Malthusian Relativityι**=7/3ψ
Population dynamics - the evolutionary extension

Sustainable exploitation

With selection there is no maximal sustainable yield

Fig. 1 Left: Density regulation: The relative growth rate r/r(max) declines monotonically with abundance (relative abundance is N/N*). Right: Selection-delayed dynamics: The relative growth rate of an increasing population when it crosses the equilibrium abundance is positively related to the magnitude of the perturbation (the perturbation from equilibrium abundance N* is given as relative abundance N/N*).

Theory on sustainability arose from ecological concepts that did not consider the importance of evolutionary dynamics; let it be on the shorter time-scale of population dynamic perturbations, or on the longer that relates to changes in equilibrium conditions when a rather stable sustainable harvest have persisted for decades or centuries.

Relating to shorter time-scales, the density regulated model introduced the concept of recruitment curves (Beverton and Holt, 1957). For constant factors extrinsic to the population, density regulation imposes a monotonic decline in the population dynamic growth rate with density (Fig. 1, left), and this translates into the recruitment curve (Fig. 2, left) that defines a maximum sustainable harvest as a function of abundance.

The growth rate and the potential sustainable harvest is no longer pre-specified by the environment when population dynamics and evolutionary processes operate on similar time-scales. For selection-delayed dynamics it is no longer the exponential growth rate, but only the acceleration of the growth rate, that can be determined by the density dependent environment. This 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. In result, there is no single curve of sustainable yield to define an optimum of maximal harvest (Witting, 2002). The actual growth rate and replacement yield is instead strongly influenced by initial conditions like density independent perturbations, with the larger growth rates following from the largest perturbations (Fig. 1, right; Witting, 2000, 2013).

Fig. 2 Left: Density regulation: The recruitment curve defines the sustainable harvest for density regulation, with the vertical line indicating the maximum sustainable yield. Right: Selection-delayed dynamics: Increased harvest selects for increase reproduction and an associated increase in the recruitment curve; illustrated here by three curves that represent three harvest levels that are defined by the slopes of the dotted lines.

On the longer time-scale, the traditional view implies that you may aim for an optimal harvest where the population is stabilised close to the abundance of the maximum sustainable yield (Fig. 2, left). With a stabilised population under selection-delayed dynamics, an increased harvest implies selection for increased reproduction, with an associated increase in the potential growth rate and in the recruitment curve of sustainable takes (Fig. 2, right; Witting, 2002). Given sufficient time with constant harvest, selection should stabilise at the equilibrium abundance of the competitive interaction fix-point. This equilibrium abundance of the exploited population should be rather similar to the abundance of the unexploited population; with the major difference being that increased harvest imposes an evolutionary decline in competitive traits like body mass. At the theoretical limit, there seems to be almost no upper limit to the long-term sustainable harvest of individuals, but there is an upper limit to the sustainable harvest of biomass (Witting, 2002).


  • Witting, L. 2000. Population cycles caused by selection by density dependent competitive interactions. Bulletin of Mathematical Biology 62:1109--1136.
  • Witting, L. 2002. Evolutionary dynamics of exploited populations selected by density dependent competitive interactions. Ecological Modelling 157:51--68.
  • Witting, L. 2013. Selection-delayed population dynamics in baleen whales and beyond. Population Ecology 55:377--401.