Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T07:24:18.299Z Has data issue: false hasContentIssue false
  • English
  • Français

Replicator Equations and Models of Biological Populations andCommunities

Published online by Cambridge University Press:  28 May 2014

G. P. Karev  and
I. G. Kareva
G. P. Karev*
Affiliation:
National Center for Biotechnology Information, National Institute of Health, Bldg. 38A, Rm. 5N511N, 8600 Rockville Pike, Bethesda, MD 20894, USA
I. G. Kareva
Affiliation:
Newman-Lakka Institute for Personalized Cancer Care, Floating Hospital for Children, Tufts Medical Center, 75 Kneeland St., Boston, MA, 02111
*
⋆Corresponding author. E-mail: karev@ncbi.nlm.nih.gov
Get access

Abstract

An overview of a general approach for mathematical modeling of evolving heterogeneouspopulations using a wide class of selection systems and replicator equations (RE) ispresented. The method allows visualizing evolutionary trajectories of evolvingheterogeneous populations over time, while still enabling use of analytical tools ofbifurcation theory. The developed theory involves introducing escort systems of auxiliary“keystone" variables, which reduce complex multi-dimensional inhomogeneous models tolow dimensional systems of ODEs that in many cases can be investigated analytically. Inaddition to a comprehensive theoretical framework, a set of examples of the method’sapplicability to questions ranging from preventing the tragedy of the commons to cancertherapy is presented.

Keywords

population heterogeneityreplicator equationselection systemtragedy of the commonscancer
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 3: Biological evolution , 2014 , pp. 68 - 95
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexandrova, R.. Tumour heterogeneity. Exper. Path. & Paras., 4 (2001), 57-67. Google Scholar
A. D. Bazykin. Nonlinear dynamics of interacting populations. World Scientific, 1998.
Berezovskaya, F., Karev, G., Snell, T. W.. Modeling the dynamics of natural rotifer populations: phase-parametric analysis. Ecol. Complex., 2 (2005), 395-409. CrossRefGoogle Scholar
F. Brauer, C. Castillo-Chavez. Mathematical models in population biology and epidemiology. Springer, 2011.
A. Cintron-Arias, F. Sanchez, X. Wang, C. Castillo-Chavez, D. M. Gorman, P. J. Gruenewald. The role of nonlinear relapse on contagion amongst drinking communities. In: Chowell, G., Hayman, J.M., Bettencourt, L.M.A., Castillo-Chavez, C. (Eds.), Mathematical and statistical estimation approaches in epidemiology, Springer Netherlands, (2009), 343-360.
Dietz, T., Ostrom, E., Stern, P. C.. The struggle to govern the commons. Science, 302 (2003), 1907–1912. CrossRefGoogle ScholarPubMed
W. Feller. An introduction to probability theory and its applications. Vol. 2. John Wiley & Sons, 2008.
R. Fisher. The genetical theory of natural selection: a complete variorum edition. Oxford University Press, 1999.
Von Foerster, H., Mora, P. M., Amiot, L. W.. Doomsday: Friday, 13 November, ad 2026. Science, 132 (1960), 1291-1295. CrossRefGoogle Scholar
Gatenby, R. A., Vincent, T. L.. Application of quantitative models from population biology and evolutionary game theory to tumor therapeutic strategies. Mol. Can. Therap., 2 (2003), 919-927. Google ScholarPubMed
Gatenby, R. A., Gillies, R. J.. Why do cancers have high aerobic glycolysis? Nature Rev. Can., 4 (2004), 891-899. CrossRefGoogle Scholar
G. Gause. The struggle for existence. Courier Dover Publications, 2003.
A. N. Gorban. Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis, Nauka, Novosibirsk, 1984.
Gorban, A. N.. Selection theorem for systems with inheritance. Math. Mod. Nat. Phen., 2 (2007), 1-45. CrossRefGoogle Scholar
J. Hofbauer, K. Sigmund. Evolutionary games and population dynamics. Cambridge University Press, 1998.
Garrett, Hardin. The tragedy of the commons. Science, 162 (1968), 1243-1248. Google Scholar
Heppner, G. H., Miller, F. R.. The cellular basis of tumor progression. Int. Rev. Cytol., 177 (1997), 1-56. CrossRefGoogle ScholarPubMed
Hu, F.B., Willett, W. C., Li, T., Stampfer, M. J., Colditz, G. A., Manson, J. E.. Adiposity as compared with physical activity in predicting mortality among women. N. Eng. J. Med., 351 (2004), 2694-2703. CrossRefGoogle Scholar
Kapitza, S.. The phenomenological theory of world population growth. Physics-Uspekhi, 39 (1996), 57-71. CrossRefGoogle Scholar
S. Kapitza. Global Population Blow-Up and After: The Demographic Revolution and Information Society. Global Marshall Plan Initiative, 2006.
Karev, G. P.. Inhomogeneous models of tree stand self-thinning. Ecol. Mod., 160 (2003), 23-37. CrossRefGoogle Scholar
Karev, G. P.. Dynamics of inhomogeneous populations and global demography models. J. Biol. Sys., 13 (2005), 83-104. CrossRefGoogle Scholar
G. P. Karev, A.R. Burk. Analytical models of forest dynamics in stable environment. In: New developments in ecology research. Nova Science Publishers, New York, USA (2006), 29-97.
Karev, G. P., Novozhilov, A. S., Koonin, E. V.. Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics. Biol. Dir., 1 (2006), 19. CrossRefGoogle ScholarPubMed
Karev, G. P., Novozhilov, A. S., Berezovskaya, F. S.. Modeling the dynamics of inhomogeneous natural rotifer populations under toxicant exposure. Ecol. Mod., 212, (2008), 80-85. CrossRefGoogle Scholar
Karev, G. P.. Inhomogeneous maps and mathematical theory of selection. Journal of Difference Equations and Applications, 14.1 (2008), 31-58. CrossRefGoogle Scholar
Karev, G. P.. On mathematical theory of selection, continuous time population dynamics. J. Math. Bio., 60 (2010), 107-129. CrossRefGoogle ScholarPubMed
Karev, G. P.. Principle of minimum discrimination information and replica dynamics. Entropy, 12 (2010), 1673-1695. CrossRefGoogle Scholar
Karev, G. P., Novozhilov, A. S., Berezovskaya, F. S.. On the asymptotic behaviour of the solutions to the replicator equation. Math. Med. & Biol., 28 (2011), 89-110. CrossRefGoogle ScholarPubMed
Kareva, I.. Prisoner’s Dilemma in Cancer Metabolism. PloS one, 6.12 (2011), e28576. Google Scholar
Kareva, I., Berezovskaya, F., Castillo-Chavez, C.. Transitional regimes as early warning signals in resource dependent competition models. Math. Biosci., 240 (2012), 114-123. CrossRefGoogle ScholarPubMed
Kareva, I., Berezovskaya, F., Castillo-Chavez, C.. Myeloid cells in tumour - immune interactions. J. Biol. Dyn., 4 (2010), 315-327. CrossRefGoogle Scholar
Kareva, I., Morin, B., Karev, G.. Preventing the tragedy of the commons through punishment of over-consumers and encouragement of under-consumers. Bull. Math. Biol., 75, (2013), 565-588. CrossRefGoogle ScholarPubMed
G. F. Khilmi. Foundations of the Physics of the Biosphere, 1967.
Komarova, N.. Mathematical modeling of tumorigenesis: mission possible. Curr. Opin. Oncol., 17 (2005), 39-43. CrossRefGoogle ScholarPubMed
Krakauer, D.C., Page, K. M., Erwin, D. H.. Diversity, dilemmas, and monopolies of niche construction. Am. Nat., 173 (2009), 26-40. CrossRefGoogle ScholarPubMed
Kuang, Y., Nagy, J. D., Elser, J. J.. Biological stoichiometry of tumor dynamics: mathematical models and analysis. Disc. Cont. Dyn. Sys. Ser. B, 4 (2004), 221-240. Google Scholar
Y.A. Kuznetsov. Elements of applied bifurcation theory, Springer-Verlag, New York, 1995.
J. Liebig. Chemistry applications to farming and physiology, 1876.
McCormick, F.. Cancer specific viruses and the development of ONYX-015. Canc. Biol. & Ther., 2 (2003), 156-159. CrossRefGoogle Scholar
Novozhilov, A. S., Berezovskaya, F. S., Koonin, E. V., Karev, G. P.. Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models. Biol. Dir., 1 (2006), 18. CrossRefGoogle ScholarPubMed
Novozhilov, A. S.. On the spread of epidemics in a closed heterogeneous population. Math. Biosci., 215 (2008), 177-185. CrossRefGoogle Scholar
Novozhilov, A. S.. Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations. Math. Mod. Nat. Phen., 7 (2012), 147-167. CrossRefGoogle Scholar
Novozhilov, A. S.. Analysis of a generalized population predator-prey model with a parameter distributed normally over the individuals in the predator population. J. Comp. Sys. Sci. Int., 43 (2004), 378-382. Google Scholar
Novozhilov, A. S.. Heterogeneous Susceptibles-Infectives model: Mechanistic derivation of the power law transmission function. Dyn. Cont., Disc. & Impul. Sys., Ser. A, Math. Anal., 16 (2009), 136-140. Google Scholar
M. Nowak, R. M. May. Virus Dynamics: Mathematical Principles of Immunology and Virology: Mathematical Principles of Immunology and Virology. Oxford University Press, 2000.
Ostrom, E.. Coping with tragedies of the commons. Ann. Rev. Pol. Sci., 2 (1999), 493-535. CrossRefGoogle Scholar
Poletaev, I.A.. On mathematical models of elementary processes in biogeocoenosis. Problems of Cibernetics, 16 (1966), 171-190. Google Scholar
Snell, T.W., Serra, M.. Dynamics of natural rotifer populations. Hydrobio., 368 (1998), 29-35. CrossRefGoogle Scholar
Tsoularis, A., Wallace, J.. Analysis of logistic growth models. Math. Biosci., 179 (2002), 21-55. CrossRefGoogle ScholarPubMed
Wright, M. E., Chang, S. C., Schatzkin, A., Albanes, D., Kipnis, V., Mouw, T., Hurwitz, P., Hollenbeck, A., Leitzmann, M. F.. Prospective study of adiposity and weight change in relation to prostate cancer incidence and mortality. Cancer, 109 (2007), 675-684. CrossRefGoogle Scholar
V.V. Zagreev. Reference Book on All-Union Forest Growth Tables. Kolos, Moscow, 1992.
Zeldovich, K. B., Chen, P., Shakhnovich, B. E., Shakhnovich, E. I.. A first-principles model of early evolution: emergence of gene families, species, and preferred protein folds. PLoS Comp. Biol., 3 (2007), e139. CrossRefGoogle ScholarPubMed