Matvey Pavlovich Kulakov

This paper explores the conditions for the spatial differentiation of traits (divergence) in a homogeneous area, where gene flow is possible between adjacent populations (local connection) and distant regions (global). It is found that the probability of divergence, which often occurs under disruptive selection, decreases as the radius of local (short-range) coupling increase. Introducing a few long-distance connections, that bring previously isolated subpopulations together, promotes divergence.

In the paper, the authors study the oscillations synchronization in a system of three limited populations coupled by migration. We constructed attraction basins for cycles of lengths 2 and 3. It is found that cycles of period 2 exhibit a partially synchronous regime, while periods of 3 or more are completely asynchronous. We have identified two types of asynchronous 3-cycle systems with different orders of point traversal and synchronous behavior in one population. These results are important for predicting the dynamics of real biological communities.

Based on a local predator-prey community model, it has been investigated the impact of predator harvest on the interacting populations dynamics. The proposed mathematical model takes into account the predator population replenishment and its reproduction, limited by the availability of food. We consider optimizing the predator harvest problem, showing that intense predator harvesting leads to a decrease in its population and, therefore, to a considerable increase in prey population.

   One of the main goals of modern evolutionary biology is to understand the mechanisms that lead to the initial differentiation (primary divergence) of populations into groups with genetic traits. This divergence requires reproductive isolation, which prevents or hinders contact and the exchange of genetic material between populations. This study explores the potential for isolation based not on obvious geographical barriers, population distance, or ecological specialization, but rather on hereditary mechanisms, such as gene drift and flow and selection against heterozygous individuals. To this end, we propose and investigate a dynamic discrete-time model that describes the dynamics of frequencies and numbers in a system of limited populations coupled by migrations. We consider a panmictic population with Mendelian inheritance rules, one-locus selection, and density-dependent factors limiting population growth. Individuals freely mate and randomly move around a one-dimensional ring-shaped habitat. The model was verified using data from an experiment on the box population system of Drosophila melanogaster performed by Yu.P. Altukhov et al. With rather simple assumptions, the model explains some mechanisms for the emergence and preservation of significant genetic differences between subpopulations (primary genetic divergence), accompanied by heterogeneity in allele frequencies and abundances within a homogeneous area. In this scenario, several large groups of genetically homogeneous subpopulations form and independently develop. Hybridization occurs at contact sites, and polymorphism is maintained through migration from genetically homogeneous nearby sites. It was found that only disruptive selection, directed against heterozygous individuals, can sustainably maintain such a spatial distribution. Under directional selection, divergence may occur for a short time as part of the transitional evolutionary process towards the best-adapted genotype. Because of the reduced adaptability of heterozygous (hybrid) individuals and low growth rates in these sites (hybrid zones), gene flow between adjacent sites with opposite genotypes (phenotypes) is significantly impeded. As a result, the hybrid zones can become effective geographical barriers that prevent the genetic flow between coupled subpopulations.

Based on a mathematical model of the predator-prey type interacting populations, we have investigated consumption of natural resources in a system with slow recovery of harvested resource. The proposed model incorporates the saturation effect in the predator population, whose reproduction is limited by the volume of available food and the way of its consumption. The focus of the study is on the problem of optimizing predator and prey production. It is shown that there is an optimal withdrawal rate that ensures maximum income with the sustainable development of the community.

The article is devoted to a review of bistability and quadro-stability phenomena found in a certain class of mathematical models of population numbers and allele frequency dynamics. The purpose is to generalize the results of studying the transition from bi- to quadro-stability in such models. This transition explains the causes and mechanisms for the appearance and maintenance of significant differences in numbers and allele frequencies (genetic divergence) in neighboring sites within a homogeneous habitat or between adjacent generations. Using qualitative methods of differential equations and numerical analysis, we consider bifurcations that lead to bi- and quadro-stability in models of the following biological objects: a system of two coupled populations subject to natural selection; a system of two connected limited populations described by the Bazykin or Ricker model; a population with two age stages and density-dependent regulation. The bistability in these models is caused by the nonlinear growth of a local homogeneous population or the phase bistability of the 2-cycle in populations structured by space or age. We show that there is a series of similar bifurcations of equilibrium states or fixed or periodic points that precede quadro-stability (pitchfork, period-doubling, or saddle-node bifurcation).

The inhomogeneous population distribution appears as various population densities or different types of dynamics in distant sites of the extended habitat and may arise due to, for example, the resettlement features, the internal population structure, and the population dynamics synchronization mechanisms between adjacent subpopulations. In this paper, we propose the model of the spatio-temporal dynamics of two-age-structured populations coupled by migration (metapopulation) with long-range displacement. We study mechanisms leading to inhomogeneous spatial distribution as a type of cluster synchronization of population dynamics. To study the spatial patterns and synchronization, we use the method of constructing spatio-temporal profiles and spatial return maps. We found that patterns with spots or stripes are typical spatial structures with synchronous dynamics. In most cases, the spatio-temporal dynamics are mixed with randomly located single populations with strong burst (outbreak) of population size (solitary states). As the coupling parameters decrease, the number of solitary states grows, and they increasingly synchronize and form the clusters of solitary states. As a result, there are the several clusters with different dynamics. The appearance of these spatial patterns most likely occurs due to the multistability of the local age-structured population, leading to the spatio-temporal multistability.

В работе изучаются механизмы, приводящие к возникновению генетической дивергенции (устойчивых генетических различий) между двумя миграционно связанными популяциями. Рассматривается «классическая» система панмиктичных популяций с менделевскими правилами наследования и монолокусным отбором, направленным против гетерозигот. Для лимитирования роста численности предложено считать, что гаметопродукция и общая фертильность (рождаемость) падает с ростом численности за счет ограниченности ресурсов. Предложена дискретная по времени нелинейная математическая модель, описывающая динамику концентрации одного из аллелей и общую численность в каждой из популяций. Описана процедура вычисления координат всех неподвижных точек, соответствующих разным типам предельной генетической структуры и соотношению численностей. Показано, что при плотностно-зависимом лимитировании рождаемости в модели возможно множество неподвижных точек, соответствующих однородному и неоднородному распределению. При однородном распределении обе популяции оказываются мономорфными, т.е. представлены особями лишь с одним генотипом. Предельные численности при этом не всегда совпадают. Неоднородность проявляется в том, что смежные популяции оказываются полиморфными, т.е. содержат особей с разными генотипами, но существенно отличаются частотами альтернативных аллелей и предельными численностями. Описаны бифуркации рождения неподвижных точек, соответствующих неоднородному распределению и генетической дивергенции. Обнаружено, что движение к одной из возможных предельных генетических структур сопровождается изменением репродуктивных возможностей популяций. Показано, что при пониженной приспособленности гетерозигот мономорфные популяции имеют более высокую рождаемость, чем полиморфные, в которых, очевидно, присутствуют особи с разными репродуктивными возможностями. В результате мономорфным и полиморфным популяциям соответствуют разные предельные численности и циклы с разными периодами и фазами колебаний в случае потери устойчивости даже при полной идентичности популяций.

Purpose is to study the mechanisms leading to genetic divergence (stable genetic differences between two adjacent populations). We considered the following classical model situation. Populations are panmictic with Mendelian rules of inheritance. The action of natural selection (differences in fitness) on each of population is the same and is determined by the genotypes of only one diallel locus. We assume that adjacent generations do not overlap and genetic transformations can be described by a discrete time model. This model describes the change in the concentration of one of the alleles in each population and the ratio (weight) of first population to the total size. Methods. We used the analogue of saddle charts to construct parametric portraits showing the domains of qualitatively different dynamic modes. The study is supplemented with phase portraits, basins of attraction and bifurcation diagrams. Results. We found that the model dynamic regimes qualitatively coincide with the regimes of a similar model with continuous time, but only for a weak migration. With a strong coupling, fluctuations of the phase variables are possible. We showed that the genetic divergence is possible only with reduced fitness of heterozygotes and is the result of a series of bifurcations: pitchfork bifurcation, period doubling, or saddle-node bifurcation. After these qualitative changes, the dynamics become bi- or quadstable. In the first case, the solutions corresponding to the genetic divergence are unstable and are just a part of the transient process to monomorphic state. In the second case, the divergence is stable and appears as 2-cycle for a strong migration coupling. Conclusion. In neighboring populations, movement towards an asymptotic genetic structure (monomorphism, polymorphism or divergence) can be strictly monotonous or in the form of damped unstable or undamped stable fluctuations with a period of 2 for biologically significant parameters. For insignificant parameters, we found a complex dynamics (chaos) that consist of divergent fluctuations around fixed points and quasi-random transitions between them.