By elementary methods of combinatorial mathematics and uniqueness of solutions systems of linear algebraic equations for non degenerate cases proved a theorem about the number and structure of the singular points of n-dimensional dynamical system of population a dynamics Lotka-Volterra model. Showed that the number of singular points for this system is equal to 2 and their structure on a combination of zero nand nonzero coordinates coincides with the binomial coefficientsyesBelgorod State Universit

Lesovik, R. V.

Merkulov, S. I.

Moskovkin, V. M.

Suleiman, B. N. E.

DSpace at Belgorod State University

World Applied Sciences Journal 25 (12): 1751-1753, 2013ISSN 1818-4952© IDOSI Publications, 2013DOI: 10.5829/idosi.wasj.2013.25.12.7081Corresponding Author: Vladimir M. Moskovkin, Belgorod State University, Russia, 308015, Belgorod, Av. Pobedi 85.1751Theorem about the Number and Structure of the Singular Points N-Dimensional Dynamical System of Population Dynamics Lotka-Volterra in Context of Informational Analysis and ModelingVladimir M. Moskovkin, Sergei I. Merkulov, Bilal N.E. Suleiman and Ruslan  V. Lesovik1 2 1 3Belgorod State University, Russia, 308015, Belgorod, Av. Pobedi 851Kursk State University, Russia, 305000, Kursk, Radisheva Street, 332Belgorod State Technological University named after V.G. Shukhov,3Russia, 308012, Belgorod, Kostyukov Street, 46Submitted: Sep 28, 2013;    Accepted: Nov 1, 2013;    Published: Nov 8, 2013Abstract: By  elementary  methods  of  combinatorial mathematics and uniqueness of solutions systems oflinear algebraic equations for non degenerate cases proved a theorem about the number and structure of thesingular points of n-dimensional dynamical system of population a dynamics Lotka-Volterra model. Showedthat  the  number of singular points for this system is equal to 2  and their structure on a combination of zeronand nonzero coordinates coincides with the binomial coefficients.Key words: Lotka-Volterra’s model  Population dynamics  Number of singular points  Binomial coefficients Solution systems of linear algebraic equationsINTRODUCTION modeling of the  interactions  of  different  types  of R  &A multidimensional model of population dynamics Lotka-Volterra  m odel, in our opinion, can be used in thewas proposed by Vito Volterra in [1, 2], but since it was simulation of competitive interactions n scientific frontsLotka who developed this type of parallel equation in within the broad area of research in which a variety ofbiophysical and chemical kinetics [3], the equations of variants for the suppression of some scientific fronts bypopulation dynamics bears the names of both scholars. the others, as well as their co-existence, will be observed.This model attracted attention of various great scientists, The following will state and prove a theorem on thee.g. G. Nicolis and Prigogine [4] R. May [5], etc. number  and  structure of   the     singular  points  of  theConsidering this model the scholars mainly studied the n-dimensional Lotka-Volterra model.stability of a non-trivial of the singular point. For example,B. Goh [6] in the study of  models of mutualism shows Main Partthat a  necessary  and sufficient condition for the local Theorem: The  number  of singular points of the n-and global stability of a non-trivial singular point of the dimensional system of nonlinear ordinary differentialLotka-Volterra model is the positivity of all the top (main) equations of  Lotka-Volterra  with positive coefficientsminors of the Jacobi matrix for this model. Later, Z. Lu and and  non-degenerate  cases  of systems of linear algebraicE. Takeuchi [7] proved a number of theorems on the equations which arise in the determination of theglobal stability of Lotka-Volterra equations. In papers on coordinates of singular points equals 2  and theireconomic  dynamics   [8,   9]   it  was  observed  that  the structure with respect to the zero and non-zeron-dimensional  system  of  equations of population coordinates coincides with binomial coefficients.dynamics of Lotka-Volterra has 2  singular points, but stillnno evidence for that was provided. The possibility of Proof: Let us consider a system of Lotka-Volterrausing of such equations in the information analysis and equations in the formD  results  is  shown  in  [10].  The   original n-dimensionaln1nii i ij jjdx x a xdt=  = −  ∑1 1 11 1 12 2 ... 1 12 1 21 1 22 2 ... 2 21 1 2 2 ...1 1 2 2 ...( ... ) 0( ... ) 0( ... ) 0( ... ) 0i i n ni i n ni i i i ii i in nn n n n ni i nn nx x x x xx x x x xx x x x xx x x x x− −− −− −− −− − − − = − − − − = − − − − =− − − − =1nnnC nn = = −  0 1 00nnC = = −  1`Cn2cnicn1nc nn− =( )( )0 1 2... ...0 1 2 1... 1 1 2 ,,i nn n n n ninn n n n n nNi n nC C C C Cn nC n ii n i i           = + + + + + + + =           −           ′′ ′′+ + + + + = + = = = ≥  − World Appl. Sci. J., 25 (12): 1751-1753, 20131752(1)For convenience, the proof of the theorem the rightsides of this system equations will be rewritten equal tozero, as follows:(2) Thus the total number of the singular points is equalWe will consider the cases of non-degenerate solutions (non-degenerate cases).solutions of systems of  linear algebraic equations, eachof them has only unique solution. CONCLUSIONFrom (2), two singular points are immediatelyidentified-the zero and non-trivial (non-zero), the latter is For an n-dimensional system of equations ofthe solution of the n-dimensional system of linear population dynamics, introduced by V. Volterra andalgebraic equations in the parentheses of the original A.Lotka in the middle of 1920s, the theorem about thesystem (2). From the point of view of combinatorial number and structure of the singular points of themathematics, these singular points correspond to the classical system of equations wasn’t proven. In thisfollowing combinations: paper,  this  theorem  was proved by elementary methodssolutions of systems  of  linear algebraic equations forof zeros from n variables; processes,  it  should  be  noted that the dynamicalcompetitive interactions in scientific fronts within theof zeros from n variables. actions, where 2  - 2 variants will be associated with theIn the first case we have only one zero singular point, competitive ones.in the second-only one non-zero singular point.Further, the number of singular points with a REFERENCEScombination of one zero coordinates of n variables isequal to , the number of singular points with a 1. Volterra, V., 1927. Variazioni  e  fluttuazioni delcombination of two zero coordinates of n variables isequal to , the number of singular points with acombination i zero coordinates of n variables is equal to, the number of singular  points  with a combination of(n-1) zero coordinates of n variables equals to .Consequently, the total number of singular points isequal to2 and their  structure with respect to the zero and non-nzero coordinates of repeats the consequent set ofcoefficients of Newton's binomial.In  this  proof  the  following statement is meant.When we take all of the singular points with zerocoordinates in the amount i, the rest of the system oflinear algebraic equations of the (n-i)-order are the uniqueof  combinatorial  mathematics and uniqueness ofnon-degenerate  cases. From the view point of informationanalysis and modeling of information systems andsystem (1) basically can simulate the process ofbroad field of scientific research. Then in such system wecan observe 2  variants of outcome of such interactionsnnsuppression of some scientific fronts by other morenumero d’individui in specie animali conviventi”Memozie del R.Comitato talassografico italiano”,Mem. CXXXI, pp: 209.2. Volterra, V., 1928. Variations and fluctuations of thenumber of individuals in animal species livingtogether. J. Cons. Int. Explor. Mer., pp: 49.3. Lotka, A.J., 1925. Elements of Physical Biology.Williams and Wilkins.World Appl. Sci. J., 25 (12): 1751-1753, 201317534. Nicolis, G. and I. Prigogine, 1979. Self-organization in 9. Moskovkin, V.M.,       A.V.        Zhuravka      andnonequilibrium systems: from dissipative structures V.S. Mikhailov, 2004. Calculation of competitiveto order in  terms  of  fluctuations. Academic Press, scenarios, cooperative  and  mixed strategies for thepp: 512. n-dimensional model of the competitive-cooperative5. May, R.M., 1976. Simple Mathematical Models with interactions in the socio-economic Systems.Very Complicated Dynamics. Nature, 261: 459-467. Economic  Cybernetics   (Donetsk),  (in  Russian)M,6. Goh, B.S., 1979.  Stability  in  models of mutualism. 5-6(29-30): 32-34.The American Naturalist, 113(2): 261-274. 10. Moskovkin,  V.M.,   N.E.     Suleiman    Bilal   and7. Lu, Z. and Y. Takeuchi,  1994. Qualitative Stability N.A. Golikov, 2011. A mathematical model ofand Global Stability for Lotka-Volterra Systems. interaction of the results of different R & D type.Journal of Mathematical Analysis and Applications, Automation Documentation and Mathematical182(1): 260-268. Linguistics, 45(1): 13-17.8. Moskovkin, V.M. and A.V. Zhuravka, 2002.Simulation of competitive cooperation interactions:(context equations of population dynamics in thesocio-economic systems). Business Inform(Kharkov), (in Russian), 5-6: 27-34.

Theorem about the number and structure of the singular points n-dimensional dynamical system of population dynamics Lotka-Volterra in context of informational analysis and modeling

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Theorem about the number and structure of the singular points n-dimensional dynamical system of population dynamics Lotka-Volterra in context of informational analysis and modeling

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