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2013, 10(3): 777-786. doi: 10.3934/mbe.2013.10.777

Equilibrium solutions for microscopic stochastic systems in population dynamics

1. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, 2, Banach Str., 02-097 Warsaw, Poland, Poland

Received  May 2012 Revised  October 2012 Published  April 2013

The present paper deals with the problem of existence of equilibrium solutions of equations describing the general population dynamics at the microscopic level of modified Liouville equation (individually--based model) corresponding to a Markov jump process. We show the existence of factorized equilibrium solutions and discuss uniqueness. The conditions guaranteeing uniqueness or non-uniqueness are proposed under the assumption of periodic structures.
Citation: MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777
References:
[1]

L. Arlotti and N. Bellomo, Solution of a new class of nonlinear kinetic models of population dynamics, Appl. Math. Lett., 9 (1996), 65-70. doi: 10.1016/0893-9659(96)00014-6.

[2]

L. Arlotti, N. Bellomo and M. Lachowicz, Kinetic equations modelling population dynamics, Transport Theory Statist. Phys., 29 (2000), 125-139. doi: 10.1080/00411450008205864.

[3]

N. Bellomo and G. Forni, Dynamics of tumor interaction with the host immune system, Math. Comput. Modelling, 20 (1994), 107-122. doi: 10.1016/0895-7177(94)90223-2.

[4]

N. Bellomo, N. K. Li and Ph.K. Maini, On the foundations of cancer modelling: Selected topics, speculations and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646. doi: 10.1142/S0218202508002796.

[5]

I. D. Chase, Models in hierarchy formation in animal societies, Behavioural Sciences, 19 (1974), 374-382. doi: 10.1002/bs.3830190604.

[6]

E. Jäger and L. Segel, On the distribution of dominance in a population of interacting anonymous organisms, SIAM J. Appl. Math., 52 (1992), 1442-1468. doi: 10.1137/0152083.

[7]

E. Geigant, K. Ladizhansky and A. Mogilner, An integro-differential model for orientational distribution of $F$-actin in cells, SIAM J. Appl. Math., 59 (1998), 787-809.

[8]

K. Kang, B. Perthame, A. Stevens and J. J. L. Valázquez, An integro-differential model for alignment and orientational aggregation, J. Diff. Eqs., 246 (2009), 1387-1421. doi: 10.1016/j.jde.2008.11.006.

[9]

M. Lachowicz, Links between microscopic and macroscopic descriptions, in "Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic" (eds. V. Capasso and M. Lachowicz), in Lecture Notes in Mathematics, 1940, Springer, (2008), 201-268. doi: 10.1007/978-3-540-78362-6.

[10]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Prob. Engin. Mech., 26 (2011), 54-60. doi: 10.1016/j.probengmech.2010.06.007.

[11]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Analysis Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014.

[12]

M. Lachowicz and D. Wrzosek, Nonlocal bilinear equations. Equilibrium solutions and diffusive limit, Math. Models Methods Appl. Sci., 11 (2001), 1375-1390. doi: 10.1142/S0218202501001380.

[13]

I. Primi, A. Stevens and J. J. I. Velázquez, Mass-selection in alignment model with non-deterministic effect, Comm. Partial Diff. Eqs., 34 (2009), 419-456. doi: 10.1080/03605300902797171.

[14]

G. Theraulaz, E. Bonabeau and J.-L. Deneubourg, Self-organization of hierarchies in animal societies: the case of the primitively eusocial wasp Polistes dominulusChrist, J. Theoret. Biol., 174 (1995), 313-323.

[15]

E. O. Wilson, "Sociobiology: The New Synthesis," Harvard University Press, Cambridge, 1975.

show all references

References:
[1]

L. Arlotti and N. Bellomo, Solution of a new class of nonlinear kinetic models of population dynamics, Appl. Math. Lett., 9 (1996), 65-70. doi: 10.1016/0893-9659(96)00014-6.

[2]

L. Arlotti, N. Bellomo and M. Lachowicz, Kinetic equations modelling population dynamics, Transport Theory Statist. Phys., 29 (2000), 125-139. doi: 10.1080/00411450008205864.

[3]

N. Bellomo and G. Forni, Dynamics of tumor interaction with the host immune system, Math. Comput. Modelling, 20 (1994), 107-122. doi: 10.1016/0895-7177(94)90223-2.

[4]

N. Bellomo, N. K. Li and Ph.K. Maini, On the foundations of cancer modelling: Selected topics, speculations and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646. doi: 10.1142/S0218202508002796.

[5]

I. D. Chase, Models in hierarchy formation in animal societies, Behavioural Sciences, 19 (1974), 374-382. doi: 10.1002/bs.3830190604.

[6]

E. Jäger and L. Segel, On the distribution of dominance in a population of interacting anonymous organisms, SIAM J. Appl. Math., 52 (1992), 1442-1468. doi: 10.1137/0152083.

[7]

E. Geigant, K. Ladizhansky and A. Mogilner, An integro-differential model for orientational distribution of $F$-actin in cells, SIAM J. Appl. Math., 59 (1998), 787-809.

[8]

K. Kang, B. Perthame, A. Stevens and J. J. L. Valázquez, An integro-differential model for alignment and orientational aggregation, J. Diff. Eqs., 246 (2009), 1387-1421. doi: 10.1016/j.jde.2008.11.006.

[9]

M. Lachowicz, Links between microscopic and macroscopic descriptions, in "Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic" (eds. V. Capasso and M. Lachowicz), in Lecture Notes in Mathematics, 1940, Springer, (2008), 201-268. doi: 10.1007/978-3-540-78362-6.

[10]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Prob. Engin. Mech., 26 (2011), 54-60. doi: 10.1016/j.probengmech.2010.06.007.

[11]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Analysis Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014.

[12]

M. Lachowicz and D. Wrzosek, Nonlocal bilinear equations. Equilibrium solutions and diffusive limit, Math. Models Methods Appl. Sci., 11 (2001), 1375-1390. doi: 10.1142/S0218202501001380.

[13]

I. Primi, A. Stevens and J. J. I. Velázquez, Mass-selection in alignment model with non-deterministic effect, Comm. Partial Diff. Eqs., 34 (2009), 419-456. doi: 10.1080/03605300902797171.

[14]

G. Theraulaz, E. Bonabeau and J.-L. Deneubourg, Self-organization of hierarchies in animal societies: the case of the primitively eusocial wasp Polistes dominulusChrist, J. Theoret. Biol., 174 (1995), 313-323.

[15]

E. O. Wilson, "Sociobiology: The New Synthesis," Harvard University Press, Cambridge, 1975.

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