Solution of the Crow-Kimura model with changing population size and Allee effect

Abstract

The Crow-Kimura model is commonly used in the modeling of genetic evolution in the presence of mutations and associated selection pressures. We consider a modified version of the Crow-Kimura model, in which population sizes are not fixed and Allee saturation effects are present. We demonstrate the evolutionary dynamics in this system through an analytical approach, examining both symmetric and single-peak fitness landscape cases. Especially interesting are the dynamics of the populations near extinction. A special version of the model with saturation and degradation on the single-peak fitness landscape is investigated as a candidate of the Allee effect in evolution, revealing reduction tendencies of excessively large populations, and extinction tendencies for small populations. The analytical solutions for these dynamics are presented with accuracy O(1/N), where N is the number of nucleotides in the genome.

Effect of harvest on MTE calculated by single step process for stochastic population model under Allee effect

“…In recent years, there has been much interest in investigating the different characteristics of the Allee effect in both ecology, evolutionary biology and genetics [1,2,3,4,5,6,7,8]. Allee effect can be defined as positive correlation between population density and mean individual fitness [9].…”

Effects of resource competition on evolution and adaptive radiation

“…number of trophic levels, amount of links per species and fractions of basal/intermediate/top species. The investigation of evolution is accessible by some analytical models 9,22 , numerical individual-based models (IBM) [23][24][25][26] , the traits in the population can be followed with the replicator equations 27,28 . Among other evolutionary phenomena the sexual selection 26,29,30 and the problem of cooperation 27,28,31 are studied.…”

“…Most number of current researches tend to study either ecological aspects (population dynamics and stability of ecosystems) or evolutionary aspects. Meanwhile, the very interesting is also an investigation of an interplay of population dynamics and evolution and their mutual effects 22,27,28,40,41 . The very comprehensive studies of the food-webs with evolving species were provided using the Webworld model 20,21 .…”