38,598 research outputs found
Selected topics on reaction-diffusion-advection models from spatial ecology
We discuss the effects of movement and spatial heterogeneity on population
dynamics via reaction-diffusion-advection models, focusing on the persistence,
competition, and evolution of organisms in spatially heterogeneous
environments. Topics include Lokta-Volterra competition models, river models,
evolution of biased movement, phytoplankton growth, and spatial spread of
epidemic disease. Open problems and conjectures are presented
Replicator equations and space
A reaction--diffusion replicator equation is studied. A novel method to apply
the principle of global regulation is used to write down the model with
explicit spatial structure. Properties of stationary solutions together with
their stability are analyzed analytically, and relationships between stability
of the rest points of the non-distributed replicator equation and distributed
system are shown. A numerical example is given to show that the spatial
variable in this particular model promotes the system's permanence.Comment: 24 page
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A HYBRID METHOD FOR STIFF REACTION-DIFFUSION EQUATIONS.
The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction-diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order hybrid IIF-ETD method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction-diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method
Global existence for semilinear reaction-diffusion systems on evolving domains
We present global existence results for solutions of reaction-diffusion
systems on evolving domains. Global existence results for a class of
reaction-diffusion systems on fixed domains are extended to the same systems
posed on spatially linear isotropically evolving domains. The results hold
without any assumptions on the sign of the growth rate. The analysis is valid
for many systems that commonly arise in the theory of pattern formation. We
present numerical results illustrating our theoretical findings.Comment: 24 pages, 3 figure
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