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Mathematical Modelling of Natural Phenomena (1)

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Replicator Equations and Models of Biological Populations andCommunities
G. P. Karev, I. G. Kareva
Journal:

Mathematical Modelling of Natural Phenomena / Volume 9 / Issue 3 / 2014

Published online by Cambridge University Press:

28 May 2014, pp. 68-95

Print publication:

2014
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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.