^{**}=7/3ψ

# Hyper-exponential increase

Natural selection generates a hyper-exponential increase in unchecked populations

The population dynamic base was laid down by Malthus in 1798 when he proposed that unchecked populations increase exponentially in numbers

_{t} = N_{0}e^{r t}

where

But recent studies have found that evolutionary changes and population dynamics can occur on similar time-scales (Thompson, 1998; Hairston et al., 2005; Saccheri and Hanski, 2006; Schoener, 2011). And with the exponential growth rate being the average fitness of the population it is the trait that is most directly affected by natural selection.

When conditions are optimal and there is no interference competition (Witting, 2000, 2002), Fisher’s fundamental theorem of natural selection (Fisher, 1930) predicts that the rate of change in average fitness

^{2}

is equal to the genetic variance (σ^{2}) in fitness so that

_{t} = r_{0}^{2}

With

_{0}^{2}

and when solved for abundance we find that the population increases hyper-exponentially in time

_{t} = N_{0} e^{ r0 t}^{2}^{2}

as illustrated by the

The fundamental theorem may thus be seen to replace the Malthusian law as the limit theorem of population dynamics; at least when the responses to selection is caused by additive genetic variation. More generally, we may think of σ^{2} as the total potential response to natural selection, a response that may include other factors like epigenetic inheritance, maternal effects, and plastic phenotypic responses.

### References

- Fisher, R.A. 1930. The genetical theory of natural selection. Clarendon, Oxford.
- Hairston, N. G.J., S.P. Ellner, M.A. Geber, T.Yoshida and J.A. Fox 2005. Rapid evolution and the convergence of ecological and evolutionary time. Ecology Letters 8:1114--1127.
- Malthus, T.R. 1798. An essay on the principle of population. Johnson, London.
- Saccheri, I., and I.Hanski 2006. Natural selection and population dynamics. Trends in Ecology and Evolution 21:341--347.
- Schoener, T.W. 2011. The newest synthesis: understanding the interplay of evolutionary and ecological dynamics. Science 331:426--429.
- Thompson, J.N. 1998. Rapid evolution as an ecological process. Trends in Ecology and Evolution 13:329--332.
- Witting, L. 2000. Interference competition set limits to the fundamental theorem of natural selection. Acta Biotheoretica 48:107--120.
- Witting, L. 2002. Two contrasting interpretations of Fisher's fundamental theorem of natural selection. Comments on Theoretical Biology 7:1--10.