{"title":"Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model","authors":"Vladislav N. Dumachev, Vladimir A. Rodin","volume":31,"journal":"International Journal of Computer and Information Engineering","pagesStart":1803,"pagesEnd":1806,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/12841","abstract":"By utilizing the system of the recurrence equations, containing two parameters, the dynamics of two antagonistically interconnected populations is studied. The following areas of the system behavior are detected: the area of the stable solutions, the area of cyclic solutions occurrence, the area of the accidental change of trajectories of solutions, and the area of chaos and fractal phenomena. The new two-dimensional diagram of the dynamics of the solutions change (the fractal cabbage) has been obtained. In the cross-section of this diagram for one of the equations the well-known Feigenbaum tree of doubling has been noted.Keywordsbifurcation, chaos, dynamics of populations, fractals","references":" [1] Computer and nonlinear phenomena. M.: Nauka, 1988. \r\n[2] E.I.Skaletskaja, E.J.Frisman, A.P.Shapiro, Discrete models of the dynamics of numbers of populations and craft optimisation. M.: Nauka. 1979. \r\n[3] A.P.Shapiro, About stability of cannibalistic populations.Mathemematical modelling of populational ecological processes. Vladivostok, 1987, p. 106-112.\r\n[5] A.P.Kuznetsov, S.P.Kuznetsov, Critical dynamics of one-dimensional displays. Izvestie vuzov. PND, 1993. Vol. \r\n[6] S.P.Kuznetsov, Dynamics of two single-directed and connected Feigenbaum systems at a hyperchaos threshold. Analysis of the renormalization group. Izvie vuzov. Radio physics. 1990. Vol.33. 7.\r\n[7] V.I. Arnold, A.N. Varchenko, S.M. Gusein-zade, Features of differentiated mapping. Classification of critical points of caustic and wave fronts. M.: Nauka. 1982.\r\n[8] T. Brecar, L. Lender, Differentiated sprouts and accidents. M.: ir,1984.\r\n[9] V.A. Rodin, V.N. Dumachev, Evolution of antagonistically interacting populations based on the two-dimensional Verhulst-Pearl model. Mathematical modeling. 2005. Vol.17, p. 11-22.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 31, 2009"}