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Extinction behaviour for competing continuous-state population dynamics

Jie Xiong, Xu Yang, Xiaowen Zhou

Abstract

We consider a system of two stochastic differential equations (SDEs) with competing two-way interactions driven by Brownian motions and spectrally positive $α$-stable random measures. Such a SDE system can be identified as a Lotka-Volterra type population model. We find nearly sharp conditions for one of the population to become extinct or extinguished.

Extinction behaviour for competing continuous-state population dynamics

Abstract

We consider a system of two stochastic differential equations (SDEs) with competing two-way interactions driven by Brownian motions and spectrally positive -stable random measures. Such a SDE system can be identified as a Lotka-Volterra type population model. We find nearly sharp conditions for one of the population to become extinct or extinguished.

Paper Structure

This paper contains 8 sections, 21 theorems, 226 equations.

Table of Contents

  1. Introduction and main result
  2. Criteria for the extinction behaviour
  3. Proofs of Theorems \ref{['t0.01']}--\ref{['t0.03']}
  4. Preliminaries
  5. Proof of Theorem \ref{['t0.01']}
  6. Proof of Theorem \ref{['t0.02']}
  7. Proof of Theorem \ref{['t0.03']}
  8. Appendix

Key Result

Theorem 1.2

$\mathbf{P}\{\tau_0^-<\infty\}=0$ iff $\theta_1,\theta_2\ge1$. Moreover, $\mathbf{P}\{\tau_0^-(X)<\infty\}=0$ for $\theta_1\ge1$ and $\mathbf{P}\{\tau_0^-(Y)<\infty\}=0$ for $\theta_2\ge1$.

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 13 more