Periodic Solutions And Stabilities Of A Periodic Switching Mosquito Population Suppression Model

Diseases,such as dengue fever,malaria,Zika and epidemic encephalitis B are all transmitted by mosquitoes,which we call mosquito-borne diseases.Nowadays,mosquito-borne diseases are prevalent in more and more areas,and do more and more harm to human health,and its prevention and control has attracted great attention of the World Health Organization.As there is neither safe and available vaccines nor specific effective medicines,controlling mosquito density is an effective way to control mosquito-borne diseases.However,traditional mosquito control methods(such as removing mosquito breeding sites,spraying insecticides,etc.)can eliminate mosquitoes locally in a short period of time,but they have disadvantages such as“environmental pollution”,“poor specificity”and“unsustainable”.New ideas and methods are urgently needed to prevent and control mosquito-borne diseases.With the rapid development of science and technology,two new mosquito-borne disease control techniques,incompatible insect technique(IIT)and sterile insect technique(SIT),emerged at the right moment,and both showed good appli-cation prospects.The main idea of both IIT and SIT is to release sterile mosquitoes(including those caused by the intracellular symbiotic bacteria Wolbachia and those caused by irradiation)reared in the laboratories or mosquito factories into the field and gradually reduce the density of wild mosquitoes by taking advantage of their low fecundity properties.In this paper,under three different release strategies of sterile mosquitoes,the interactive dynamics between wild and sterile mosquitoes are studied,and the relevant conclusions on the exact number of periodic solutions of the model and their corresponding stabilities under various parameter settings are obtained,which is expected to provide a theoretical reference for the formulation of the optimal release strategy of sterile mosquitoes.Specifically,based on the existing models and considering the actual strategy of releasing sterile mosquitoes from a mosquito factory,we use the idea of dimension reduction:taking the number of sterile mosquitoes released into the field as a known and non-negative function,and combined with the fact that the density dependence of mosquitoes mainly occurs in the first three stages of their life cycle,we establish a class of periodic switching models with density-dependent survival probability.For a long time,due to the lack of specific mathematical theories and methods,there are few literatures on the determination of the exact number of periodic solutions of a given model.We adopt the idea of transformation:the number of periodic solutions of the model is transformed into the number of fixed points of the corresponding Poincarémap-ping,firstly,the Poincarémapping related to the model understudied is defined,then it is qualitatively analyzed,and our conclusions are proved by mathematical ideas and methods such as reduction to absurdity and variable separation method.The first chapter takes dengue fever as an example,expounds the general symptoms of mosquito-borne diseases,and introduces the traditional and new prevention and control methods of mosquito-borne diseases.Then,three key pa-rameters that determine the suppression effect of wild mosquitoes are introduced:the release period T(the waiting time between two adjacent releases),the re-lease amount in a batch c,and the duration of mating competitiveness of sterile mosquitoes(?).Obviously,there are three different relations between T and(?):T>(?),T=(?),and T<(?).Under the assumptions of T>(?) and T<(?),this paper discusses the model dynamics under some parameter settings.Finally,by defining a threshold g~*for the release amount and an upper bound G~*related to release period T for the release amount,respectively,we give the periodic switching models in this paper.In Chapter 2,we discuss the dynamics of the model under the release strategy of T>(?) and c>g~*.We find a threshold T~*for the release period and the other threshold c~*with c~*>g~*for the release amount,and draw the following conclu-sions:the model has a unique periodic solution,and it is globally asymptotically stable if T>T~*or T=T~*and g~*<c<c~*;the model has no periodic solutions and the origin is globally asymptotically stable provided T?T~*and c?c~*;if g~*<c<c~*,then the origin is locally asymptotically stable if and only if T<T~*.Chapter 3 considers the dynamics of the model under the release strategy of T>(?) and c?g~*.We find that our model has at least one and at most two periodic solutions,in this case,the exact number of periodic solutions depends on the stabilities of the origin:if the origin is unstable,then the model has a unique periodic solution,and it is globally asymptotically stable;if the origin is asymptotically stable,then the model has exactly two periodic solutions,and the one with a larger initial value is asymptotically stable,the other is unstable.Chapter 4 studies the dynamics of the model under the release strategy of T<(?) and c?G~*.We find that the origin is always asymptotically stable.In addition,the model has exactly two periodic solutions,the one with a larger initial value is asymptotically stable and the other is unstable.The end is the summary of the paper and the prospect of the future work.