3 research outputs found
Multiple wave solutions in a diffusive predator-prey model with strong Allee effect on prey and ratio-dependent functional response
A thorough analysis is performed in a predator-prey reaction-diffusion model
which includes three relevant complex dynamical ingredients: (a) a strong Allee
effect; (b) ratio-dependent functional responses; and (c) transport attributes
given by a diffusion process. As is well-known in the specialized literature,
these aspects capture adverse survival conditions for the prey, predation
search features and non-homogeneous spatial dynamical distribution of both
populations. We look for traveling-wave solutions and provide rigorous results
coming from a standard local analysis, numerical bifurcation analysis, and
relevant computations of invariant manifolds to exhibit homoclinic and
heteroclinic connections and periodic orbits in the associated dynamical system
in . In so doing, we present and describe a diverse zoo of traveling wave
solutions; and we relate their occurrence to the Allee effect, the spreading
rates and propagation speed. In addition, homoclinic chaos is manifested via
both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An
actual computation of global invariant manifolds near a focus-focus homoclinic
bifurcation is also presented to enravel a multiplicity of wave solutions in
the model. A deep understanding of such ecological dynamics is therefore
highlighted.Comment: 35 pages, 22 figure
Conditions for Turing and wave instabilities in reaction-diffusion systems
Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reactionâdiffusion system with n components and diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to [Formula: see text] , gives a constructive method for choosing diffusion constants. The results are illustrated on a 3-component FitzHughâNagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4-components