Kravchenko et al. (2011) Study of the stability of hemiclonal population systems...

Kravchenko M. A., Shabanov D. A., Vladimirova M. V., Zholtkevych G. N. Investigation of the stability of hemiclonal population systems of the hybrid complex of green frogs using simulation modeling // Visnyk Dnipropetrovskyi University. Biology. Ecology. – 2011. – Issue 19, vol. 1. – pp. 51–64. UDC: 004.942:597.851 M. A. Kravchenko, D. A. Shabanov, M. V. Vladimirova, G. N. Zholtkevych Kharkiv National University named after V. N. Karazin INVESTIGATION OF THE STABILITY OF HEMICLONAL POPULATION SYSTEMS OF THE HYBRID COMPLEX OF GREEN FROGS USING SIMULATION MODELING The concept of stability in ecology and mathematics is analyzed. Stability is interpreted as the ability of a system to remain within a bounded region of phase space corresponding to a particular type of systems under consideration. This approach is applied to describe changes in the composition of hemiclonal population systems of the hybrid complex of green frogs (Pelophylax esculentus complex). A simulation model built on recurrent difference equations in MS Excel was used. The dynamics of transitions in a part of the phase space of hemiclonal population systems of green frogs are described. The typology of stability states of biosystems is analyzed. M. O. Kravchenko, D. A. Shabanov, M. V. Vladimirova, G. M. Zholtkevych V. N. Karazin Kharkiv National University The concept of stability used in ecology and mathematics is analyzed. Stability is interpreted as the ability of the system to remain in a restricted zone of a phase space that corresponds to a certain type of systems. This approach is applied to describe changes in the structure of hemiclonal population systems of the hybridogenous complex of water frogs (Pelophylax esculentus complex). A simulation model of these population systems built on recursive difference equations in MS Excel is used. The dynamics of transitions in the part of phase space of hemiclonal populations of water frogs is described. Typology of stable states of biosystems is analyzed. Introduction For the overwhelming majority of sexually reproducing organisms, species populations are formed, which constitute biotic communities. Individuals within a population share a common gene pool and jointly participate in population reproduction. Populations in a community interact in various ways, from competition to exploitation and mutualism. A portion of the community that unites competing populations using the same resource can form guilds. The listed levels of organization of biosystems (population, guild, community) can be considered typical. However, for some groups of organisms other, unusual levels of organization occur. Among them are the European green frogs, Pelophylax esculentus complex. The name Pelophylax esculentus (Linnaeus, 1758) does not refer to a species but to a hybrid that arises from crossing two parental species: Pelophylax lessonae (Camerano, 1882) and Pelophylax ridibundus (Pallas, 1771). P. esculentus is characterized by hemiclonal inheritance, in which either the genome of P. lessonae or the genome of P. ridibundus is transmitted to the gametes, rather than recombinant genomes composed of a mixture of parental genomes, as occurs in the overwhelming majority of sexually reproducing organisms [17, 20]. We denote the genome of P. lessonae as L and that of P. ridibundus as R, and we indicate clonality by enclosing the genome symbol in parentheses: (L) or (R). When hybrids that transmit a heterospecific genome (i.e., the genome of the other parental species) mate with individuals of the parental species, all offspring are hybrids: RR × R(L) → R(L). The clonal genome (L) remains the same in both generations, whereas recombinant genomes—R—are “reassembled” anew during transmission from generation to generation in individuals of the parental species. Green frogs characteristically form population systems in which both parental species and various hemiclonal hybrids coexist and reproduce. For example, in the North‑Donetsk center of green frog diversity [17], in addition to individuals with genotype RR (i.e., P. ridibundus), diploid hybrids (L)R, L(R), (L)(R) and triploid hybrids LLR and LRR are found. Among non‑reproductive individuals, solitary tetraploids LLRR and individuals LL (i.e., P. lessonae) occur. Importantly, P. lessonae individuals arising from crosses of (L)R are non‑viable. Non‑viability of parental‑species individuals that receive both of their genomes from hybrid parents has also been recorded in other regions [20]. Clearly, such systems, in which both clonal and recombinant genomes of different species are transmitted across generations, belong to a special group of biosystems and deserve a distinct designation. We have proposed [16, 17] to call them hemiclonal population systems—HPS. HPS are not populations, because they include individuals of different species. HPS are not guilds or communities, because they are united by the process of joint reproduction. The dynamics of HPS differ from the dynamics of populations, guilds, and communities. One consequence of HPS being a distinct level of biosystem organization, whose existence has been recognized only recently, is their considerably lower level of study compared with typical biosystem levels. The aim of this work is to analyze various states of HPS from the perspective of their stability. A prerequisite for such analysis is a consideration of the meaning embedded in the term “stability.” Analysis of the concept “stability” The concept of stability became one of the most important ecological notions early in the development of the discipline. It was crucial for F. Clements’ climax theory of plant communities, which emerged at the beginning of the 20th century [19, cited in 13]. V. N. Sukachev [14] defined the stability of a plant community as the ability “to preserve its composition and structure over a prolonged period. This stability is due to the community’s capacity for self‑renewal.” Despite the interest in stability in ecology, it is premature to consider the concept clear. Two authoritative statements illustrate this: “Unfortunately, the term ‘stability’ has often remained vague and undefined” [11, p. 324]. “Ecologists are remarkably ignorant of everything concerning the stability of natural systems: what internal mechanisms of communities are involved and how they operate?” [12, p. 394]. R. Ricklefs conducts an analysis of the concept “stability.” He defines the stability inherent to a system as the ratio between environmental variability and variability within the system itself [12, p. 378], emphasizing that this definition (essentially describing buffering properties) is difficult to apply to populations and communities. The approach is easier to grasp with an organism‑level example: Ricklefs presents data on climate parameter dynamics and tree‑ring width dynamics, showing that trees can be more or less stable to external changes. Damping of perturbations (a 50 % drop in precipitation leading to a 25 % reduction in plant production and a 10 % decline in herbivore numbers) indicates the operation of internal mechanisms that maintain ecosystem stability. Ricklefs distinguishes three types of stability‑related states: stable equilibrium (perturbations are compensated), unstable equilibrium (perturbations are amplified), and neutral equilibrium (perturbations cause changes insignificant for system functioning). Ultimately, Ricklefs formulates a definition reflecting the duality of the concept: “Stability is an inherent capacity of a system to withstand externally induced change, or to recover after it” [12, p. 379]. R. Wittekir interprets stability primarily as buffering. “We will call the processes that reduce population losses when the environment becomes more adverse buffering of populations” [15, p. 58]. The duality of the stability phenomenon highlighted by Ricklefs is expressed by Y. Odum [10, p. 66], who treats resistance (the ability to withstand perturbations) and resilience (the ability to recover after disturbance) as manifestations of stability. M. Bigon and co‑authors [3, pp. 325‑327, as well as 19, p. 576‑577], analyzing the concept of stability, identify three pairs of notions: — resilience (resilence) as a measure of the ability to return to the original state and resistance as a measure of the ability to avoid change; — local and global stability, differing by the magnitude of perturbations that the systems can compensate; — dynamic fragility or robustness, reflecting the relationship of the system to the external environment: the ability of the system to persist in a narrow set of external conditions or across a wide range. The authors provide graphical analogies to clarify the terms they use. As Bigon and co‑authors note, the study of community stability is largely conducted using mathematical modeling. V. G. Storozhenko [13] lists an extensive set of parameters whose preservation reflects the stability of biogeocenoses, including species numbers and the species themselves, internal connections, population sizes, functional traits, suppression of harmful factors, etc. The main task of Storozhenko’s analysis is to find criteria for a stable forest community. He does not consider or classify system states that are far from climax. As the review shows, most works on ecological stability describe conditions under which particular communities (especially forest ones) are stable. However, besides the state of true stability (stable equilibrium), which is the most valuable state of a biosystem, other possible states should also be investigated. Can the set of possible system states be divided into types that differ in terms of their dynamics and stability? To address this problem, the apparatus of stability analysis developed in mathematics must be employed. The concept under consideration suffers here as well: “stability – a term lacking a clearly defined content” [9, p. 604]. The classic definition of “stability” was given in 1892 by the founder of the theory of motion stability, Aleksandr Mikhailovich Lyapunov (associate professor, later professor at Kharkiv University, now named after V. N. Karazin). Lyapunov defined stability in terms of the effect of perturbations on motion with given initial conditions [8]. Applying this approach to biosystem dynamics encounters significant difficulties and does not fully align with established tradition. In our view, among mathematical approaches to the concept for ecology, the representation of stability by J. L. Lagrange, which is a particular case of Lyapunov’s stability [5], is most useful. From this perspective, stability is the ability of a system to remain within a bounded region of its phase space. Phase space here is the set of all possible system states described by a set of state variables. Choosing variables that define phase space allows, for example, to consider community stability both in terms of species number and in terms of structural constancy. The essence of the stability phenomenon remains the same; only the coordinate set in which the trajectory of the studied system is examined changes. Such a notion of stability fits well with its use in ecological research. For instance, although F. Clements did not formulate his views in these terms, the stability of a climax community according to Clements can be defined as the community’s ability to keep its type constant over time (regardless of the typology used to delineate this type). A similar approach is found in the treatment of geosystem stability by A. D. Armand [1]. Armand distinguishes stability I, i.e., invariance, inertia, and stability II, i.e., the ability to resist perturbations. Stability II is associated with the concept of a basin of stability: “a region in an abstract multidimensional space built on axes of state variables, from all points of which the transition process brings the system to the same equilibrium or homeostatic state.” Armand identifies several equilibrium states in which the system does not undergo self‑development: stable, unstable, indifferent, cyclic, dynamic, and mobile equilibria. The first three correspond to those distinguished by Ricklefs. In cyclic equilibrium the system regularly returns to the same state. The last two are more complex: “dynamic equilibrium is a state in which oppositely directed processes (import and export of matter and energy, organization and disorganization) mutually compensate. Mobile equilibrium is the trajectory of the system in state space along which the system follows changes in input variables (factors), maintaining at each moment a state of stable equilibrium” [1]. For biosystems, any equilibrium state is dynamic. The alternative to dynamic equilibrium is stability I, i.e., staticity. From our perspective, to describe the difference between stable and mobile equilibrium, it is necessary to separate external and internal factors influencing the system. Stable equilibrium results from balancing internal factors, while mobile equilibrium arises from their displacement by external factors. Thus, the concept of system stability turned out to be closely linked to the typology of biosystem states. By dividing the system’s state space into basins of stability (separated by stability boundaries [2]), a dynamic typology of its states can be obtained. For solving this task for the HPS of Pelophylax esculentus complex, a simulation model of the considered systems will be useful. Simulation modeling of HPS Pelophylax esculentus complex To model the dynamics of HPS of green frogs we developed a set of user requirements (URD, User Requirements Definition). The dynamics of HPS of green frogs were studied by simulation modeling. For verification of the modeling results we created two divergent simulation models. Together with A. A. Lucic we built a multi‑agent simulation model of HPS [4]. More convenient in practice, due to ease of modification, proved to be the HPS model based on recurrent difference equations, implemented in Microsoft Office Excel [6]. One version of this model (named Batrachometrics‑2010) is freely available on the website of the Central Scientific Library of Kharkiv National University named after V. N. Karazin at https://dspace.univer.kharkov.ua/handle/123456789/2037. It is important to emphasize that both implementations of the model, the multi‑agent one and the difference‑equation one in MS Excel, generate qualitatively similar dynamics of the simulated systems. The model postulates the existence of an HPS composed of various frog forms. In the HPS genomes differing in species affiliation (P. lessonae and P. ridibundus), mode of transmission (clonal and recombinant) and sex (female and male) are transferred. From the genome (set of genotypes) and the age of an individual depend its life‑history parameters: maximum lifespan, age at sexual maturity, survival in the absence of competition, competitiveness, attractiveness to partners, fecundity and, for males, the probability of repeated amplexus (pair formation). One model step corresponds to one year. During each simulated year the model imitates the decline in the number of individuals of each age class of each genotype during winter. Environmental carrying capacity limits the total number or biomass of all individuals in the HPS. If population size exceeds the carrying capacity, competitive reduction occurs. The number of each group of individuals declines inversely proportional to the competitiveness of its members. Forming amplexus pairs depends on the attractiveness of individuals of each group. All members of the sex that, given the observed HPS numbers, can engage in fewer amplexus events, as well as the corresponding number of individuals of the opposite sex, participate in reproduction.For all possible combinations of parental genotypes, the genotypes of the resulting offspring are defined. After simulating survival in hibernation, competitive reduction, pair formation for reproduction, and offspring production, the model proceeds to the next step (in the following year), and the entire described sequence of calculations repeats. To run the model, specific viability parameter values must be set for all considered forms of frogs (see [6] for details). In this article, except for the noted cases, the default values from Batrachometrics‑2010 are used. Phase space and typology of the RE‑type population system (PPS) of the Pelophylax esculentus complex The fact that green frogs form population systems that differ in composition has been known for several decades. The most widely used typology of frog population systems was proposed in 1975 with the participation of the discoverer of the hybrid nature of P. esculentus, L. Berger [21]. In this typology the letter L denotes the presence of P. lessonae in the population system, R denotes the presence of P. ridibundus, and E denotes the presence of P. esculentus. The presence of triploid individuals among hybrids is indicated by the letter t, the presence of females only by the letter f, and males only by the letter m [7]. For example, one can state that in the floodplain of the Seversky Donets River near the biological station of Kharkiv University (village Haidary, Zmiievskyi district, Kharkiv region) a RE‑type population system occurs [17].

Phase space and typology of the *Pelophylax esculentus* complex The fact that green frogs form population systems that differ in their composition has been known for several decades. The most common typology of frog population systems was proposed in 1975 with the participation of L. Berger [21], the discoverer of the hybrid nature of *P. esculentus*. In this typology, the letter L denotes the presence of *P. lessonae* in the population system, R - the presence of *P. ridibundus*, and E - the presence of *P. esculentus*. The presence of triploid individuals among hybrids is indicated by the letter t, the presence of exclusively females by the letter f, and exclusively males by the letter m [7]. For example, it can be stated that in the floodplain of the Seversky Donets River near the biological station of Kharkiv University (village of Gaidary, Zmiiv district, Kharkiv region), an REt-type population system lives [17].

The described approach is widely used in studying the distribution of different forms of green frogs, but it does not take into account that *P. esculentus* individuals can produce gametes with different clonal genomes. The composition of the gamete determines both the composition of future offspring and the dynamics of the considered population system.

We propose to use the composition of transmitted genomes to characterize hybridogenetic populations (HPPs). In this work, we will consider RE-type HPPs, where recombinant genomes R are transmitted from generation to generation, and clonal genomes (L) and (R) can also be transmitted. We will construct a two-dimensional phase space, with axes representing the proportion of female clonal genomes (XL) and (XR) in the total number of genomes (Fig. 1). The origin of the coordinate system corresponds to an R-type population consisting of individuals of *P. ridibundus*. On the line connecting the points on the coordinate axes corresponding to 100% (R) and 100% (L), HPPs of the E-type are located. Points above this line are undefined, as the sum of transmitted clonal genomes for them would exceed 100%. Other points in the described plane correspond to RE-type HPPs.

Fig. 1. Types of population systems in the phase space of a population system consisting of *P. ridibundus* individuals, as well as hybrids that transmit female clonal genomes of both species

The set of population system states shown in Fig. 1 is only part of the possible diversity of green frog population systems. In population systems with one parental species, both female and male clonal genomes can be transmitted. In addition to RE-type population systems, LE-type (whose set of phase states is symmetrical to the set of RE-type states) and LER-type can also be considered. Finally, various triploid hybrids can be present in population systems.

The characterization of the concept of stability requires specification of the perturbations affecting the system under consideration. For the composition of a population system, a perturbation is a change in the number of forms of individuals represented in it, caused by external circumstances. In the phase space under consideration, such perturbations are the addition to the population system of *P. esculentus* individuals that transmit female clonal genomes: ♀♀(XL)XR, ♂♂(XL)YR, ♀♀XL(XR), and ♂♂YL(XR).

Transformations of RE-type population systems with female clonal genomes An R-type population can exist in a stationary state indefinitely. When ♀♀XL(XR) or ♂♂YL(XR) are added, the system returns to its previous state after a few years. In this regard, the population system is in a state of stable equilibrium with respect to this perturbation.

When at least one individual transmitting the (XL) genome enters the *P. ridibundus* population, irreversible transformations begin. All offspring from the crossbreeding of individuals of the parental species and hybrids turn out to be hybrid; as a result, the proportion of hybrids increases from generation to generation. As hybrids carrying the (XL) genome accumulate in the population system, an increasing proportion of crosses occur between them, leading to the appearance of non-viable individuals (XL)(XL). The described process ends with the death of the population system (Fig. 2).

[IMG_2] Fig. 2. Transformations of a PPS derived from a P. ridibundus population into which one ♀ (XL)XR was added in the first simulation year; the PPS collapses around year 140

Fig. 2 illustrates the transition of the system from a state of unstable to additive genome (XL) equilibrium into a state of targeted change, characterized by an increase in the proportion of individuals (XL)XR and (XL)YR. The result of this process is the death of the entire system.

As the share of (XL) genomes grows, the PPS loses stability to the introduction of (XR) genomes, because, according to the default assumptions of Batrachometrics‑2010, individuals (XL)(XR) are viable. To explain the system’s properties in this region of phase space, we follow M. Bigon and co‑authors [3] and use a physical analogy (Fig. 3). The initial state in Fig. 3 is a stable equilibrium with respect to the addition of (XR) genomes and an unstable one with respect to (XL) genomes. Overall, this state should be described as an unstable equilibrium. When (XL) genomes are introduced, the system “rolls” toward a higher share of them and loses stability to (XR) genomes.

At the 90th year of simulated introduction into the GPS shown in Fig. 2, neither ♀ XL(XR) nor ♂ YL(XR) leads to any significant changes. At the 100th and 110th years, both influences cause the GPS to transition to another state, in which a significant number of clonal genomes of both species are transmitted (Fig. 4).

Fig. 3. Physical analogy of the properties of the phase space of the marsh frog (see Fig. 1) near the origin (population *P. ridibundus*, shown as a ball)

[IMG_4] Fig. 4. Dynamics of the PPS in phase space. At the start, one ♀ (XL)XR is added to the P. ridibundus population; in the 100th simulation year another ♀ XL(XR) is added. The displayed events span 500 simulation years

As Fig. 4 shows, a PPS transmitting clonal genomes of both species undergoes damped cyclic changes in its composition. An analysis of 500 simulation years could suggest that the PPS eventually reaches a cyclic equilibrium, moving along a closed trajectory. By analogy with other biosystem dynamics, such a cyclic equilibrium corresponds to cyclic successional stages of biogeocenoses or undamped cyclic oscillations of two species in a predator‑prey system with delayed responses to each other’s population changes [11, p. 231].

However, in the case of the considered GPS model, the equilibrium is not cyclical, and by approximately the 1000th year, the GPS reaches a stable equilibrium point. The composition of the spawning stock at this point is as follows: 10.5% ♀♀ XRXR; 6.9% ♂♂ XRYR; 53.9% ♀♀ (XL)XR; 16.7% ♂♂ (XL)YR; 8.6% ♀♀ XL(XR); 3.4% ♀♀ (XL)(XR). This GPS composition is very far from sex equality: it contains only 23.6% males and 76.4% females. Nevertheless, all GPS located within the loop-shaped part of the trajectory shown in Fig. 4, arrive at this very state.

What happens to PPSs lying outside the “loop” shown in Fig. 4? Consider the case where genome (XR) is added only in the 120th year (Fig. 5). [IMG_5] Fig. 5. Dynamics of the PPS in phase space. At the start a ♀ (XL)XR is added to the P. ridibundus population, and in year 120 a ♀ XL(XR) is added. After describing a loop in phase space, the PPS reaches the right corner in the 215th simulation year—that is, the point where 100 % of transmitted genomes are (XL)—and collapses

[IMG_5] Fig. 5. Dynamics of the PPS in phase space. At the start a ♀ (XL)XR is added to the P. ridibundus population, and in year 120 a ♀ XL(XR) is added. After describing a loop in phase space, the PPS reaches the right corner in the 215th simulation year—that is, the point where 100 % of transmitted genomes are (XL)—and collapses

Diversity of green‑frog PPS states from the perspective of stability Based on the foregoing, we can identify equilibrium positions and their basins on the studied phase space (Fig. 6). The boundary between the basin of stable equilibrium and the basin of PPS extinction lies between the loop described in Fig. 4 and the larger loop described in Fig. 5. By placing points corresponding to initial PPS states on the unexplored part of the phase space, we can determine to which basins they belong. PPSs lying on the ordinate axis move to the origin, i.e., to the P. ridibundus population. The development of all other PPSs, surprisingly, ends in their extinction at the point corresponding to 100 % (XL).

Fig. 6. Zones (equilibrium positions and their basins) of the studied phase space of the green frog hybrid complex

Fig. 6 shows three basins of attraction (one of them corresponds to the y-axis) and three equilibrium positions: unstable, stable, and system extinction. Paradoxical as it may sound, the state of unstable equilibrium also has its basin. This is because the population of *P. ridibundus*, while being unstable to some disturbances, demonstrates stability to others.

Can a PPS exist in other stability states besides those shown in Fig. 6? As noted above, A. D. Armand [1] distinguishes indifferent and mobile equilibria as separate categories. The figures 2, 4, 5 and 6 were obtained using the default viability parameters of Batrachometrics‑2010. Changing the relative viability of the different frog forms alters the properties of the phase space. For instance, when the relative viability of hybrid frogs is reduced, the reproductive advantage of hybrids is compensated by the higher viability of parental‑type individuals. Adding a ♀ (XL)XR to a P. ridibundus population (i.e., under conditions analogous to those illustrated in Fig. 2) shifts the PPS into a mobile equilibrium. Recall that, from our viewpoint, mobile equilibrium can be discussed only when an external factor opposes the internal developmental processes of the system. Here we treat the reduced hybrid viability as an environmental (i.e., external) influence on PPS dynamics.

The Batrachometrics‑2010 model is deterministic: with identical initial conditions it always reaches the same state. Indifferent equilibrium is attainable only under stochastic, random PPS dynamics, which requires the introduction of a random component into the model.

In the next version, Batrachometrics‑2011, the size of each frog group is rounded to an integer once per year, with probabilistic rounding (e.g., a value of 1.7 is rounded to 1 with probability 0.3 and to 2 with probability 0.7). Because of this probabilistic rounding, repeated simulations with the same initial conditions can lead to different outcomes. Figure 7 shows the dynamics of the Batrachometrics‑2011 model under reduced hybrid viability. From the perspective of the ratio between parental‑type individuals and hybrids, the system moves into a mobile equilibrium: the share of recombinant parental genomes oscillates, often exceeding 50 %. The clonal genomes (XL) in Fig. 7 are represented by three hemiclones that are identical in viability and initial size. Different simulation runs with the same initial conditions yield different fates for these hemiclones. The figure depicts a case where, due to random causes, the size of one hemiclone falls to zero and it disappears from the PPS.

[IMG_7] Fig. 7. Dynamics of the Batrachometrics‑2011 model (with probabilistic rounding of group sizes). Reduced hybrid viability leads to a mobile equilibrium between parental‑type individuals and hybrids. Three heterospecific clonal hemiclones are in a state of indifferent equilibrium; at the beginning of the fourth century of simulation one of them disappears from the PPS due to random causes

Fig. 7. Dynamics of the Batrachometrics-2011 model (with probabilistic rounding of population sizes). Reduced hybrid viability leads to a mobile equilibrium between individuals of the parent species and hybrids. Three heterospecific clonal hemiklines are in a state of indifferent equilibrium; at the beginning of the 4th simulation century, one of them disappears from the gene pool for random reasons.

Types of Biosystem Stability Summarizing the study of the dynamics of the green frog's reproductive system using simulation models, a classification of biosystem stability types can be proposed. This classification mainly corresponds to the views of A. D. Armand [1]. Types of stability can be divided into two groups (Fig. 8). The first group includes states that can be

Fig. 8. Types of ecosystem stability (physical analogies)

Transient states form basins of attractor states. These are unstable equilibrium, directed change, and indifferent equilibrium. The boundaries between basins (stability boundaries) are represented by states of unstable equilibrium.

As the examples discussed in this article show, the described types of stability are not absolute. Their identification depends on which parameters of the system's state are monitored (in which coordinate system the phase space of its states is constructed). Thus, the GPS achieved in Fig. 7 can be interpreted ambiguously. From the perspective of the relationship between recombinant and clonal genomes, it is a state of mobile equilibrium, while from the perspective of the relationship between different hemiclones, it should be considered a neutral equilibrium.

Conclusions Hemi-clonal population systems (HPS) of hybridogenic complexes are a group of biosystems that have been discovered quite recently, and their study is only beginning. This study not only provides valuable material for the cognition of the studied groups of organisms but also expands the understanding of the general properties of biosystems.

In this work, the stability of a biosystem is considered as the ability to remain in a certain part of its phase space. This part of the phase space corresponds to a specific type of system under consideration. The entire state space of a biosystem can be divided into stability basins, which correspond to specific attractor states. The proposed approach is suitable for the phase space of the hybridogenetic system (HGS) of green frogs of the RE-type, which was studied using simulation modeling. Simulation models built on different computational foundations demonstrated qualitatively similar dynamics of the studied systems. The influence of the conditions under which the considered HGS develops on the location and configuration of stability basins is shown.

This work demonstrates six types of biosystem stability states; the seventh type is known from the literature. In total, these types form two groups: attractors, the "centers" of stability basins, and transitional states, which over time transfer the system under consideration to one or another attractor state.

Bibliographic references 1.       Armand A. D. Mechanisms of geosystem stability. – M.: Nauka, 1992. – 208 p. 2.       Arnold V. I. Theory of catastrophes. – M.: Nauka, 1990. – 128 p. 3.       Bigon M. Ecology. Individuals, populations and communities: in 2 vols. Vol. 2. / Bigon M., Harper J., Townsend K. // M.: Mir, 1989. – 477 p. 4.       Vladimirova M. V. Study of integral properties of biosystems on the example of simulation modeling of hybridogenic population systems of green frogs / Vladimirova M. V., Zholtkevych G. N., Lutsik A. A., Shabanov D. A. // Visnyk Kharkiv National University. Series “Mathematical Modeling. Information Technologies. Automated Control Systems” – 2007, No 780, p. 61–70. 5.       Kalitin B. S. Qualitative theory of stability of dynamical system motion. – Minsk: BSU, 2002. – 198 p. 6.       Kravchenko M. A. Modeling transformations of hemiclone population systems of green frogs (Pelophylax esculentus complex; Amphibia, Ranidae) using recurrent difference equations / Kravchenko M. A., Shabanov D. A. // Visnyk Kharkiv National University named after V. N. Karazin. Series: Biology. – 2010. – Issue 12 (No 920). – p. 70–82. 7.       Lada G. A. Central‑European green frogs (hybridogenic complex Rana esculenta): introduction to the problem // Flora and Fauna of the Chernozem Region. – Tambov, 1995. – p. 88–109. 8.       Lyapunov A. M. General problem of stability of motion. M‑L.: Gostehizdat, 1950. – 472 p. 9.       Mathematical Encyclopedic Dictionary. – M.: Soviet Encyclopedia, 1988. – 847 p. 10.    Odum Y. Ecology: in 2 vols. Vol. 1. – M.: Mir, 1986. – 328 p. 11.    Pianka E. Evolutionary ecology. – M.: Mir, 1981. – 400 p. 12.    Ricklefs R. Foundations of general ecology. – M.: Mir, 1979. – 424 p. 13.    Storozhenko V. G. Stable forest communities. Theory and experiment. – Tula: Griff and Co., 2007 – 192 p. 14.    Sukachev V. N. Selected works. – L.: Nauka, 1972. – Vol. 1, 343 p. 15.    Witteker R. Communities and ecosystems. – M.: Progress, 1980. – 327 p. 16.    Shabanov D. A. Which green frogs inhabit Kharkiv region? Terminological and nomenclatural aspects of the problem / Shabanov D. A., Korshunov O. V., Kravchenko M. O. // Biology and Valiology. – Kharkiv: HDPU, 2009. – Issue 11. – p. 164–125. 17.    Shabanov D. A. Green frogs: life without rules or a special mode of evolution? / Shabanov D. A., Lytvynchuk S. N. // Nature. – 2010. – No 3. – p. 29–36. 18.    Begon M. Ecology. From individuals to ecosystems. / Begon M., Townsend C. R., Harper J. L. // Blackwell Publishing, 2006. – 738 p. 19.    Clements F. E. Plant succession. An analysis of the development of vegetation. – Washington: Carnegie Inst., 1916 – 242 p. 20.    Plötner J. Die westpaläarktischen Wasserfrösche. Bielefeld: Laurenti‑Verlag, 2005. – 161 p. 21.    Uzzell T. M.{"translated_text":"Electrophoretic phenotypes of Rana ridibunda, Rana lessonae and their hybridogenic associate Rana esculenta / Uzzell T. M., Berger L. // Proc. Acad. nat. Sci. Phila. — 1975. — Vol. 127. — P. 13-24."}