Traveling Wave Solutions of a Delayed Cooperative System

Abstract

This paper deals with the dynamics of a delayed cooperative system without quasimonotonicity. Using the contracting rectangles, we obtain a sufficient condition on the stability of the unique positive steady state of the functional differential system. When the spatial domain is whole R , the existence and nonexistence of traveling wave solutions are investigated, during which the asymptotic behavior is investigated by the contracting rectangles.

Asymptotic spreading for a time-periodic predator-prey system

“…By directly calculating, we can prove that these continuous functions (ϕ 1 , ϕ 2 ), (ϕ 1 , ϕ 2 ) satisfy (18)- (19) and are a pair of generalized upper and lower solutions for (17). Due to the classical theory of reaction-diffusion system (see [31,43]), we obtain…”

“…Because of the uniform boundedness of u, v and w i , ϕ i (x, t) is bounded and well defined for all t > 0, x ∈ R, i = 1, 2. To estimate (ϕ 1 , ϕ 2 ), we introduce a pair of generalized upper and lower solutions (ϕ 1 , ϕ 2 ) and (ϕ 1 , ϕ 2 ) for (17), which satisfy…”

“…Then c * is called the spreading speed of u(x, t) if (a) lim the works cited above include many important results with applications. When a scalar equation is not monotone due to time delay or discrete, we may refer to [12,16,17,23,42,44] and references cited therein. In spatial ecology, the spatial propagation dynamics of predator-prey system has attracted considerable attention [6,9,10,25,27].…”

Spreading Speed in A Nonmonotone Equation with Dispersal and Delay

“…The nonmonotonicity may have originated from time delay in reaction-diffusion equations, see their complex dynamics by Wu [20]. When the spatial propagation of nonmonotone models is concerned, much attention has been paid to the study of traveling wave solutions, see several recent papers [21,22] and references cited therein. Besides the traveling wave solutions, the entire solutions of some nonmonotone models were also studied [23][24][25][26].…”

Minimal Wave Speed in a Competitive Integrodifference System without Comparison Principle

“…When the temporal variable is discrete, the deficiency of monotonicity is also universal, e.g., the discrete logistic model may lead to rich dynamics ( [11], Chapter 2). When spatial propagation dynamics are considered, we may refer to some results on the propagation dynamics of non-monotone integrodifference systems by Hsu and Zhao [12], Li et al [13], Lin [14], Pan and Lin [15], Pan and Zhang [16], and very recent papers [17,18] and references cited therein for other non-monotone diffusion systems. In these works, to establish the minimal wave speed, a general recipe is to pass to a limit function from the results of large wave speeds.…”