Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats

Abstract

MSC: 45C05 45G10 45M20 47G10 92D25 Keywords: Monostable equation Nonlocal dispersal Random dispersal Periodic habitat Spreading speed Spreading speed interval Traveling wave solution Principal eigenvalue Principal eigenfunction Variational principleThe current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established.

Traveling Wave Solutions in Nonlocal Dispersal Models With Nonlocal Delays

“…In literature, the corresponding propagation theory of nonlocal dispersal models (1.1) and (1.2) has been widely S. PAN studied, and most results are indexed by the traveling wave solutions and asymptotic speed of spreading. We refer to Bates et al [2], Carr and Chmaj [3], Chen [4], Coville and Dupaigne [5,6], Jin and Zhao [9], Li et al [10], Pan [14], Pan et al [15,16], Shen and Zhang [17], Sun et al [20], Wu and Liu [21], Xu and Weng [23], Zhang et al [26], Zhang et al [27] and the references therein.…”

Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal

“…Regarding the existence of principal eigenvalue of O in (2.7), the following proposition is proved in (Shen & Zhang, 2010;Shen & Zhang, 2012). Letâ be the average of a(·), that is,â…”

Spreading speeds of a partially degenerate reaction–diffusion system in a periodic habitat

“…Zhao's research was supported in part by the NSERC of Canada and the URP research fund of Memorial University. and empirically; see [1][2][3][4][5]7,[10][11][12][13][15][16][17]19,24,26,[31][32][33], and references therein. Homogeneity and heterogeneity are concepts referring to qualitatively uniform and varying habitats for populations, respectively.…”

“…Weinberger [31] studied the spreading speed and traveling waves for a recursion with a periodic order-preserving compact operator, which can be regarded as a general model of time evolution in population genetics or population ecology in a periodic habitat (see also [32]). Recently, Shen and Zhang [26] investigated the spreading speed for a nonlocal dispersal equation in a periodic habitat. Differently from the construction of spreading speeds in [30,31,19], they developed a new approach to spreading speeds (see also [15]) to overcome the difficulty induced by the non-compactness of solution operators.…”

“…Differently from the construction of spreading speeds in [30,31,19], they developed a new approach to spreading speeds (see also [15]) to overcome the difficulty induced by the non-compactness of solution operators. We will study the global dynamics and spreading speeds of system (1.3) by combining the theory of monotone dynamical systems and the ideas in [26]. To characterize the spreading speed for system (1.3), we need to use the principal eigenvalues of a class of degenerate elliptic eigenvalue problems subject to the periodic boundary condition.…”