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### Open Access Journals

DCDS

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.

PROC

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.

DCDS

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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DCDS

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.

CPAA

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.

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