^{**}=7/3ψ

# Selection-delayed dynamics

Selection-delayed dynamics; a single-species mechanism for population dynamic cycles

Population dynamic modelling have typically assumed that population dynamic growth is regulated by the exploitation of the resource

N_{t+1} = N_{t} λ_{m} f(N_{t})

where N_{t} is the abundance at time t, λ_{m} > 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.

But as the population dynamic growth rate of an individual is fitness, we find that natural selection is also regulating the growth rates of populations. And when the life histories that are selected by the population dynamic feed-back of interactive competition are allowed to be perturbed away from the population dynamic equilibrium (N^{*}), we have selection-delayed dynamics

N_{τ+1} = N_{τ} λ_{τ} (N_{τ} / N^{*})^{-γ}

where the per generation (τ) growth rate

λ_{τ} = λ_{τ-1} (N_{τ-1} / N^{*})^{-γq}

is accelerated when then abundance is below the equilibrium [ N < N^{*} ] and decelerated when the abundance is above [ N > N^{*} ].

The resulting dynamics is cyclic, with population cycles that are either damped (Fig. 1, left) or stable (Fig. 1, middle). And as the 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-delayed 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-delayed 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 maybe first proposed by Ginzburg in 1972 from analogy to Newton's laws of motion (Newton, 1687).

### References

- 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.
- Newton, I. 1687. Philosophiæ Naturalis Principia Mathematica. London.
- Witting, L. 2000. Population cycles caused by selection by density dependent competitive interactions. Bulletin of Mathematical Biology 62:1109--1136.
- Witting, L. 2013. Selection-delayed population dynamics in baleen whales and beyond. Population Ecology 55:377--401.