Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment

Asymptotic propagations of a nonlocal dispersal population model with shifting habitats

“…and denote the maximum point of v(x; μ) by σ (μ), that is, v(σ (μ); μ) = max where C(x) is a continuous and symmetric cut-off function introduced by [37] as follows:…”

“…In addition, some other factors such as Allee effect, seasonal succession and intra-species competition were considered in the framework of shifting environments, see for example [1,6,11,29,45]. For recent studies on the asymptotic propagations and forced waves for the 'shifting environment' problem, one can refer to [4,9,14,20,46] on scalar equations and [3,27,32,33,37,43,44] on competition/cooperative systems. Motivated by [15,42] and as a complement to [23], we continue to study the nonlocal model (1.1) under more general assumptions (J) and (R) in contrast with [23].…”

“…Particularly, for the persistence of species, we find that it won't make any differences if the unfavourable environment is not so hostile, that is r(−∞) = 0 instead of r(−∞) < 0. Meanwhile, by virtue of a proper truncation function introduced by [37], we remove the compacted supporting condition on the kernel J (x) in our previous work [23]. The methods adopted here mainly depend on constructing several kinds of appropriate sub-and super-solutions as well as the comparison principle.…”

Wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse

“…As we all know, due to the heterogeneity of the environment the species living and the large mobility of individuals in an area or even worldwide, which leads to that spatial uniform models are not sufficient to give a realistic picture of disease's transmission, we should and indeed must distinguish the spatial locations in the mathematical models. In recent years, the following convolution operator J * u(x, t) − u(x, t) = ∞ −∞ J(x − y)u(y, t)dy − u(x, t) has been widely introduced to biological models (see, e.g, [16,5,42,20,38,30]) and infectious disease problems (see, e.g, [13,21,22,40,29]). This type of diffusion operator can describe the free and large-range migration of species, one called as nonlocal dispersal, in which the transition probability from one location to another depends on the distance the organisms traveled.…”

Spreading speeds for a class of non-local convolution differential equation

“…(2)-(4) as special cases. More works on the wave propagation dynamics of integral equations and integro-differential equations can be found in [4,6,17,7,8,24,23,12,16,22,18,14,21,25,15,27,13,31,29] and references therein.…”