Probabilistic Model for the Lotka-Volterra System with Cross-Diffusion
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Two approaches that allow to construct a probabilistic representation of a generalized solution of the Cauchy problem for a system of quasilinear parabolic equations are proposed. The system under consideration describes a population dynamics model for a prey-predator population. The stochastic problem associated with this parabolic system is presented in two forms, which give a way to derive the required probabilistic representation. Bibliography: 16 titles.
KeywordsCauchy Problem Stochastic System Parabolic System Distributional Sense Evolution Family
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- 2.G. Galiano, M. Garzron, and A. Jüngel, “Analysis and numerical solution of a nonlinear cross-diffusion model arising in population dynamics,” Rev. Real Acad. Ciencias, Serie A. Mat., 95, 281–295 (2001).Google Scholar
- 4.A. Jüngel, “Diffusive and nondiffusive population models,” in: Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (2010), pp. 397–425.Google Scholar
- 8.Ya. Belopolskaya and W. Woyczynski, “Generalized solutions of nonlinear parabolic equations and diffusion processes,” Acta Appl. Math., 96, 55–69 (2007).Google Scholar
- 9.Ya. Belopolskaya and W. Woyczynski, “Generalized solutions of the Cauchy problem for systems of nonlinear parabolic equations and diffusion processes,” Stoch. Dyn., 12, No. 1, 1–31 (2012).Google Scholar
- 10.Ya. Belopolskaya, “Markov processes associated with fully nondiagonal systems of parabolic equations,” Markov Process. Relat. Fields, 20, No. 3, 452–478 (2014).Google Scholar
- 11.Ya. Belopolskaya, “Probabilistic counterparts for strongly coupled parabolic systems,” in: Springer Proc. Math. Stat., 114 (2014), pp. 33–42.Google Scholar
- 12.E. Pardoux and S. Peng, “Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations,” in: Lecture Notes CIS, 176, Springer-Verlag (1992), pp. 200–217.Google Scholar
- 13.J. Fontbona and S. Meleard, “Non local Lotka–Volterra system with cross-diffusion in an heterogeneous medium,” J. Math. Biology, 20–39 (2014).Google Scholar
- 14.P. Protter, Stochastic Integration and Differential Equations, Springer (2010).Google Scholar
- 16.Ya. Belopolskaya, “Forward–Backward Stochastic Equations Associated with Systems of Quasilinear Parabolic Equations and Comparison Theorems,” J. Math. Sci., 204, No. 1, 7–27 (2015).Google Scholar