m.r.Life ι**=7/3ψ

Selection-regulated dynamics

Selection-regulated dynamics provide a single-species mechanism for the population cycles that have fascinated biologists for decades

Fig. 1 Projections of selection-regulated dynamics. Left: The dynamics of an introduced species with strongly damped dynamics (solid curve is abundance, and dashed curve is carrying capacity). Middle: Stable cycles in abundance (solid curve) and carrying capacity (dashed curve) following a perturbation. Right: A damped cycle in abundance (solid curve) and body mass (dashed curve) following a perturbation. All horizontal lines are evolutionary equilibria. From Witting (2000, 2013).

Population dynamic modelling have typically assumed that population dynamic growth is density-regulated

Nt+1 = Nt λm f(Nt)

where Nt is the abundance at time t, λm > 1 the maximal growth rate, and f(N) the density-regulation function that declines monotonically from one to zero as N increases from zero to infinity.

Selection-regulation was integrated into the model of density-regulated growth by Witting in 1997. Starting from the natural selection attractor of the life history, i.e., from the competitive interaction fixpoint, the population dynamic feed-back selection of interactive competition defines the equilibrium abundance of the population (N*). When the abundance and life histories that are naturally selected are allowed to be perturbed away from the evolutionarily determined population dynamic equilibrium, we obtain selection-regulated dynamics

Nτ+1 = Nτ λτ (Nτ / N*)

where the per generation (τ) growth rate

λτ = λτ-1 (Nτ-1 / N*)q

is accelerated when then abundance is below the equilibrium [ N < N* ] and decelerated when the abundance is above [ N > N* ] (Witting, 1997, 2000).

The resulting dynamics is cyclic, with population cycles that are either damped (Fig. 1, left), stable (Fig. 1, middle), or unstable. As these cycles are generated by changes in the intrinsic growth rate, they are associated with cyclic changes in the carrying capacity (as defined traditionally by density regulated growth, Fig. 1, left and middle) and life history traits like body mass (Fig. 1, right).

Under selection-regulated dynamics, it is no longer possible to determine the per capita growth rate, but only the acceleration of the growth rate, as a function of the density dependent environment. Traditional ecological thinking based on density regulated growth assumes that given environmental conditions (including density and inter-specific interactions) define a specific growth rate for a population. But selection-regulated dynamics implies that a population can have a large, if not infinite, number of growth rates, often with opposite signs, associated with the same environmental conditions. This conceptual change was proposed by Ginzburg in 1972 from analogy to Newton's laws of motion (Newton, 1687).


Selection-regulated dynamics predict the cyclic dynamics that is widespread in natural populations. This is illustrated in Fig. 2, where the dynamics of the larch budmoth in the Upper Engadine valley is shown together with a projection of a selection-regulated model.

Fig. 2 The curve is a projection of a selection-regulated model with discrete generations and the diamonds the yearly densities of the lack budmoth in the Upper Engadine valley from 1950 to 1985. Data from Baltensweiler and Fischlin (1988); figure from Witting (2000).

But how widespread is selection-regulated dynamics compared to density-regulated growth across natural populations? By applying AIC model-selection to 462 population trajectories from the North American Breeding Bird Survey (Sauer et al. 2017), selection-regulated dynamics was found to be 25,000 times more probable than density-regulated growth (Witting, 2021). Selection was essential in 94% of the best models explaining 82% of the population dynamics variance across the North American continent. Similar results were obtained for 111 populations of British birds (BTO 2020), 215 populations of Danish birds (DOF 2020), and 420 populations of birds and mammals in the Global Population Dynamic Database (GPDD 2010). Three examples of selection-regulated dynamics are shown in Fig. 3.

Fig. 3 Three examples of selection-regulated dynamics from Witting (2021). Yellow-breasted chat and painted bunting from the North American Breeding Bird Survey (Sauer et al., 2017), and American beaver from the Global Population Dynamic Database (2010). The blue inserts are n(t+1)-n(t) plots, with grey lines being the selection gradients that cause the acceleration and deceleration of the growth rate. Red lines are the equilibrium abundance, which is allowed a linear change in some models.

When tested against density-regulated growth and predator-prey dynamics, selection-regulated dynamics was the only sufficient mechanism to explain the long-term dynamics of baleen whales (Witting, 2013), with predicted trajectories of three species shown in Fig. 4.

Fig. 4 Estimated selection-regulated trajectories for three species of baleen whales, following the commercial whaling in past centuries (with catch histories shown by bars in bottom plots). Data obtained from the International Whaling Commission, with plots generated from online simulations at mrLife.org.

For the simple case of non-overlapping generations and stable population cycles there is a theoretical relationship between the cycle period in generations and the response parameter (γ) of the selection-regulated model (Witting, 1997). This relation is shown in the left plot of Fig. 5 together with empirically observed periods for forest insects with non-overlapping generations.

When observed as a function of mass, the predicted period in physical time will tend to increase to the 1/4, or 1/6, power of mass, because of the predicted body mass allometry for generation time. This increase is known as the Calder allometry (Calder, 1983), and it is shown for mammals and birds in the middle plot in Fig. 5.

Fig. 5 Left: The period of the population cycles in generations against the response parameter (γ) of selection-regulated cycles. The curve is defined by the population dynamic equations, and the numbered diamonds represent the following species: (1) spruce budworm, (2) southern pine beetle, (3) douglas-fir tussock moth, (4) larch budmoth, (5) fall webworm, (6) nun moth, (7) pine looper moth, (8) larch cone fly, and (9) wasp. Spp. Middle: The period of population cycles against body mass for terrestrial homoiotherms on double logarithmic scale, with the line being the regression line. Right: The dynamics in the density (N, solid curve) and body mass (w, dashed curve) of a Daphnia population against time. From Witting (1997, 2000).

Other evidence on the causal mechanism of population cycles includes observations on the presence versus absence of associated cycles in life history parameters like body mass. These life history cycles are predicted by selection-regulated dynamics, and they operate against the expectations of density regulation, in the sense that the largest body masses are predicted in the late peak phase of a cycle, where body masses should be smallest if controlled by density regulation.

The life history cycles that are predicted by selection-regulated dynamics are widespread in natural populations: Cycles in competitive quality occur in side-blotched lizard with selection-regulated dynamics (Sinervo et al., 2000), and the abundance cycle in the Daphnia experiments of Murdoch and McCauley (1985) had an associated cycle in body mass, with the larger Daphnia occurring mainly during the late peak phase of the cycle (Fig. 5, right). In fact, such body mass cycles appear to be the rule, rather than the exception, in species with cyclic population dynamics. They are widespread in voles and lemmings with cyclic dynamics (e.g., Chitty, 1952; Hansson, 1969; Krebs and Myers, 1974; Mihok et al., 1985; Stenseth and Ims; Norrdahl and Korpimäki, 2002; Lambin et al., 2006), and they have been observed in snowshoe hare (Hodges et al, 1999) and cyclic forest insects (Myers, 1990; Simchuk et al., 1999). Quite generally, it is observed that voles and lemmings are small, non-aggressive, and that they have a high reproductive rate when the abundance is low and increasing. When the abundance is high and declining, they are instead aggressive, and 20 to 30 percent larger with a delayed and low reproductive rate.

Download publications

bioRxiv 2021.11.27.470201 (2021)Download

On selection-regulated population dynamics in birds and mammals

Population Ecology 55:377-401 (2013)Download

Selection-delayed population dynamics in baleen whales and beyond

The Journal of Cetacean Research and Management 5:45-54 (2003)Download

Reconstructing the population dynamics of eastern Pacific gray whales over the past 150 to 400 years

Ecological Modelling 157:51-68 (2002)Download

Evolutionary dynamics of exploited populations selected by density dependent competitive interactions

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.


  • Baltensweiler, W., and A.Fischlin 1988. The larch budmoth in the alps. pp. 331--351, In: A. A. Berryman (ed.) Dynamics of forest insect populations. Patterns, causes, implications. Plenum Press, New York.
  • Calder, W. A.I. 1983. An allometric approach to population cycles of mammals. Journal of Theoretical Biology 100:275--282.
  • Chitty, D. 1952. Mortality among voles ( Microtus agrestis) at Lake Vyrnwy, Montgomeryshire in 1936-9. Philosophical Transactions of the Royal Society of London 236:505--552.
  • Ginzburg, L.R. 1972. The analogies of the ``free motion'' and ``force'' concept in population theory (in Russian). pp. 65--85, In: V. A. Ratnar (ed.) Studies on theoretical genetics. Academy of Sciences of the USSR, Novosibirsk.
  • 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.
  • Hansson, L. 1969. Spring populations of small mammals in central Swedish Lapland in 1964-1968. Oikos 20:431--450.
  • Hodges, K.E., C.I. Stefan and E.A. Gillis 1999. Does body condition affect fecundity in a cyclic population of snowshoe hares?. Canadian Journal of Zoology 77:1--6.
  • Krebs, C.J., and J.Myers 1974. Population cycles in small mammals. Advances in Ecological Research 8:267--399.
  • Lambin, X., V.Bretagnolle and N.G. Yoccoz 2006. Vole population cycles in northern and southern Europe: Is there a need for different explanations for single pattern?. Journal of Animal Ecology 75:340--349.
  • Mihok, S., B.N. Turner and S.L. Iverson 1985. The characterization of vole population dynamics. Ecological Monographs 55:399--420.
  • Murdoch, W.W., and E.McCauley 1985. Three distinct types of dynamic behavior shown by a single planktonic system. Nature 316:628--630.
  • Myers, J.H. 1990. Population cycles of western tent caterpillars: experimental introductions and synchrony of fluctuations. Ecology 71:986--995.
  • Newton, I. 1687. Philosophiæ Naturalis Principia Mathematica. London.
  • ki, 2002Norrdahl:Korpimaki:2002Norrdahl, K., and E.Korpimaki 2002. Changes in individual quality during a 3-year population cycle of voles. Oecologia 130:239--249.
  • Sauer, J.R., D.K. Niven, J.E. Hines, D.J. Ziolkowski, K.L. Pardieck, J.E. Fallon and W.A. Link 2017a. 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.
  • Sauer, J.R., K.L. Pardieck, D.J. Ziolkowski, A.C. Smith, M.R. Hudson, V.Rodriguez, H.Berlanga, D.K. Niven and W.A. Link 2017b. The first 50 years of the North American Breeding Bird Survey. The Condor 119:576--593.
  • Simchuk, A.P., A.V. Ivashov and V.A. Companiytsev 1999. Genetic patters as possible factors causing population cycles in oak leafroller moth, Tortrix viridana L.. Forest Ecology & Management 113:35--49.
  • Stenseth, N.C., and R.Ims, eds. 1993. The biology of lemmings. Academic Press, San Diego.
  • 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. Population cycles caused by selection by density dependent competitive interactions. Bulletin of Mathematical Biology 62:1109--1136, https://doi.org/10.1006/bulm.2000.0200.
  • Witting, L. 2013. Selection-delayed population dynamics in baleen whales and beyond. Population Ecology 55:377--401, https://dx.doi.org/10.1007/s10144--013--0370--9.
  • Witting, L. 2021. Selection-regulated population dynamic in birds and mammals. Preprint at bioRxiv https://dx.doi.org/10.1101/2021.11.27.470201.