Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats Jia-Bing Wang and Wan-Tong Li Abstract. In this paper, we study a Lotka–Volterra cooperative system with nonlocal dispersal under worsening habitats. By constructing appropriate vector super-/subsolutions combined with the monotone iteration scheme, we obtain the existence of bounded and positive forced waves connecting zero equilibrium to the coexistence state of the limiting system corresponding to the most favorable resource with the wave speed at which the habitat is worsening. Compared with the existence result of traveling wave to the homogeneous system, we find that the wave phenomenon is more likely to occur in such shifting environment. Here, we remove the common assumption that the dispersal kernels are compactly supported and symmetric. Further, we investigate the tail behavior of the forced waves, especially for the rate of convergence to the extinction state, and derive the long-time dynamics of two species. Our result shows that weak interspecies cooperation and nonlocal diffusion pattern cannot prevent species from disappearing eventually under such a worsening habitat. AMS Subject Classification (2010). 35K57, 35C07, 45K05, 92D25. Keywords. Nonlocal dispersal, Worsening habitats, Forced waves, Extinction.
1. Introduction In recent years, to study the climate change (e.g., global warming [14,31]) and the pathogen spread [6,12], much attention has been focused on the following nonlinear evolution equations ∂t u(t, x) = du(t, x) + f (x − ct, u(t, x)),
(1.1)
where the number c represents that the reaction of the populations will change with the moving habitat of speed c due to some external environment change, and the Laplacian operator u denotes the diffusion originating from Fick’s law of diffusion which can only happen between adjacent spatial locations. Problem (1.1) with various nonlinear terms modeling different shifting habitats has been widely studied for certain classical reaction–diffusion equations in a continuous/discrete medium, or with a free boundary as well as integro-difference equations, one refers to Berestcki et al. [4], Berestcki and Fang [5], Du et al. [11], Hu and Li [16], Hu and Zou [17], Lewis et al. [20], Li et al. [21,22], Potapov and Lewis [29], Vo [32], Zhou and Kot [44] and the references cited therein. One critical issue is whether a given species is capable of moving along with the continuously changing climate or whether the pathogen spread can keep pace with its host invasion. As is well known, there exist some limitations and shortcomings for Laplacian diffusion models to study spatial long-range interaction of species but the nonlocal dispersal represented by an integral (convolution) operator J(y)u(t, x − y)dy − u(t, x) [J ∗ u](t, x) − u(t, x) := R
has obvious advantages in this respect, see Hutson et al. [18] and Murray [28]. In mathematical biology, nonlocal dispersal is usually used to characterize the so-called long-di
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