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This paper is concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIR model with nonlocal delayed transmission. The threshold dynamics are determined by the so-called basic reproduction number of the corresponding reaction system and the critical wave speed. Our results imply that (i) the diffusion ability of the infected individuals can accelerate the wave speed; and (ii) the incubation period can slow down the propagation while the non-locality of interaction would speed up the spread of the disease. In particular, we remove the usual condition that the nonlocal dispersal kernel function is compacted supported. [ABSTRACT FROM AUTHOR]

• The novelty of this work lies in the construction of super-sub solutions and the proof of the complete continuity of operator. • Most importantly and differently, the authors obtain the existence of wave front with critical speed, by verifying the boundedness of component R. • The result provides a method for some non-monotone systems. It is well-known that the nonlocal dispersal operator has the advantage of capturing short-range as well as long-range factors for the dispersal of the spices by choosing the kernel function properly, and is also capable to include spatial dispersal strategies of the species beyond random (local) diffusion. This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with non-monotone incidence, which is a non-monotone system. The method of sub and super solutions combined with Schauder's fixed-point theorem is applied to establish the existence of positive traveling waves as the wave speed is over critical speed. We further prove the existence of traveling waves with critical speed and the nonexistence of bounded positive traveling waves by the delicate analysis method. The main difficulty is to get the boundedness of traveling waves caused by the nonlocal dispersal operator. [ABSTRACT FROM AUTHOR]

• The novelty of this work lies in the construction of super-sub solutions and the proof of the complete continuity of operator. • The minimal speed of traveling wave solutions of the model are determined. • The result provides an estimation of the invasion speed. This paper is concerned with the existence of traveling wave solutions for a predator-prey model with nonlocal dispersal. By applying Schauder's fixed point theorem and a cross-iteration technique, we reduce the existence of traveling wave solutions to the existence of a pair of super-sub solutions. More precisely, we proved that there exists a positive constant c * such that when c > c *, the nonlocal dispersal predator-prey system admits a traveling wave solution. In particular, the existence of traveling wave solution for c = c * is also established by asymptotic spreading theory and comparison principle. Furthermore, by investigating the non-existence of traveling wave solution, we determine the minimal speed of traveling wave solution for this model. This provides an estimation of the invasion speed. The novelty of this work lies in the construction of super-sub solutions and the proof of the complete continuity of operator. [ABSTRACT FROM AUTHOR]