Stochastic competition between two populations in space

We present a model describing spatial competition between two biological populations. Individuals belonging to the two populations diffuse in space, reproduce, and die as effect of competitions; all these processes are implemented stochastically. We focus on how the macroscopic equations for the densities of the two species can be derived within the formalism of the chemical master equations. We also compare the case in which the total density of individuals is kept fixed by constraint with a case in which it can fluctuate.

[1]  F. Toschi,et al.  Particle algorithms for population dynamics in flows , 2011 .

[2]  R. Benzi,et al.  Population genetics in compressible flows. , 2011, Physical review letters.

[3]  W. Young,et al.  A master equation for a spatial population model with pair interactions. , 2006, Theoretical population biology.

[4]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

[5]  M. Kimura,et al.  The Stepping Stone Model of Population Structure and the Decrease of Genetic Correlation with Distance. , 1964, Genetics.

[6]  Cristóbal López,et al.  Clustering, advection, and patterns in a model of population dynamics with neighborhood-dependent rates. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  U. Dieckmann,et al.  POPULATION GROWTH IN SPACE AND TIME: SPATIAL LOGISTIC EQUATIONS , 2003 .

[8]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[9]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[10]  K. Korolev,et al.  Genetic demixing and evolution in linear stepping stone models. , 2010, Reviews of modern physics.

[11]  J. Ross,et al.  Enhanced (hydrodynamic) transport induced by population growth in reaction-diffusion systems with application to population genetics. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[12]  F. Toschi,et al.  Growth, competition and cooperation in spatial population genetics. , 2012, Theoretical population biology.

[13]  Charles R. Doering,et al.  Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality , 2003 .

[14]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .