A Reaction-diffusion Population Growth Equation With Multiple Pulse Perturbations

Integrated pest management has played a more and more important role in agricultural production.Therefore,how to describe and evaluate the effect of control strategy to the growth of pest population and the loss to agriculture production is one of the core issues.It is inefficient to pest control if we only rely on past experience,so the more precise mathematical model and theoretical and numerical analyse have become one of the most important evaluating methods for integrated pest control.There always exist a number of internal and external perturbations including birth pulses and impulsive pesticide applications,which result in a complex growth pattern within each pest growth generation due to the variation of perturbation timings for the pest population with non-overlap generations.On the other hand,the dispersal and the spatial factors related to the birth rate and the killing rate will also have an important influence on the growth of the pest.Thus,we have extended the classical Fisher reaction-diffusion equation by involving instant birth and control perturbations with aims to analyze the impact of these multi pulse perturbations on the dynamic behavior of the pest population.In the second chapter,we mainly discuss the growth of pest population under a monotonic pulse growth function.The results show that the perturbations will have a significant effect on the existence and stability of a spatially homogeneous periodic solution.In particular,the birth rate and killing efficacy can affect the traveling wave and its spreading speed,while the timing of pesticide application does not make any effect on those threshold conditions.However,extensively numerical investigations reveal that the timing of pesticide applications can significantly affect the distributions of the pest population,i.e.the earlier or later pesticide application could result in a faster growth and wider spread,and the optimal timing for pest control is the middle of each generation if the efficacy of the pesticide is relatively low.In the third chapter,we focus on the growth of pest population under a non-monotonic pulse growth function.Because the difficulty of mathematical analysis under non-monotonic growth function,in this section,we mainly employ numerical bifurcation methods to reveal how the multiple pulse perturbations affect on the dynamical behaviour for our model.The bifurcation analysis indicates that the pest population will produce diverse spatial patterns for a continuously changing pest birth rate or killing rate.Specifically,the birth rate is too small or the killing rate is too large,the pest population will present spatially stable distribution,when the birth rate increases or decreases in killing rate,stable distribution gradually from the symmetric form to irregular complex space form.We also reveal that even generations and odd generations display a different spatial pattern,which depict some important issues related to the pest control and pest management,for example the different control strategies should be adopted for even and odd generations.In the fourth chapter,we focus on population growth patterns under spatial heterogeneous cases.We find that the spatial factors will affect the dynamical behaviour of pest population by the bifurcation analysis for spatial parameters which can characterize the spatial heterogeneity of the birth rate and kill rate.Therefore,we should try to reduce the homogeneity of birth rate in integrated pest management so that the pest population can be concentrated in a region.To address how the spatial heterogeneity affects on the cost of the pest control,we further formulate the cost function to investigate the optimal pest control strategy,and the optimal spatial dependent killing rates which minimize the cost have been obtained numerically.