How to solve replicator equations?
by
Georgy P. Karev
NCBI, NIH

We suggest a general approach to a wide class of replicator equations (RE) and corresponding self-regulated selection systems, which mathematically are systems of differential equations in Banach space. The developed theory gives methods for reducing complex inhomogeneous selection models to the “escort systems” of ODEs that, in many cases, can be explored analytically or solved numerically with high precision. It allows us to compute explicitly the evolution of distributions, which solve the RE, and all statistical characteristics of interest of the system. The considered examples show how different can be the global dynamics of a selection system depending on the initial distribution even for the same dynamical model. The approach was applied to a general inhomogeneous logistic equation; implicit solutions to its particular versions used in the literature were derived. Similar approach to discrete-time models was also developed. The methods were applied to models in global demography, ecology, cancer modeling, epidemiology, etc. and have a potential for different other applications.

PDF

Date received: February 15, 2008