Forced waves in a Lotka-Volterra competition-diffusion model with a shifting habitat

Recent developments on spatial propagation for diffusion equations in shifting environments

“…with h being a indicator function over the interval [−L/2, L/2], we can obtain the particular form (20). The authors of [116,117] focused on determining the critical speed for extinction and the role that the dispersal and growth play in persistence by analyzing some examples.…”

“…Regarding the case that two competing species have same habitat preference, that is both growth functions r 1 (•) and r 2 (•) satisfy (H 0 ), there are several work focusing on such a Lotka-Volterra competition system (24) as well as its discrete and nonlocal dispersal versions, see Dong et al [20,22,23], Meng et al [72], Wu et al [102], Yuan et al [111], Yuan and Zou [112], Zhang et al [114] and Zhu et al [118] from different motivations and viewpoints. In the following, we summarize the main results involving the spreading dynamics and forced wave solutions of (24) as well as its discrete and nonlocal dispersal versions.…”

“…and faster diffuser being stronger competitor (d 1 < d 2 , a 1 ≥ 1 > a 2 ), species u 1 is always going to extinction while species u 2 will go extinct provided that c * 2 (∞) < c but will persist provided that c * 2 (∞) > c. (iii) [112] For the case of faster diffuser being weaker competitor (d 1 < d 2 , a 1 < 1 ≤ a 2 ), then two competing species will coexist by spreading to the right provided that 0 < c < c * 1 (∞); and species u 1 dies out while species u 2 will persist if c * 1 (∞) < c < c * 2 (∞); and both competing species will go to extinction if c > c * 2 (∞). Regarding the forced waves of (24) and its discrete and nonlocal dispersal versions, we have [20,22,118] (i) For the weak competition case, for any c > 0, there exists a forced wave solution…”

“…In the past years, two-species competition system with monostable structure [4,20,95,98,102,111,112,114], cooperation system of weak type [52,97,108] and chemotaxis models [88,89] in shifting environments have been widely considered and studied. In future research, one should turn more attention to the strong competition case (which leads to a bistable system) and general cooperation situation.…”

Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal

“…As far as systems involving several species are concerned, the existence and asymptotics of forced wave solutions is studied for a cooperative model in [37]. For a Lotka-Volterra type competition model, the existence of forced waves was shown by Dong et al [12], and the persistence and extinction of species were established in [36,40]. Forced waves and gap formation in a competition model was also studied when the species' favorable habitats shift with opposite directions [2].…”