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The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach
About this Title
Isabel Averill, Department of Mathematics, Bryn Mawr College, King-Yeung Lam, Department of Mathematics, Ohio State University and Yuan Lou, Institute for Mathematical Sciences, Renmin University of China and Department of Mathematics, Ohio State University
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 245, Number 1161
ISBNs: 978-1-4704-2202-8 (print); 978-1-4704-3611-7 (online)
DOI: https://doi.org/10.1090/memo/1161
Published electronically: July 26, 2016
Keywords: Reaction-diffusion,
advection,
evolution of dispersal,
principal eigenvalue,
global bifurcation
MSC: Primary 35J57, 35B32, 92D25
Table of Contents
Chapters
- 1. Introduction: The role of advection
- 2. Summary of main results
- 3. Preliminaries
- 4. Coexistence and classification of $\mu$-$\nu$ plane
- 5. Results in $\mathcal {R}_1$: Proof of Theorem
- 6. Results in $\mathcal {R}_2$: Proof of Theorem 2.11
- 7. Results in $\mathcal {R}_3$: Proof of Theorem
- 8. Summary of asymptotic behaviors of $\eta _*$ and $\eta ^*$
- 9. Structure of positive steady states via Lyapunov-Schmidt procedure
- 10. Non-convex domains
- 11. Global bifurcation results
- 12. Discussion and future directions
- A. Asymptotic behavior of $\tilde {u}$ and $\lambda _u$
- B. Limit eigenvalue problems as $\mu ,\nu \to 0$
- C. Limiting eigenvalue problem as $\mu \to \infty$
Abstract
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 we 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 we consider the bifurcation diagram of positive steady states by using the advection rate as the bifurcation parameter. This approach enables us to capture the change of dynamics from weak advection to strong advection. We will determine three different types of bifurcation diagrams, depending on the difference of diffusion rates. Some exact multiplicity results about bifurcation points will also be presented. Our results can unify some previous work and, as a case study about the role of advection, also contribute to our understanding of intermediate (relative to diffusion) advection in reaction-diffusion models.- Lee Altenberg, Resolvent positive linear operators exhibit the reduction phenomenon, Proc. Natl. Acad. Sci. USA 109 (2012), no. 10, 3705–3710. MR 2903373, DOI 10.1073/pnas.1113833109
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