Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Frequency-dependent models of population dynamics
Georgiy P Karev

Last modified: 2014-03-28


Consider a model of inhomogeneous population composed on clones each of which has its own reproduction rate a; dynamics of the clone densities l(t,a) is given by the equation

dl(t,a)/dt=l(t,a)P(t,a)                                  (1)

where P(t,a)=l(t,a)/N(t) is the current frequency of a-clone, and  N(t) is the total population size. Model (1) is called to F-model (after frequency-dependent model).

Here I present a method of solving the F-model based on a general approach developed in [1]. The dynamics of F-model crucially depends on the initial distribution P(0,a).  In particular, if P(0,a) is the Gamma-distribution, then the total population size solves the power equation suggested in [2]  for modeling of prebiological evolution:

dN/dt=kN^p                                                (2)

where p>0 is determined by the parameters of the Gamma-distribution .

The models (2) for p>1 (hyperbolic case)  and p<1 (parabolic case) demonstrate deviation from Darwinian “survival of the fittest” [2].

I prove that inhomogeneous model (1) demonstrates Darwinian “survival of the fittest” if and only if the moment generating function M(x) of the initial distribution of the reproduction rate is finite for all x. In other case the model demonstrates non-Darwinian “survival of everybody” in the sense that the ratio  0<l(t,a)/l(t,b) is bounded for all t, although the frequency of every clone tends to 0 with course of time. Evidently, only the first case corresponds to real situations because the reproduction rate should be bounded.

Acknowledgement: this research was supported by the Intramural Research Program of the NIH, NCBI.



  1. G. Karev. On mathematical theory of selection: continuous time population dynamics, Journal of Mathematical Biology 60, pp.107-129 (2010).
  2. E. Szathmary, M. Smith. From Replicators to Reproducers: the First Major Transitions Leading to Life, J. Theor. Biol., 187, pp. 555-571(1997).
  3. G. Karev. Non-Linearity and Heterogeneity in Modeling of Population Dynamics (submitted)


frequency dependent model; dynamics of distribution; non-Darwinian selection.