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

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 D0 > 0 such that if trueT̂, the time period of the ODE, satisfies trueT̂MathClass-rel>δ, then the solution to the original system with D ≥ D0 cannot approach the stationary state. Copyright © 2012 John Wiley & Sons, Ltd.

Global-in-time behavior of the solution to a Gierer-Meinhardt system

“…In the following sections, first, we confirm that H(u, v) defined by (17) is a Hamiltonian of an ODE system associated with (7) for (16) and then show that H(u(·, t), v(·, t)) defined by (34) and (17) acts as a Lyapunov function to (9) with (10). This propery implies Theorem 1.2 immediately.…”

“…Preliminaries. The parabolic strong maximum principle to (9), (10), and (14) guarantees u(·, t) > 0 in Ω × (0, +∞), provided that u 0 ≡ 0 on Ω. Hence we shall treat positive solutions to (9) with (10) and to (7), mostly.…”

“…where a, b, c, d > 0 are constants (see [1,10]), that is, the ODE part takes always time-periodic orbits and the PDE solution is absorbed into one of them. We have, actually, common mathematical structures between these two models.…”

IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics

“…A partitioned Runge-Kutta scheme is employed to solve the above model: the dynamics of y 1 is approximated by a diagonally implicit method, whereas the evolution of y 2 and y 3 variables is described by a partitioned symplectic Runge-Kutta method defined as (5) This choice is motivated by specific features that characterize the problem (see [19]). In particular it is known that, when functions g u and g v are defined according to the Lotka-Volterra model, then the problem becomes spatially homogeneous as t → ∞ and it is controlled by the classical Lotka-Volterra ODE…”

Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions

“…2 . In the case N = 3, under such a δ-closeness assumption, (32) then implies directly u i ∈ L 2,∞ and we can apply Lemma 3.6 in order to establish global classical solutions in 3D, see also [16]. In order to show classical solutions in higher space dimensions N > 3, we can proceed similar by considering (32) for suitable exponents p > 2.…”