A Mathematical Model of Biological Evolution

In order to understand generally how the biological evolution rate depends on relevant parameters such as mutation rate, intensity of selection pressure and its persistence time, the following mathematical model is proposed: dNn(t)/dt = (mn(t) - mu)Nn(t) + muNn-1(t) (n = 0,1,2,3,...), where Nn(t) and mn(t) are respectively the number and Malthusian parameter of replicons with step number n in a population at time t and mean is the mutation rate, assumed to be a positive constant. The step number of each replicon is defined as either equal to or larger by one than that of its parent, the latter case occurring when and only when mutation has taken place. The average evolution rate defined by v infinity identical to lim t leads to infinity sigma infinity n = o nNn(t)/t sigma infinity n = o Nn(t) is rigorously obtained for the case (i) mn(t) = mn is independent of t (constant fitness model), where mn is essentially periodic with respect to n, and for the case (ii) mn(t) = s(-1) n+[t/tau] (periodic fitness model), together with the long time average -m infinity of the average Malthusian parameter -m identical to sigma infinity n = o mn(t)Nn(t)/sigma infinity n = o Nn(t). The biological meaning of the results is discussed, comparing them with the features of actual molecular evolution and with some results of computer simulation of the model for finite populations.