Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case

Abstract

This paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatiotemporal delay, i.e., solutions that are defined in the whole space and for all time t ∈ R. We first derive a particular model for a single species and show how such systems arise from population biology. Then we construct some new types of entire solutions other than traveling wave solutions and equilibrium solutions of the equation under consideration with quasi-monotone and non-quasi-monotone nonlinearities. Various qualitative properties of the entire solutions are also investigated. In particular, the relationship between the entire solutions and the traveling wave fronts which they originated is considered. Our main arguments are based on the comparison principle, the method of super-and sub-solutions, and the construction of auxiliary control systems.

Entire Solutions with Annihilating Fronts to a Nonlocal Dispersal Equation with Bistable Nonlinearity and Spatio-Temporal Delay

“…Applying the theory of abstract functional differential equations [19,Corollary5], we have the following result (see also [34]). …”

“…Lee et al [14] and Murray [22]. Taking this fact into account, the authors of [34] recently introduced a nonlocal dispersal equation for a structured population with spatio-temporal delay. The governing equation is (1.2) where (J * u)(x, t) − u(x, t) means the "nonlocal dispersal operator" and (J * u)(x, t) is a "spatial convolution operator" defined by In biological and epidemiological models, the existence of traveling wave solutions is an important issue due to their significant applications.…”

“…As mentioned in [21], we see that such entire solutions also play important roles in some other areas, such as, transient dynamics and distinct history of two solutions, etc.. For Eq. (1.2), Wu and Ruan [34] recently established the existence and qualitative properties of entire solutions under the monostable assumption of birth functions. However, for the case of bistable birth functions, the study for entire solutions of (1.2) other than traveling wave solutions still remains open.…”

Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence

“…Furthermore, the state of time delay exists universally in the objective material world [12,16]. In addition, the general incidence is more extensive to illustrate the disease spread process than the special standard incidence.…”

Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity

“…In particular, when the nonlinearity is homogeneous (i.e. L = 0 and f 1 = f 2 ), the theories of traveling waves and entire solutions for nonlocal dispersal equation 1 with various types of nonlinearities have been well established, the related results refer to [2,6,7,8,9,15,18,23,24,26,27] and references therein. Specifically, when the kernel J(x) is compactly supported, Sun et al [23] constructed a two-dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions for bistable nonlocal dispersal equation.…”