m.r.Life ι**=7/3ψ

Hyperexponential growth

Natural selection generates hyperexponential growth in unchecked populations

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

Nt = N0er t

where N is abundance, r the exponential growth rate, and t is time (curve a in figure). Exponential increase, and basically all subsequent population dynamic models, assumes that evolutionary changes occur much slower than population dynamics so that the exponential growth rate is constant over time under optimal conditions.

As the exponential growth rate is the average fitness of the population it is the trait that is most directly affected by natural selection. This lead Witting to integrate the natural selection changes in the growth rate into the exponential equation, deducing hyperexponential growth in 1997.

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

d r / d t = σ2

is equal to the genetic variance (σ2) in fitness so that r increases linearly

rt = r0 + σ2 t .

With r being the per capita rate of increase, the change in population is

d N / d t = (r0 + σ2 t) N

and when solved for abundance we find that the population increases hyperexponentially in time

Nt = N0 e r0 t + σ2t2

as illustrated by the b curve in the figure.

The fundamental theorem is thus replacing the Malthusian law as the limit theorem of population dynamics; at least when the response to selection is caused by additive genetic variation. More generally, we should think of σ2 as the total potential response to natural selection, a response that includes other factors like epigenetic inheritance, social inheritance, maternal effects, and plastic phenotypic responses.

Fig. 1 Three examples of hyperexponential growth from Witting (2021). White-eyed vireo and black vulture from the North American Breeding Bird Survey (Sauer et al., 2017), and American marten from the Global Population Dynamic Database (2010).


There is plenty of evidence for evolutionary processes that operate on population dynamic timescales (see reviews by Thompson, 1998; Hairston et al., 2005; Saccheri and Hanski, 2006; Schoener, 2011). This includes a natural selection acceleration of the population dynamic growth rate of aphids by up to 40% over few generations at low densities (Turcotte et al., 2011a,b). And the evolution of faster-spreading Covid-19 variants (Halley et al., 2021; Kupferschmidt, 2021) generating hyperexponential growth curves (Baruah, 2020; Pavithran and Sujith, 2022).

Applying AIC model selection to 1200 population dynamic trajectories for birds and mammals, Witting (2021) found hyperexponential population models to be selected more than twice as often as exponential models.

Download publications

bioRxiv 2021.11.27.470201 (2021)Download

On selection-regulated population dynamics in birds and mammals

Comments on Theoretical Biology 7:1-10 (2002)Download

Two contrasting interpretations of Fisher's fundamental theorem of natural selection

Acta Biotheoretica 48:107-120 (2000)Download

Interference competition set limits to the fundamental theorem of natural selection

Bulletin of Mathematical Biology 62:1109-1136 (2000)Download

Population cycles caused by selection by density dependent competitive interactions

Peregrine Publisher, Aarhus (1997)Download

A general theory of evolution. By means of selection by density dependent competitive interactions.


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