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Full Description
The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been studied. In contrast, the role of intermediate advection remains poorly understood. For example, concentration phenomena can occur when advection is strong, providing a mechanism for the coexistence of multiple populations, in contrast with the situation of weak advection where coexistence may not be possible. The transition of the dynamics from weak to strong advection is generally difficult to determine.
In this work the authors consider a mathematical model of two competing populations in a spatially varying but temporally constant environment, where both species have the same population dynamics but different dispersal strategies: one species adopts random dispersal, while the dispersal strategy for the other species is a combination of random dispersal and advection upward along the resource gradient. For any given diffusion rates the authors consider the bifurcation diagram of positive steady states by using the advection rate as the bifurcation parameter. This approach enables the authors to capture the change of dynamics from weak advection to strong advection.
The authors determine three different types of bifurcation diagrams, depending on the difference of diffusion rates. Some exact multiplicity results about bifurcation points are also presented. The authors' results can unify some previous work and, as a case study about the role of advection, also contribute to the understanding of intermediate (relative to diffusion) advection in reaction-diffusion models.
Contents
Introduction: The role of advection
Summary of main results
Preliminaries
Coexistence and classification of $\mu$-$\nu$ plane
Results in $\mathcal {R}_1$: Proof of Theorem 2.10
Results in $\mathcal {R}_2$: Proof of Theorem 2.11
Results in $\mathcal {R}_3$: Proof of Theorem 2.12
Summary of asymptotic behaviors of $\eta _*$ and $\eta ^*$
Structure of positive steady states via Lyapunov-Schmidt procedure
Non-convex domains
Global bifurcation results
Discussion and future directions
Appendix A, Asymptotic behavior of $\tilde{u}$ and $\lambda _u$
Appendix B. Limit eigenvalue problems as $\mu ,\nu \to 0$
Appendix C, Limiting eigenvalue problem as $\mu \to \infty$
Bibliography.
