Abstract
In this paper, we study a time-periodic and delayed reaction-diffusion system with quiescent stage in both unbounded and bounded habitat domains. In unbounded habitat domain R, we first prove the existence of the asymptotic spreading speed and then show that it coincides with the minimal wave speed for monotone periodic traveling waves. In a bounded habitat domain Omega subset of R-N (N >= 1), we obtain the threshold result on the global attractivity of either the zero solution or the unique positive time-periodic solution of the system.
Highlights
In population ecology, dormancy or quiescence plays an important role in the growing process of some species such as reptiles and insects, which is an attractive biological phenomenon
∂ ∂t v(t, x) γu(t, x) βv(t, x), where f (u(t, x), u(t − τ, x)) is the reproduction function, τ is a nonnegative constant. They established the existence of the minimal wave speed and further studied the asymptotic behavior, monotonicity and uniqueness of the traveling wave fronts
The purpose of this paper is to investigate the asymptotic behavior of system (1.4)
Summary
Dormancy or quiescence plays an important role in the growing process of some species such as reptiles and insects, which is an attractive biological phenomenon. Βv(t, x), where f (u(t, x), u(t − τ, x)) is the reproduction function, τ is a nonnegative constant They established the existence of the minimal wave speed and further studied the asymptotic behavior, monotonicity and uniqueness of the traveling wave fronts. In the case where the spatial domain is bounded, a threshold result on the global attractivity of either zero or positive periodic solution was established. Due to the zero diffusion arising from the quiescent stage, the system (1.4) has a weak regularity, which leads to a difficulty in obtaining the existence of traveling waves To overcome this problem, we adopt the ideas involving the minimal wave speeds for monotone and “point-α-contraction” systems with monostable structure developed in [3]. The frameworks, concepts and results presented by this section are adapted from [3, 12, 13]
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