Travelling waves and paradoxical effects in a discrete-time growth-dispersal model

Integrodifference equations have been widely used to describe the spatiotemporal dynamics of a population. Here, we first extended the discrete Beverton-Holt model, to model the effects of an instantaneous control measure within each generation on a population's growth. Then we investigated the effects of instantaneous killing rate and timing of pesticide application on a pest's dynamics, together with the effects of a spatial factor on the existence of a travelling wave solution, its spreading speed and asymptotic stability, and the occurrence of paradoxical effects. The main results indicate that the instantaneous killing rate significantly influences the existence of a travelling wave and its speed of spread, while the timing of pesticide applications does not. In order to address how such dynamic complexities affect the above results, we included an interruption constant into the model which can generate chaotic dynamics. The results indicate that the pesticide efficacy not only affects the existence of a travelling wave and its speed of spread, but it can also produce paradoxical effects, i.e. low pesticide efficacy can result in an increase, not a decrease, of the stable population size, while high pesticide efficacy could inhibit the pests growth. Furthermore, different patterns of stable population sizes between odd and even generations can be produced. The results reveal that the pest control tactics should be designed according to the spatial characteristics and the pest generation.