Existence and stability of stationary waves of a population model with strong Allee effect

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

“…The numerical solutions of the traveling and stationary solutions strongly suggest that these solutions are locally and globally stable [3,6]. The existence, stability and behavior of the stationary and wave solution of the model (36) have several biological implications which have been discussed in [3]. For instance, it has been shown that the threshold property of single species with Allee effect does not hold when the diffusion is present (i.e., when D m > 0; see figures 3(d) and 4(d) of [3]).…”

“…The traveling wave solutions of these models have been studied by several researchers (see [52] for a review). Also, the existence of stationary wave solutions of models (33) and (34) has been proven in [3,6] and their stability has been numerically examined. Formation of a traveling wavefront may resemble the spatial dispersal of invasive species, whereas a stationary wavefront or wave pulse relates to space-dependent extinction-survival (see Table 3 of [6]).…”

“…The wave solution of model (36) was also approximated using a boundary layer method combined with the asymptotic expansion [2,5]. Moreover, the existence of stationary wave solutions and the convergence of the w(x, t) solution of model (36) to the stationary wave solutions were investigated in [3,6,39]. The numerical solutions of the traveling and stationary solutions strongly suggest that these solutions are locally and globally stable [3,6].…”

“…Moreover, the existence of stationary wave solutions and the convergence of the w(x, t) solution of model (36) to the stationary wave solutions were investigated in [3,6,39]. The numerical solutions of the traveling and stationary solutions strongly suggest that these solutions are locally and globally stable [3,6]. The existence, stability and behavior of the stationary and wave solution of the model (36) have several biological implications which have been discussed in [3].…”

“…The existence, stability and behavior of the stationary and wave solution of the model (36) have several biological implications which have been discussed in [3]. For instance, it has been shown that the threshold property of single species with Allee effect does not hold when the diffusion is present (i.e., when D m > 0; see figures 3(d) and 4(d) of [3]). The rest of this section is focused on the reduced model (30).…”