# American Institute of Mathematical Sciences

• Previous Article
Spatial stochastic models of cancer: Fitness, migration, invasion
• MBE Home
• This Issue
• Next Article
On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth
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. 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. 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. 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. doi: 10.1142/S0218202508002796. [5] I. D. Chase, Models in hierarchy formation in animal societies,, Behavioural Sciences, 19 (1974), 374. 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. 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. [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. doi: 10.1016/j.jde.2008.11.006. [9] M. Lachowicz, Links between microscopic and macroscopic descriptions,, in, 1940 (2008), 201. 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. 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. 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. 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. 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. [15] E. O. Wilson, "Sociobiology: The New Synthesis,", Harvard University Press, (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. 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. 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. 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. doi: 10.1142/S0218202508002796. [5] I. D. Chase, Models in hierarchy formation in animal societies,, Behavioural Sciences, 19 (1974), 374. 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. 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. [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. doi: 10.1016/j.jde.2008.11.006. [9] M. Lachowicz, Links between microscopic and macroscopic descriptions,, in, 1940 (2008), 201. 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. 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. 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. 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. 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. [15] E. O. Wilson, "Sociobiology: The New Synthesis,", Harvard University Press, (1975).
 [1] Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541 [2] Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057 [3] Xu Chen, Jianping Wan. Integro-differential equations for foreign currency option prices in exponential Lévy models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 529-537. doi: 10.3934/dcdsb.2007.8.529 [4] Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [5] Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160 [6] Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191 [7] Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 [8] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [9] Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693 [10] Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119 [11] Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015 [12] Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053 [13] Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 911-923. doi: 10.3934/dcdss.2020053 [14] Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249 [15] Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677 [16] Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026 [17] Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517 [18] Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069 [19] Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741 [20] Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17

2017 Impact Factor: 1.23