#### For citation:

Frisman E. Y., Kulakov M. P. On the genetic divergence of two adjacent populations living in a homogeneous habitat. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2021, vol. 29, iss. 5, pp. 706-726. DOI: 10.18500/0869-6632-2021-29-5-706-726

# On the genetic divergence of two adjacent populations living in a homogeneous habitat

The purpose is to study the mechanisms leading to the genetic divergence, i.e. stable genetic differences between two adjacent populations coupled by migration of individuals. We considered the case when the fitness of individuals is strictly determined genetically by a single diallelic locus with alleles A and a, the population is panmictic and Mendel's laws of inheritance hold. The dynamic model contains three phase variables: concentration of allele A in each population and fraction (weight) of the first population in the total population size. We assume that the numbers of coupled populations change independently or strictly synchronously. In the first case, the growth rates are determined by fitness of homo- and heterozygotes, the mean fitness of the each population and the initial concentrations of alleles. In the second case, the growth rates are the same. Methods. To study the model, we used the qualitative theory of differential equations studies, including the construction of parametric and phase portraits, basins of attraction and bifurcation diagrams. We studied local bifurcations that provide the fundamental possibility of genetic divergence. Results. If heterozygote fitness is higher than homozygotes, then both populations are polymorphic with the same concentration of homologous alleles. If the heterozygotes fitness is reduced, then over time the populations will have the same monomorphism in one allele, regardless of the type of population changes. In this case, the dynamics is bistable. We showed that the divergence in the model is a result of subcritical pitchfork bifurcation of an unstable polymorphic state. As a result, the genetic divergent state is unstable and exists as part of the transient process to one of monomorphic state. Conclusion. Divergence is stable only for populations that maintain a population ratio in a certain way. In this case, it is preceded by a saddle-node bifurcation and dynamics is quad-stable, i.e. depending on the initial conditions, two types of stable monomorphism and divergence are possible simultaneously.

- Lyubich YI. Basic concepts and theorems of the evolutionary genetics of free populations. Russian Math. Surveys. 1971;26(5):51–123. DOI: 10.1070/RM1971v026n05ABEH003829.
- Bazykin AD. Disadvantages of heterozygotes in a system of two adjacent populations. Sov. Genet. 1974;8(11):1453–1457.
- Bazykin AD. Selection and genetic divergence in local systems populations and populations with a continuous areal (mathematical model). Evolution Problems. 1973;3:231–241 (in Russian).
- Frisman EY, Shapiro AP. Selected Mathematical Models of Divergent Evolution of Populations. Moscow: Nauka; 1977. 152 p. (in Russian).
- Frisman EY. Primary Genetic Divergence (Theoretical Analysis and Modeling). Vladivostok: Far East Scientific Center of the Academy of Sciences of the USSR; 1986. 160 p. (in Russian).
- Arnold VI, Afraimovich VS, Ilyashenko YS, Shilnikov LP. Bifurcation Theory (Dynamical Systems - 5). In: Results of Science and Technology. Series Modern Problems of Mathematics. Fundamental Directions. Vol. 5. Moscow: VINITI; 1986. P. 5–218 (in Russian).
- Shilnikov LP, Shilnikov AL, Turaev DV, Chua LO. Methods of Qualitative Theory in Nonlinear Dynamics. Part I. Singapore: World Scientific; 1998. 416 p. DOI: 10.1142/3707.
- Dhooge A, Govaerts W, Kuznetsov YA, Meijer HGE, Sautois B. New features of the software MatCont for bifurcation analysis of dynamical systems. Mathematical and Computer Modelling of Dynamical Systems. 2008;14(2):147–175. DOI: 10.1080/13873950701742754.
- Stewart I, Elmhirst T, Cohen J. Symmetry-Breaking as an Origin of Species. In: Buescu J, Castro SBSD, da Silva Dias AP, Labouriau IS, editors. Bifurcation, Symmetry and Patterns. Trends in Mathematics. Basel: Birkhauser; 2003. P. 3–54. DOI: 10.1007/978-3-0348-7982-8_1.
- Burger R. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B. 2014;19(4):883–959. DOI: 10.3934/dcdsb.2014.19.883.
- Yamamichi M, Ellner SP. Antagonistic coevolution between quantitative and Mendelian traits. Proc. R. Soc. B. 2016;283(1827):20152926. DOI: 10.1098/rspb.2015.2926.
- Telschow A, Hammerstein P, Werren JH. The effect of Wolbachia on genetic divergence between populations: Models with two-way migration. The American Naturalist. 2002;160(S4):S54–S66. DOI: 10.1086/342153.
- Zhdanova OL, Frisman EY. Dynamic regimes in a model of single-locus density-dependent selection. Russ. J. Genet. 2005;41(11):1302–1310. DOI: 10.1007/s11177-005-0233-3.
- Altrock PM, Traulsen A, Reeves RG, Reed FA. Using underdominance to bi-stably transform local populations. J. Theor. Biol. 2010;267(1):62–75. DOI: 10.1016/j.jtbi.2010.08.004.
- Yeaman S, Otto SP. Establishment and maintenance of adaptive genetic divergence under migration, selection, and drift. Evolution. 2011;65(7):2123–2129. DOI: 10.1111/j.1558-5646.2011.01277.x.
- Laruson AJ, Reed FA. Stability of underdominant genetic polymorphisms in population networks. J. Theor. Biol. 2016;390:156–163. DOI: 10.1016/j.jtbi.2015.11.023.
- Yamamichi M, Hoso M. Roles of maternal effects in maintaining genetic variation: Maternal storage effect. Evolution. 2016;71(2):449–457. DOI: 10.1111/evo.13118.
- Wakeley J. The effects of subdivision on the genetic divergence of populations and species. Evolution. 2000;54(4):1092–1101. DOI: 10.1111/j.0014-3820.2000.tb00545.x.
- Frisman EY, Zhdanova OL, Kulakov MP, Neverova GP, Revutskaya OL. Mathematical modeling of population dynamics based on recurrent equations: Results and prospects. Part II. Biol. Bull. Russ. Acad. Sci. 2021;48(3):227–240. DOI: 10.1134/S1062359021030055.
- Neverova GP, Zhdanova OL, Frisman EY. Effects of natural selection by fertility on the evolution of the dynamic modes of population number: bistability and multistability. Nonlinear Dyn. 2020;101(1):687–709. DOI: 10.1007/s11071-020-05745-w.
- Shea K, Metaxas A, Young CR, Fisher CR. Processes and Interactions in Macrofaunal Assemblages at Hydrothermal Vents: A Modeling Perspective. In: Lowell RP, Seewald JS, Metaxas A, Perfit MR, editors. Magma to Microbe: Modeling Hydrothermal Processes at Ocean Spreading Centers. Vol. 178 of Geophysical Monograph Series. Washington, DC: Blackwell Publishing Ltd; 2008. P. 259–274. DOI: 10.1029/178GM13.
- Sundqvist L, Keenan K, Zackrisson M, Prodohl P, Kleinhans D. Directional genetic differentiation and relative migration. Ecol. Evol. 2016;6(11):3461–3475. DOI: 10.1002/ece3.2096.

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