Mathematical Models Of Evolutionary Biology And Dynamical Behaviors

In this paper, mathematical models of the coevolution of biological phenotypical traits and dynamical behaviors are investigated. In the first part of this paper, we introduce some general theory of evolutionary dynamics. In the second part of this paper, we consider the coevolution of phenotypical traits in a community comprising the populations of predators and prey with Holling typeâ…¡. We investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. It is shown that the branching in the prey can induce the secondary branching in the predators. Furthermore, it is shown that the evolutionary dynamics admits a stable limit cycle. The evolutionary cycle is a likely outcome of the process, which requires higher evolutionary speed of the prey than that of predator. It is also found that different evolutionary rates and conversion efficiencies can influence the lengths of evolutionary cycles.In the third part of this paper, we consider the coevohition of phenotypical traits in a community comprising the populations of predator and prey subject to Allee effects. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching under symmetrical interactions. It is shown that the branching in the prey can induce the secondary branching in the predator. Secondly, it is shown that the evolutionary dynamics undergoes a supercritical Hopf bifurcation and a subcritical Hopf bifurcation under symmetrical interactions. Evolutionary cycle is a likely outcome of the process, which requires higher evolutionary speed of the prey than that of predator and smaller Allee effects constant. Thirdly, we show that evolutionary suicide is impossible if the interaction coefficients are symmetrical, however, evolutionary suicide can occur for prey populations when they are subject to Allee effects and the interaction coefficients are asymmetrical.In the fourth part of this paper, we consider the coevolution of phenotypical traits in a community comprising two competitive species subject to Allee effects. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching under symmetrical competition. Secondly, we find that evolutionary suicide is impossible when the two-species undergo symmetrical competition, however, evolutionary suicide can occur in an asymmetrical competition model with Allee effects. Thirdly, it is found that evolutionary bistability is a likely outcome of the process under both symmetrical and asymmetrical competition, which depends on the properties of symmetrical and asymmetrical competition. Fourthly, under asymmetrical competition, we find that evolutionary cycle is a likely outcome of the process, which depends on the properties of both intraspecific and interspecific competition: Under asymmetrical competition, we also find that the evolutionary dynamics may admit two limit cycles or bistable equilibria with an unstable limit cycle or a stable equilibrium and a stable limit cycle at the same time.