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Role of advection in a two-species competition model : a bifurcation approach / Isabel Averill, King-Yeung Lam and Yuan Lou.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society ; v 245, no 1161.Publication details: Providence : American Mathematical Society, 2017.Description: v, 106 pages : illustrations ; 26 cmISBN:
  • 9781470422028 (alk. paper)
Subject(s): DDC classification:
  • 510 23 Am512
Contents:
Chapter 1. Introduction: The role of advection; Chapter 2. Summary of main results; 2.1. Existence of positive steady states of (2.1); 2.2. Local stability of semi-trivial steady states; 2.3. Global bifurcation results; Chapter 3. Preliminaries; 3.1. Abstract Theory of Monotone Dynamical Systems; 3.2. Asymptotic behavior of and as →0; Chapter 4. Coexistence and classification of -- plane; 4.1. Coexistence: Proof of Theorem 2.2; 4.2. Classification of -- plane: Proof of Theorem 2.5; 4.3. Limiting behavior of ; Chapter 5. Results in ℛ₁: Proof of Theorem 2.10 5.1. The case when ( , )∈ℛ₁ and ( )/( ) is sufficiently large5.2. The one-dimensional case; 5.3. Open problems; Chapter 6. Results in ℛ₂: Proof of Theorem 2.11; 6.1. Proof of Theorem 2.11(b); 6.2. Open problems; Chapter 7. Results in ℛ₃: Proof of Theorem 2.12; 7.1. Stability result of ( ,0) for small ; 7.2. Stability result of (0, ); 7.3. Open problems; Chapter 8. Summary of asymptotic behaviors of _{*} and *; 8.1. Asymptotic behavior of *; 8.2. Asymptotic behavior of _{*}; Chapter 9. Structure of positive steady states via Lyapunov-Schmidt procedure Chapter 10. Non-convex domainsChapter 11. Global bifurcation results; 11.1. General bifurcation theorems; 11.2. Bifurcation result in ℛ₁; 11.3. Bifurcation result in ℛ₃; 11.4. Bifurcation result in ℛ₂; 11.5. Uniqueness result for large , ; Chapter 12. Discussion and future directions; Appendix A. Asymptotic behavior of and ᵤ; A.1. Asymptotic behavior of when →∞; A.2. Asymptotic behavior of and its derivatives as →0; A.3. Asymptotic behavior of ᵥ as , →∞; Appendix B. Limit eigenvalue problems as , →0; Appendix C. Limiting eigenvalue problem as →∞; Acknowledgements
Summary: 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.
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"Volume 245, number 1161 (sixth of 6 numbers), January 2017."

Includes bibliographical references.

Chapter 1. Introduction: The role of advection; Chapter 2. Summary of main results; 2.1. Existence of positive steady states of (2.1); 2.2. Local stability of semi-trivial steady states; 2.3. Global bifurcation results; Chapter 3. Preliminaries; 3.1. Abstract Theory of Monotone Dynamical Systems; 3.2. Asymptotic behavior of and as →0; Chapter 4. Coexistence and classification of --
plane; 4.1. Coexistence: Proof of Theorem 2.2; 4.2. Classification of --
plane: Proof of Theorem 2.5; 4.3. Limiting behavior of ; Chapter 5. Results in ℛ₁: Proof of Theorem 2.10 5.1. The case when ( , )∈ℛ₁ and ( )/( ) is sufficiently large5.2. The one-dimensional case; 5.3. Open problems; Chapter 6. Results in ℛ₂: Proof of Theorem 2.11; 6.1. Proof of Theorem 2.11(b); 6.2. Open problems; Chapter 7. Results in ℛ₃: Proof of Theorem 2.12; 7.1. Stability result of ( ,0) for small ; 7.2. Stability result of (0, ); 7.3. Open problems; Chapter 8. Summary of asymptotic behaviors of _{*} and *; 8.1. Asymptotic behavior of *; 8.2. Asymptotic behavior of _{*}; Chapter 9. Structure of positive steady states via Lyapunov-Schmidt procedure Chapter 10. Non-convex domainsChapter 11. Global bifurcation results; 11.1. General bifurcation theorems; 11.2. Bifurcation result in ℛ₁; 11.3. Bifurcation result in ℛ₃; 11.4. Bifurcation result in ℛ₂; 11.5. Uniqueness result for large , ; Chapter 12. Discussion and future directions; Appendix A. Asymptotic behavior of and ᵤ; A.1. Asymptotic behavior of when →∞; A.2. Asymptotic behavior of and its derivatives as →0; A.3. Asymptotic behavior of ᵥ as , →∞; Appendix B. Limit eigenvalue problems as , →0; Appendix C. Limiting eigenvalue problem as →∞; Acknowledgements

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.

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