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Extinction behaviour for mutually enhancing continuous-state population dynamics

Jie Xiong, Xu Yang, Xiaowen Zhou

Abstract

In this paper, we study a two-dimensional process arising as the unique nonnegative solution to a system of two stochastic differential equations (SDEs) with mutually enhancing two-way interactions driven by independent Brownian motions and spectrally positive $α$-stable random measures. Such a SDE system can be identified as a continuous-state Lotka-Volterra type population model. Extinction properties of the populations are studied for different choices of the coefficients involved in the SDEs.

Extinction behaviour for mutually enhancing continuous-state population dynamics

Abstract

In this paper, we study a two-dimensional process arising as the unique nonnegative solution to a system of two stochastic differential equations (SDEs) with mutually enhancing two-way interactions driven by independent Brownian motions and spectrally positive -stable random measures. Such a SDE system can be identified as a continuous-state Lotka-Volterra type population model. Extinction properties of the populations are studied for different choices of the coefficients involved in the SDEs.
Paper Structure (11 sections, 25 theorems, 195 equations)

This paper contains 11 sections, 25 theorems, 195 equations.

Table of Contents

  1. Introduction and main results
  2. Criteria for extinction
  3. Proofs of the main results
  4. Proofs of Proposition \ref{['tp1.2']} and Theorems \ref{['t1.1a']}--\ref{['t1.1']} and \ref{['t1.4']}
  5. Preliminaries
  6. Proofs of Proposition \ref{['tp1.2']} and Theorems \ref{['t1.1a']}--\ref{['t1.1']} and \ref{['t1.4']}
  7. Proofs of Theorems \ref{['t1.2a']}, \ref{['t1.2']} and \ref{['t1.4bb']}
  8. Proof of Theorem \ref{['t1.2a']}
  9. Preliminaries
  10. Equations (\ref{['1.1']}) with dominate drift terms
  11. Equations (\ref{['1.1']}) with dominant diffusion or jump terms

Key Result

Proposition 1.2

If $r_1,r_2\ge0$, then $\mathbf{P}\{\tau_0<\infty\}=0$.

Theorems & Definitions (29)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.10
  • Theorem 1.11
  • Remark 1.12
  • ...and 19 more