Stochastic delay Lotka-Volterra model

Arifah Bahar, Xuerong Mao

Research output: Contribution to journalArticle

156 Citations (Scopus)

Abstract

We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model ̇ x(t) = diag(x1 (t),..., xn(t))[b + Ax(t - τ)] into the Itô form dx(t) = diag(x1 (t),..., xn(t) [(b + Ax(t - τ))dt + σ x(t) dw(t)], and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.

LanguageEnglish
Pages364-380
Number of pages17
JournalJournal of Mathematical Analysis and Applications
Volume292
Issue number2
DOIs
Publication statusPublished - 15 Apr 2004

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Lotka-Volterra Model
Delay Equations
Stochastic Equations
Explosion
Explosions
Infinity
Form

Keywords

  • Brownian motion
  • explosion
  • Itô's formula
  • stochastic differential delay equation
  • ultimate boundedness

Cite this

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title = "Stochastic delay Lotka-Volterra model",
abstract = "We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model ̇ x(t) = diag(x1 (t),..., xn(t))[b + Ax(t - τ)] into the It{\^o} form dx(t) = diag(x1 (t),..., xn(t) [(b + Ax(t - τ))dt + σ x(t) dw(t)], and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.",
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Stochastic delay Lotka-Volterra model. / Bahar, Arifah; Mao, Xuerong.

In: Journal of Mathematical Analysis and Applications, Vol. 292, No. 2, 15.04.2004, p. 364-380.

Research output: Contribution to journalArticle

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AB - We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model ̇ x(t) = diag(x1 (t),..., xn(t))[b + Ax(t - τ)] into the Itô form dx(t) = diag(x1 (t),..., xn(t) [(b + Ax(t - τ))dt + σ x(t) dw(t)], and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.

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