions in terms of diffusion, reproduction, self-limitation, and competition rates are established for existence, uniqueness, and stability of coexistence states as well as for the extinction of one species. The focus is on environments with hostile surroundings. In this case, our results correspond to those for random dispersal under Dirichlet boundary conditions. Similar results hold for environments with non-flux boundary and for periodic environments, which correspond to those for random dispersal under Neumann boundary conditions and periodic boundary conditions, respectively.
keywords: extinction comparison principle. principal eigenvalue nonlocal dispersal coexistence Volterra-Lotka competition model

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