eBook ISBN:  9781470436117 
Product Code:  MEMO/245/1161.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470436117 
Product Code:  MEMO/245/1161.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 245; 2016; 106 ppMSC: Primary 35; 92
The effects of weak and strong advection on the dynamics of reactiondiffusion 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 reactiondiffusion 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 LyapunovSchmidt procedure

10. Nonconvex 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 $


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The effects of weak and strong advection on the dynamics of reactiondiffusion 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 reactiondiffusion 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 LyapunovSchmidt procedure

10. Nonconvex 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 $