Transient and asymptotic dynamics of a prey-predator system with diffusion

Evangelos Latos, Takashi Suzuki, Yoshio Yamada

    Research output: Contribution to journalArticle

    11 Citations (Scopus)

    Abstract

    In this paper, we study a prey-predator system associated with the classical Lotka-Volterra nonlinearity. We show that the dynamics of the system are controlled by the ODE part. First, we show that the solution becomes spatially homogeneous and is subject to the ODE part as t → ∞. Next, we take the shadow system to approximate the original system as D → ∞. The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as D → ∞. Although the asymptotic dynamics of the original system are also controlled by the ODE, the time periods of these ODE solutions may be different. Concerning this property, we have that any δ > 0 admits D 0 > 0 such that if TÌ, the time period of the ODE, satisfies TÌ>δ, then the solution to the original system with D ≥ D 0 cannot approach the stationary state.

    Original languageEnglish
    Pages (from-to)1101-1109
    Number of pages9
    JournalMathematical Methods in the Applied Sciences
    Volume35
    Issue number9
    DOIs
    Publication statusPublished - 2012 Jun

    Keywords

    • Hamiltonian system
    • Lotka-Volterra dynamics
    • Lyapunov functional
    • shadow system

    ASJC Scopus subject areas

    • Mathematics(all)
    • Engineering(all)

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