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Transient and asymptotic dynamics of a prey–predator system with diffusion
Author(s) -
Latos Evangelos,
Suzuki Takashi,
Yamada Yoshio
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2524
Subject(s) - ode , mathematics , transient (computer programming) , shadow (psychology) , nonlinear system , dynamical system (definition) , mathematical analysis , dynamical systems theory , computer science , physics , psychology , quantum mechanics , psychotherapist , operating system
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 T ̂ , the time period of the ODE, satisfies T ̂ > δ , then the solution to the original system with D ≥ D 0 cannot approach the stationary state. Copyright © 2012 John Wiley & Sons, Ltd.