Existence of traveling waves for integral recursions with nonmonotone growth functions

Abstract

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c * , and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme [10] for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers.

On the existence of traveling waves for delayed reaction–diffusion equations

“…Hsu and Zhao [8] also established the existence of traveling waves for a class of nonmonotone discrete-time integrodifference equation models by the Schauder's fixed point theorem. Other related results for nonmonotone equations can also be found in [4,9,15].…”

Traveling waves for nonlinear cellular neural networks with distributed delays

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Spreading speed and traveling wave solutions in impulsive reaction–diffusion models

“…There is a large body of literature that studies continuous models [1][2][3][4][5] and discrete models [2,6,7]. Recently, researches have found out that some species such as fishes or large mammal populations exhibit a birth pulse growth pattern [8].…”