Monotonicity and uniqueness of traveling waves for a reaction–diffusion model with a quiescent stage

Existence and uniqueness of traveling waves for non-monotone integral equations with application

“…The approach was firstly developed by Ma [12] for a nonlocal delayed reaction-diffusion equation without quasi-monotonicity, and recently employed by Hsu and Zhao [8] to prove the existence of traveling waves for a non-monotone discrete-time integro-difference equation. For other related results, we refer to Wang [16], Li et al [10], Wu et al [17] and Zhang and Li [22].…”

Wave propagation for a reaction–diffusion model with a quiescent stage on a 2D spatial lattice

“…In recent years, several authors investigated single-species population reaction-diffusion models where only part of the population is migrating and another part is stationary; see [5,6,3,[7][8][9] and the references therein. In [5], Hadeler and Lewis presented and discussed briefly the following model with a quiescent stage:…”

“…Such solutions, in many situations, determine the long time behavior of other solutions, and account for phase transitions between different states of physical systems, and domain invasion of species in population biology. In the past few decades, the study on traveling wave solutions of reaction-diffusion systems (both continuous and discrete) has been extensive and intensive, and has led to many interesting and significant results; see, e.g., [10][11][12]2,13,14,3,15,7,[16][17][18][19][20][21][22][23]8,9,24]. In [8], Zhang and Zhao considered the asymptotic behavior of system (1.1).…”

“…More precisely, they established the existence of the spreading speed and showed that it coincides with the minimal wave speed for monotone traveling wave solutions. Zhang and Li [9] further considered the monotonicity and uniqueness of traveling wave solutions.…”

Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

“…1), the problem of traveling waves is quite well understood since the whole interaction term is quasi-monotone and the solution semiflow is monotone. Actually, in the monostable case, the existence, uniqueness and stability of monotone traveling waves have been established by many researchers for far more general reaction-diffusion systems with local or non-local delays, see eg., [3,10,[12][13][14][15][16][21][22][23][24]27,[29][30][31][32][33][34][37][38][39][40], and the recent surveys of Gourley and Wu [9] and Gourley et al [7].…”