Mathematical Madeling Of The Predator-prey Systems With Size Refuge And Evolutionary Dynamics

Predation is a major cause of early-stage mortality for prey individuals,which are often forced to use refuges in order to reduce the risk of being consumed.In evolutionary dynamics of population,effects of size refuge specificity on the system is one of interesting research fields.The paper mainly studies the ability of the mutant to invade the resident population depends strongly on the trade-off of different sizes of prey individuals leading to different successful predation,which is characterized by a specificity parameter.First of all,our paper introduces the biological significance and the theoretical knowledge of ordinary differential equations.On the basis of the predator-prey dynamics together with population genetic phenotypes,we thereby introduce the theoretical knowledge of evolutionary dynamics.Secondly,the ability of certain genotypes of a prey population inducing different sizes of prey individuals leads to different successful predation.We then investigate how this specificity behavior affects the evolution of a given population by using a modified population model,in which the proportion of each genotype available for predation consists of two components: an intrinsic part and a combination from all genotypes present in the population.The trade-off of these components is characterized by a specificity parameter.Based on the sufficient conditions for mutant invasion by using the theory of ordinary differential equations,We obtain the results of only mutants evolve without predators,only mutants evolve with predators,residents and mutants both evolve and are stable without predator,residents and mutants are stable with predators.Thirdly,considering the size refuge that occurs by the single phenotypic trait,we assume that the carrying capacity,predation rate and size refuge which of them depend directly on the phenotypic trait.Thus,we propose another predator-prey model to investigate the effect of different phenotypic traits on population evolutionary dynamics.Furthermore,we can find,from an evolutionary point of view,that the evolutionary singular traits that determine different successful predation undergo evolutionary unstable strategy,continuously stable strategy,evolutionary branching point,and some conditions for possibility of evolutionary suicide,which characterized by a specificity parameter.Finally,we summarize the paper according to the research contents.