Hyperexponential growth

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

N_t = N₀e^{r 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

r_t = 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

N_t = 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).

Evidence

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.

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References

• Baruah, H.K. 2020. Hyper-exponential growth of COVID-19 during resurgence of disease in Russia. Preprint at medRxiv https://dx.doi.org/10.1101/2020.10.26.20219626.
• Fisher, R.A. 1930. The genetical theory of natural selection. Clarendon, Oxford.
• GPDD 2010. The Global Population Dynamics Database v2.0. NERC Centre for Population Biology, Imperial College, http://www.sw.ic.ac.uk/cpb/cpb/gpdd.html.
• 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.
• Halley, J.M., D.Vokou, G.Pappas and I.Sainis 2021. SARS-CoV-2 mutational cascades and the risk of hyper-exponential growth. Microbial Pathigenesis 161:https://doi.org/10.1016/j.micpath.2021.105237.
• Kupferschmidt, K. 2021. Fast-spreading U.K. virus variant raises alarm. εm Science 371:9--10.
• Malthus, T.R. 1798. An essay on the principle of population. Johnson, London.
• Pavithran, I., and R.I. Sujith 2022. Extreme COVID-19 waves reveal hyperexponential growth and finite-time singularity. Chaos 32:041104.
• Saccheri, I., and I.Hanski 2006. Natural selection and population dynamics. Trends in Ecology and Evolution 21:341--347.
• Sauer, J.R., D.K. Niven, J.E. Hines, D.J. Ziolkowski, K.L. Pardieck, J.E. Fallon and W.A. Link 2017. The North American Breeding Bird Survey, Results and analysis 1996 -- 2015. Version 2.07.2017. USGS Patuxent Wildlife Research Center, Laurel, Maryland, Available at www.mbr-pwrc.usgs.gov/bbs/bbs.html.
• 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.
• Turcotte, M.M., D.N. Reznick and J.D. Hare 2011a. Experimental assessment of the impact of rapid evolution on population dynamics. Evolutionary Ecology Research 13:113--131.
• Turcotte, M.M., D.N. Reznick and J.D. Hare 2011b. The impact of rapid evolution on population dynamics in the wild: experimental test of eco-evolutionary dynamics. Ecology Letters 14:1084--1092.
• Witting, L. 1997. A general theory of evolution. By means of selection by density dependent competitive interactions. Peregrine Publisher, Århus, 330 pp, URL https://mrLife.org.
• Witting, L. 2000. Interference competition set limits to the fundamental theorem of natural selection. Acta Biotheoretica 48:107--120, https://doi.org/10.1023/A:1002788313345.
• Witting, L. 2002. Two contrasting interpretations of Fisher's fundamental theorem of natural selection. Comments on Theoretical Biology 7:1--10.
• Witting, L. 2021. Selection-regulated population dynamic in birds and mammals. Preprint at bioRxiv https://dx.doi.org/10.1101/2021.11.27.470201.