## Design and Stability Analysis of NondeterministicMultidimensional Populations Dynamics Models

**Authors:**Demidova AV^{1}, Druzhinina OV^{2}^{,3}, Masina ON^{4}**Affiliations:**- Department of Applied Probability and Informatics Peoples’ Friendship University of Russia (RUDN University)
- Federal Research Center “Computer Science and Control” of RAS
- V. A. Trapeznikov Institute of Control Sciences of RAS
- Yelets State Ivan Bunin University

**Issue:**Vol 25, No 4 (2017)**Pages:**363-372**Section:**Mathematical Modeling**URL:**http://journals.rudn.ru/miph/article/view/17430**DOI:**https://doi.org/10.22363/2312-9735-2017-25-4-363-372- Cite item

# Abstract

The multidimensional models of the population dynamics are considered in the paper. Thesemodels are the generalizations of the Lotka-Volterra model in case of interaction of the ﬁnitenumber of populations. The deterministic description of the models is given by the systemsof the ordinary nonlinear diﬀerential equations presented in the paper in the form of themultidimensional vector diﬀerential equations. The qualitative properties of the speciﬁed modelsare suﬃciently well studied by means of Lyapunov methods. However, the probabilistic factorsinﬂuencing on the behavior of models are not taken into account at the deterministic descriptionof models. The new approaches to the modeling and stability analysis are of theoretical andapplied interest in the nondeterministic case.In this paper, the methods for design of multidimensional nondeterministic models ofinteraction of populations are considered. The ﬁrst method is connected with the transitionfrom the vector nonlinear ordinary diﬀerential equation to the corresponding vector diﬀerentialinclusions, fuzzy and stochastic diﬀerential equations. On the basis of the reduction principle,which makes it possible to reduce the problem of the stability of solutions of a diﬀerentialinclusion to the problem of stability of solutions of other types of equations, stability conditionsfor the constructed models are obtained. The second method is connected with the technique ofdesign of the self-consistent stochastic models. The scheme of interaction is received on the basisof this technique. This scheme includes a symbolical record of possible interactions between thesystem elements. The structure of the multidimensional stochastic Lotka-Volterra models isdescribed, and the transition to the corresponding Fokker-Planck vector equations is carriedout by means of the system state operators and the system state change operator. The rules forthe transition to the multidimensional stochastic diﬀerential equation in the Langevin form areformulated. The execution of the numerical experiment with the application of the developedprogram complex for solving the systems of the stochastic diﬀerential equations is possible forthe models which are the concretizations of the studied general models. The described approachto the modeling of the stochastic systems can be applied in the problems of comparing of thequalitative properties of the models in deterministic and stochastic cases. The obtained resultsare aimed at the development of the methods for the analysis of nondeterministic nonlinearmodels.

# Full Text

1. Introduction The stability research of the models of the population dynamics is an important problem. Some directions of the solution of the speciﬁed problem are presented in the works [1-7]. The questions of existence and stability of the solutions of the models described by the diﬀerential equations of various types were considered in [8-12] and in other works. Lyapunov functions method is one of the widely used methods of the stability research [10, 11]. The stability of the solutions of classical and generalized models of the population dynamics by Lyapunov function method was considered in [1]. The systemic approach is described in [2, 6-9] which allows us to consider properties of stability of the models described by the diﬀerential equations of various types from the uniﬁed point of view. The speciﬁed approach is based on the transition from the deterministic description of the model to stochastic description and on the principle of reduction of the stability problem of solutions of diﬀerential inclusion to the stability problem of other types of the equations. The approach allows us to study the stability properties of solutions of diﬀerential inclusions, the fuzzy and stochastic diﬀerential equations from the uniﬁed point of view. In this work we consider a nonlinear multidimensional model of the dynamics of the populations interaction. The determined description of model is given by system of the ordinary nonlinear diﬀerential equations. The transition from the speciﬁed model to the corresponding nondeterministic models given by means of ﬁnite-dimensional diﬀerential inclusions, the fuzzy and stochastic diﬀerential equations is performed. The stability analysis is performed on the basis of the reduction principle in this work. It is known [5, 13-15] that in the deterministic description of the model the probabilistic factors aﬀecting the behavior of the model are not taken into account. In this connection, an important problem is construction and study of adequate stochastic models, as well as a comparative analysis of the properties of deterministic and corresponding stochastic models. The technique of design of the self-consistent stochastic models [14] allows us to take into account stochastics in the structure of the model without adding additive stochastic terms. In this work the structure of the multidimensional stochastic Lotka- Volterra models is described, and the transition to the corresponding Fokker-Planck vector equations is carried out by means of the system state operators and the system state change operator. The rules for the transition to the multidimensional stochastic diﬀerential equation in the Langevin form are formulated. It is shown that the used approach to construct multidimensional stochastic models can ﬁnd application in problems of comparing the qualitative properties of the generalized Lotka-Volterra models. 2. Deterministic Models We consider the model described by the system of diﬀerential equations of the form [1, 3]: where is density of -th population in moment and are growth coeﬃcients of -th population in the absence of others, constants at ̸= characterize the inﬂuence of interaction between populations on the rate of growth, is interaction matrix. The model (1) is the classical Lotka-Volterra model for the -dimensional case. This model describes the dynamics of the biological community under the following conditions: 1) the relative growth rate of each population does not depend on the intrapopulation structure; 2) this rate depends linearly on the number of populations in the community. These conditions, characteristic of the Lotka-Volterra equation, represent a simpliﬁed hypothesis about the nature of the interactions between populations in the community. This hypothesis, known as the principle of pair interactions, suggests the additivity of each population contribution to the relative growth rate, which is reasonably well founded biologically. However, the linear nature of this contribution is much worse in the processes occurring in biological communities, and can be taken into account in approximating the equilibrium state in some neighborhood [1]. In this connection, the study of the model (1) can be considered as an important stage preceding the study of the models that are generalizations of the model (1). A generalization of the model (1) is a model of the following form: The stability conditions for solutions of the model (1), (2) on the basis of the Lyapunov functions method are obtained in [1]. The stability conditions on the basis of the divergent method are obtained in [3] for indicated models. The transition is possible from the deterministic model (1), (2) to diﬀerent types of the corresponding nondeterministic models. 3. Design of Nondeterministic -dimensional Lotka-Volterra Models and Stability Analysis Based on the Reduction Principle The nonlinear model (1) is presented in the form of the vector equation where - -fold Cartesian product of the set + on itself, For the model (3) the coeﬃcients and , , = 1, can take diﬀerent values from the corresponding intervals [ 1 , 2 ] and [ 1 , 2 ] taking into account the ecological meaning accordingly. The transition from the model (3) to the ﬁnitedimensional diﬀerential inclusion is the following The model (4) in the vector form is presented as follows: where The introduced sets and deﬁne the sets of values of the corresponding parameters and . Subsets represent the narrower sets that we obtain when we take into account the additional conditions ∈ (0, ], that aﬀect the interaction of the components and, consequently, the stability of the model (3). Then equation (3) can be replaced by a fuzzy ﬁnite-dimensional diﬀerential equation where : is the set of all fuzzy subsets of + . The diﬀerential inclusion corresponding to the equation (6) has the form ˙ ∈ (), where ∈ (0, 1], The following stability conditions of the diﬀerential inclusion (5) and the fuzzy equation (6) we formulate by means of the principle of reduction [7, 8] and by means of the transition from model (1) to models (5) and (6): 1) if there is a Lyapunov function for the closed set ⊂ + regarding the inclusion (5), such that the inequality is satisﬁed, where () is upper derivative of Lyapunov function, set (, ) is neighborhood of the set , then the set is stable in small regarding this inclusion; 2) if the inequality is satisﬁed, where function : is the continuous and positive function in then the set is asymptotically stable in small regarding the inclusion (5); 3) if there is a Lyapunov function regarding the equation (6) for the closed set where ( + ) is the set of all fuzzy subsets of + , such that the inequality is satisﬁed then the set -stable regarding this equation; 4) if the inequality is satisﬁed, where : the continuous and positive function, then the set is asymptotically stable regarding the equation (6). In this work we consider the generalization of the model (3) to the stochastic case, namely, the transition is carried out from the equation (3) to the corresponding stochastic diﬀerential equation where () is the stochastic function. By means of the principle of reduction we formulate the stability conditions of the fuzzy equation (6) and the stochastic equations (7). It is shown that if the trivial solution of a fuzzy equation (6) is -stable (asymptotically -stable) for every ∈ (0, 1], then the trivial solution of the corresponding stochastic equation (7) is stable on probability (asymptotically stable on probability). In addition, the conditions of almost surely stability and stability on average we give by the aid of the principle of reduction. The comparative analysis of the qualitative properties of the deterministic and stochastic models is given on the basis of the obtained suﬃcient stability conditions. System (2) is represented as a nonlinear vector equation We consider the transition from this vector equation to the nondeterministic models described by diﬀerential inclusion, fuzzy and stochastic diﬀerential equations. This transition is similar to the transition from the vector equation (3) to the models (5)-(7). The stability conditions of the indicated diﬀerential inclusion, fuzzy diﬀerential equation and stochastic diﬀerential equation are obtained using the reduction principle. 4. Design of the Self-Consistent -dimensional Lotka-Volterra Stochastic Models The synthesis of some models of population dynamics on the basis of the method of construction of self-consistent stochastic models [14] is implemented in [5,6]. According to the main idea of the method it is possible for the system under consideration to describe the scheme of interaction in the form of symbolic representation of all possible interaction between the system elements. The operators of the system state and the operator of change of the system state are used for this purpose. Then we give the intensities of transitions and master equation, for which we can obtain an approximate Fokker-Planck equation by the aids of formal series expansion. It is not diﬃcult to transit from the Fokker-Planck equation to the equivalent stochastic diﬀerential equation in Langevin form: where () ∈ is the vector of the system state, (, ()) is the vector of demolition, (, ()) is the diﬀusion matrix. In addition, in equation (8) ∈ is the standard -dimensional Wiener process. In practice, the stochastic diﬀerential equation can be written immediately after the representation of the interaction scheme. It is connected with the fact that for the obtained coeﬃcients of the Fokker-Planck equation it is necessary to know only the intensities of transitions and operators of state changes. We present the scheme of interaction elements for the system (1) in the form: where , = 1, . . . , . Thus, the scheme (9) describes the system of species in which individuals can interact ( + 1) various ways. The ﬁrst row of the scheme of interaction corresponds to natural reproduction -th species in the absence of other factors. The second row corresponds to intraspeciﬁc competition at = , corresponds to interspeciﬁc competition at ̸= . The operator of state change is presented in the form: The state of the system can be described by means of vector = ( 1 , . . . , ). The following relations are given for intensities of transitions from the state to the state + in the unit of the time: Let us present Fokker-Planck equation corresponding to the model in the form: We have the following relations for the coeﬃcients of stochastic equation (8) and coeﬃcients of Fokker-Planck equation: The obtaining of analytical solution for the constructed self-consistent stochastic Lotka-Volterra model is diﬃcult, however, for special cases of the general -dimensional model, it is possible to conduct a numerical experiment using the developed software package for the solution systems of stochastic diﬀerential equations [16, 17]. In the future, it is planned to conduct a numerical analysis to obtain numerical solutions for the obtained models, as well as to reveal the inﬂuence of stochastics on the behavior of the system. The investigation of the obtained stochastic diﬀerential equation in Langevin form allows us to study the inﬂuence of stochastics on the behavior of the considered system. This approach to the construction and analysis of nonlinear models can serve to the solving of problems aimed at the comparative analysis of deterministic and stochastic models. Note that the technique of design of the self-consistent stochastic models can be applied to the system (2), but for this it is necessary to specify the form of the functions ( ) taking into account the physical sense. In the problems of constructing of selfconsistent stochastic models that generalize models (1) and (2), the consideration of two-dimensional, three-dimensional and four-dimensional models is of prime interest, and it becomes necessary to compare stability properties in deterministic and stochastic cases. 5. Conclusions The principle of reduction allowed us to obtain the conditions of stability of the multidimensional model of the population dynamics with the transition to the diﬀerential inclusion, fuzzy and stochastic diﬀerential equations. The speciﬁed transition takes into account the changing parameters of the model and allows us to perform a comparative analysis of the properties of the models based on the principle of reduction. The stability conditions can be used to study the population dynamics models. The application of selfconsistent stochastic models construction method for Lotka-Volterra multidimensional systems allows us to estimate the impact of the introduction of stochastics onto the behavior of these systems.The obtained results are aimed at further development of methods of design and stability analysis of stochastic models.

# About the authors

### A V Demidova

Department of Applied Probability and Informatics Peoples’ Friendship University of Russia (RUDN University)
**Author for correspondence.**

Email: demidova_av@rudn.university

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

### O V Druzhinina

Federal Research Center “Computer Science and Control” of RAS; V. A. Trapeznikov Institute of Control Sciences of RAS
Email: ovdruzh@mail.ru

40 Vavilov Str., Moscow, 119333, Russian Federation; 65 Profsoyuznaya St., Moscow, 117997, Russian Federation

### O N Masina

Yelets State Ivan Bunin University
Email: olga121@inbox.ru

28 Communards St., Yelets, 399770, Russian Federation

# References

- Y. A. Pykh, Equilibrium and Stability in Models of Population Dynamics, Nauka, Moscow, 1983, in Russian.
- O. V. Druzhinina, O. N. Masina, Methods of Stability Research and Controllability of Fuzzy and Stochastic Dynamic Systems, Dorodnicyn Computing Center of RAS, Moscow, 2009, in Russian.
- O. V. Druzhinina, O. N. Masina, E. V. Igonina, Stability Research of Equilibrium States of the Ecological Equations by the Index-Divergent Method. Qualitative Properties, Asymptotics and Stabilization of Nonlinear Dynamical Systems, Publishing house of the Mordovian University, Saransk, 2010, pp. 105–111, in Russian.
- O. V. Druzhinina, O. N. Masina, A. V. Shcherbakov, Structure and qualitative analysis of mathematical models of population dynamics in the presence of mutualism, Nonlinear World 14 (6) (2016) 32–42, in Russian.
- V. Demidova, O. V. Druzhinina, O. N. Masina, Stability Research of Population Dynamics Model on the Basis of Construction of the Stochastic Self-Consistent Models and the Principle of the Reduction, Bulletin of Peoples Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (3) (2015) 18–29, in Russian.
- V. Demidova, O. V. Druzhinina, M. Jacimovic, O. N. Masina, Construction and Analysis of Nondeterministic Models of Population Dynamics, Vol. 678, Springer, Cham, 2016, pp. 498–510.
- O. V. Druzhinina, O. N. Masina, System Approach to Stability Research of the Models Described by the Diﬀerential Equations of Diﬀerent Types, Bulletin of the Russian Academy of Natural Sciences. Diﬀerential Equations. (3) (2015) 24–30, in Russian.
- Y. Katz, N. N. Krasovskii, On the Stability of Systems with Random Parameters, Journal of Applied Mathematics and Mechanics 24 (5) (1960) 809–823.
- F. Kozin, Stability of the Linear Stochastic Systems, Lecture Notes in Mathematics 294 (1972) 189–192.
- A. Shestakov, Generalized Direct Lyapunov Method for Systems with Distributed Parameters, URSS, Moscow, 2007, in Russian.
- Yu. N. Merenkov, Stability-Like Properties of Diﬀerential Inclusions, Fuzzy and Stochastic Diﬀerential Equations, PFUR, Moscow, 2000, in Russian.
- O. N. Masina, On the existence of solutions of diﬀerential inclusions, Diﬀerential Equations 44 (6) (2008) 845–847, in Russian.
- P. Pavlotsky, V. M. Suslin, Stochastic model of evolution of populations in space, Mathematical modeling 6 (3) (1994) 9–24, in Russian.
- V. Demidova, M. N. Gevorgyan, A. D. Egorov, D. S. Kulyabov, A. V. Korolkova, L. A. Sevastyanov, Inﬂuence of Stochastization on One-Step Models, Bulletin of Peoples Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (1) (2014) 71–85.
- V. Korolkova, E. G. Eferina, E. B. Laneev, I. A. Gudkova, L. A. Sevastianov, D. S. Kulyabov, Stochastization of One-Step Processes in the Occupations Number Representation, in: Proceedings 30th European Conference on Modelling and Simulation, Springer, Regensburg, Germany, 2016, pp. 698–704. doi: 10.7148/2016- 0698. URL http://www.scs-europe.net/dlib/2016/2016-0698.htm
- M. N. Gevorkyan, T. R. Velieva, A. V. Korolkova, D. S. Kulyabov, L. A. Sevastyanov, Stochastic Runge–Kutta Software Package for Stochastic Diﬀerential Equations, Dependability Engineering and Complex Systems (470) (2016) 169–179.
- E. G. Eferina, A. V. Korolkova, M. N. Gevorkyan, D. S. Kulyabov, L. A. Sevastyanov, One-Step Stochastic Processes Simulation Software Package, Bulletin of Peoples Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (3) (2014) 46–59.