The Study Of Evolution In Three Species Population Systems With Indirect Effects

The study of evolutionary consequence in predator-prey system is more com-mon. The theoretical study of evolutionary consequence in three species system or more complex ecosystem is seldom. Hence, based on the three species trophic cas-cade system and intraguild predation system, the ecological models are constructed from the perspective of rapid evolution and long-term evolution respectively, the evolutionary consequence of ecosystem and the influence of trait evolution on the structure and function of ecosystem are systemically studied.Firstly, starting with three species trophic cascade system. By applying the quantitative trait evolutionary theory and the adaptive dynamic theory, we formu-late and explore two different types of eco-evolutionary resource-consumer-predator trophic cascade model. First, an eco-evolutionary model incorporating the rapid evolution is formulated to investigate the effect of rapid evolution of the consumer's body size, and to explore the impact of density-mediate indirect effect on the pop-ulation dynamics and trait dynamics. Next, by employing the adaptive dynamiic theory, a long-term evolutionary model of consumer body size is formulated to evaluate the effect of long-term evolution on the population dynamics and the ef-fect of trait-mediate indirect effect. Those models admit rich dynamics that has not been observed yet, in empirical studies. It is found that, the body size of con-sumer in predator-consumer-resource interaction (indirect effect) evolves smaller than that in consumer-resource and predator-consumer interaction (direct effect)and the evolution of consmer body size contributes to avoiding consumer extinc-tions (i.e., evolutionary rescue). The trait-mediate and density-mediate effects may produce opposite evolutionary response.secondly, by integrating the evolutionary dynamics with the ecological stoichio-metric theory, we build an ecological and an evolutionary trophic cascade model.The analysis are divided into two parts, that are the consumer is C-limitation and both C-limitation and P-limitation. Our outcomes expound the theorems of convergence stability?continuous stability and evolutionary branching. Our study reveals that total phosphorus asserts a considerable impact on both the ecologi-cal dynamics and the evolutionary dynamics. The studies show that the higher total phosphorus is always in favor of the coexistence of three species system in the ecological dynamics without considering the evolutionary processes. Increasing the density of total phosphorus may produce complex evolutionary dynamics and trigger the dimorphic populations, or sensitive dependence on the initial conditions.Next, by using the adaptive dynamic theory, and based on the ecological model of intraguild predation system, two evolutionary model are constructed, one is trait evolutionary model of top predator and the other is trait evolutionary model of intermediate predator, where the trade-off function between top predator and intermediate predator are considered. In particular, the critical function analy-sis is applied. This study reveals that the evolutionary dynamics of the intraguild predation system is completely characterized by the concavity of the trade-off func-tion and admits trichotomous dynamic scenarios. Consequently, the evolution may promote the diversity of both the intermediate predator and top predator and the persistence of the system on the evolutionary timescale.At the end of this paper, the remark of main results and further work are presented, many related references are listed.