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The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach
 
Isabel Averill Bryn Mawr College, PA
King-Yeung Lam Ohio State University, Columbus
Yuan Lou Renmin University, China
The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach
eBook ISBN:  978-1-4704-3611-7
Product Code:  MEMO/245/1161.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach
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The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach
Isabel Averill Bryn Mawr College, PA
King-Yeung Lam Ohio State University, Columbus
Yuan Lou Renmin University, China
eBook ISBN:  978-1-4704-3611-7
Product Code:  MEMO/245/1161.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2452016; 106 pp
    MSC: Primary 35; 92

    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.

  • 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 $
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2452016; 106 pp
MSC: Primary 35; 92

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.

  • 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 $
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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