Study On The Population Control And Evolution Using Sex-and Stage-Structured Mathematical Models

The relationship between humans and other populations are both interrelated and mutually restricted.For better survival and development,humans inevitably take some control measures to other populations.Conversely,what evolutionary strategy will be adopted by these populations for survival under the control of humans? This paper spreads out academic study from two aspects at the same time.On one hand,taking the mosquito population control through the sterile insect technique（SIT）as an example,we establish sexstructured models to investigate the optimal mosquito control strategy and sex ratio evolutionary strategy.On the other hand,taking the harvest-induced evolution of fish and deer as another example,we establish a general stage-structured model and use the adaptive dynamics theory to investigate the effects of harvesting adults on the reproduction age of the populations.From these two aspects,we study the following four specific contents:（Ⅰ）The dynamics of mosquito population models with sex structure.We establish new sex-structured models to study the dynamics of the interactions between wild mosquitoes and sterile males.We consider two different strategies of releasing sterile males,that is,the constant release strategy and the proportional release strategy.By modeling analysis,we give the sufficient conditions for the existence and stability of all equilibria and obtain the optimal mosquito control strategy.We find that the mosquito population goes extinct as soon as the amount of release of the sterile males exceeds a certain threshold.Finally,we illustrate the dynamic characteristics of these models through numerical simulations.（Ⅱ）The evolution of sex ratio strategies in wild mosquitoes with the release of sterile male mosquitoes.We study the population dynamics with the constant and the proportional release strategies of sterile mosquitoes separately.We obtain the conditions of existence and stability of the equilibria by qualitative analysis of the two-dimensional sex-structured model.Moreover,we investigate the evolution of sex ratio strategies for wild mosquitoes by analyzing the key threshold.Our results show that the wild mosquitoes will adopt a femalebiased sex ratio strategy with the constant release of sterile males,or a male-biased sex ratio strategy with the proportional release of sterile males.Our study demonstrates that wild mosquitoes will change their sex ratio to adapt to the control measures of humans.（Ⅲ）The evolution of maturation strategies of population with stage structure.We develop a stagestructured population model with two maturation strategies and obtain some sufficient conditions of the coexistence or competition for the same population with two different maturation strategies.We also find the eventual evolutionarily stable strategy of the population.Our results show that the optimal maturation strategy is the evolutionarily stable strategy,i.e.the population will eventually evolve to the optimal maturation strategy when there are two different maturation strategies.（Ⅳ）The effect of harvesting adults on the reproduction age of population.We establish a general stagestructured model to derive an adaptive dynamical system for the evolution of reproduction age.We consider the trade-offs between maturity rate and fecundity,juvenile mortality and adult mortality,respectively.Our results show that the reproduction age of the population is not influenced by the harvest under the tradeoff between maturity rate and juvenile mortality.Under the other two tradeoffs,the reproduction age is delayed if and only if there is strong adult protection for juveniles.Otherwise,harvesting adults always leads to the advance of reproduction age.This study suggests that the protection for juveniles provided by adults plays an important role in population evolutionary strategies.