### Spreading Speeds and Traveling Waves for Nonmonotone Integrodifference Equations

Sze-Bi Hsu, Xiao-Qiang Zhao
2008 SIAM Journal on Mathematical Analysis
The spreading speeds and traveling waves are established for a class of non-monotone discrete-time integrodifference equation models. It is shown that the spreading speed is linearly determinate and coincides with the minimal wave speed of traveling waves. 1. Introduction. The invasion speed is a fundamental characteristic of biological invasions, since it describes the speed at which the geographic range of the population expands, see, e.g., [6, 8, 9, 15] and references therein. Aronson and
more » ... nberger [1, 2] first introduced the concept of the asymptotic speed of spread (in short, spreading speed) for reaction-diffusion equations and showed that it coincides with the minimal wave speed for traveling waves under appropriate assumptions. Weinberger [20] and Lui [13] established the theory of spreading speeds and monostable traveling waves for monotone (order-preserving) operators. This theory has been greatly developed recently in [21, 10, 11, 12] to monotone semiflows so that it can be applied to various discrete-and continuous-time evolution equations admitting the comparison principle. It is known that many discrete-and continuous-time population models with spatial structure are not monotone. For example, scalar discrete-time integrodifference equations with non-monotone growth functions, and predator-prey type reactiondiffusion systems are among such models. The spreading speeds were obtained for some non-monotone continuous-time integral equations and time-delayed reactiondiffusion models in [17, 19] , and a general result on the nonexistence of traveling waves was also given in [19, Theorem 3.5]. The existence of monostable traveling waves were established for several classes of non-monotone time-delayed reactiondiffusion equations in [22, 4, 16, 14] . For certain types of non-monotone discrete-time integrodifference equation models, non-monotone traveling waves and even traveling cycles were observed in [7] by numerical simulations. In [7, 9, 15, 8] , the monotone linear systems, resulting from the linearization of the non-monotone discrete-time models at zero, were used to estimate spreading speeds. It is worthy to find sufficient conditions under which the spreading speed is linearly determinate for these non-monotone systems. The purpose of our current paper is to study the spreading speeds and traveling waves for non-monotone discrete-time systems. As a starting point, we consider scalar integrodifference equations with non-monotone growth functions. The key techniques are to sandwich the given growth function in between two appropriate nondecreasing functions (for spreading speeds) and to construct a closed and convex subset in an