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Approximation of the generalized principal eigenvalue of cooperative nonlocal dispersal systems and applications

Mingxin Wang, Lei Zhang

Abstract

It is well known that, in the study of the dynamical properties of nonlinear evolution system with nonlocal dispersals, the principal eigenvalue of linearized system play an important role. However, due to lack of compactness, in order to obtain the existence of principal eigenvalue, certain additional conditions must be attached to the coefficients. In this paper, we approximate the generalized principal eigenvalue of nonlocal dispersal cooperative and irreducible system, which admits the Collatz-Wielandt characterization, by constructing the monotonic upper and lower control systems with principal eigenvalues; and show that the generalized principal eigenvalue plays the same role as the usual principal eigenvalue.

Approximation of the generalized principal eigenvalue of cooperative nonlocal dispersal systems and applications

Abstract

It is well known that, in the study of the dynamical properties of nonlinear evolution system with nonlocal dispersals, the principal eigenvalue of linearized system play an important role. However, due to lack of compactness, in order to obtain the existence of principal eigenvalue, certain additional conditions must be attached to the coefficients. In this paper, we approximate the generalized principal eigenvalue of nonlocal dispersal cooperative and irreducible system, which admits the Collatz-Wielandt characterization, by constructing the monotonic upper and lower control systems with principal eigenvalues; and show that the generalized principal eigenvalue plays the same role as the usual principal eigenvalue.

Paper Structure

This paper contains 13 sections, 21 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thA']}
  3. Threshold dynamics for cooperative systems
  4. Some properties of nonnegative solutions of \ref{['1.8']}
  5. The global dynamics of \ref{['1.1']}---Proof of Theorem \ref{['thB']}
  6. A West Nile virus model
  7. The existence and uniqueness of positive solutions of \ref{['4.5']}
  8. Dynamical properties of \ref{['4.2']}
  9. Discussions
  10. General discussion
  11. Partially degenerate case
  12. More applications
  13. Appendix

Key Result

Theorem A

(Approximation and characterization of the generalized principal eigenvalue) Assume that the condition (B) holds. Then there exist $\underline B^\varepsilon(x)=(\underline b^\varepsilon_{ik}(x))_{n\times n}$ and $\overline B^\varepsilon(x)=(\bar{b}^\varepsilon_{ik}(x))_{n\times n}$, with $\varepsil such that

Theorems & Definitions (39)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['thA']}
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 29 more