^{**}=7/3ψ

# Selection-delayed population dynamics

When the population dynamic feed-back selection of interactive competition is integrated into population dynamics, we obtain selection-delayed dynamics as a more general version of density regulated growth. This explains the cyclic dynamics that have fascinated ecologists for decades, and it provides new insights into the limitation and exploitation of natural populations.

## Hyper-exponential increase

The Malthusian law of exponential increase describes unchecked dynamics at the limit of zero population density. It assumes that evolutionary changes occur much slower than population dynamics, but recent evidence have found that this is not always the case.

Natural selection at zero density follows Fisher’s fundamental theorem of natural selection, predicting a linearly increasing exponential growth rate and a hyper-exponentially increasing population.

## Selection-delayed dynamics

For density regulated growth we need to include the density dependent selection on the exponential growth rate. This induces a selection-delayed feed-back that accelerates growth in populations that are below the population dynamic equilibrium, and decelerates growth above. The result is a single-species mechanism for the population cycles that have fascinated ecologists for decades; with growth rate acceleration (and not the growth rate) being a function of the density dependent environment.

## Population limitation

With density dependent competitive interactions there is selection for a green and balanced world, where overexploited resources are rare because species have intermediate growth rates with intermediate population densities.

The evolutionarily determined equilibrium abundance is the abundance that generates the interactive competition of the competitive interaction fix-point for the selection of mass. This equilibrium selects for an allometric solution, with Damuth's -3/4 power decline in abundance with mass following as an essential prediction.

## Sustainable exploitation

Theory on sustainable use is usually based on density regulated growth, where the growth rate is a function of the environment and the sustainable yield has a maximum at an optimal population density. For selection-delayed dynamics it is not possible to determine the growth rate, but only the acceleration of the growth rate, as a function of the density dependent environment. This implies that there is no maximal sustainable yield at an optimal abundance, but a range of possible replacement yields at any given abundance.

For harvest situations where populations are allowed sufficient time to equilibrate at the evolutionary equilibrium, the abundance of the exploited population will be relatively independent of the harvest, but the intrinsic growth rate will increase, and the average body mass decline, with increased harvest.