Density regulation depends on the life history and the geometry of ecological interactions
Density regulation [ ε = εmax f(N) ] causes the net energy (ε) of the average individual to decline monotonically with an increase in the density (N) of the population. This regulation
f(N) = fe(Nwβ) fι(NVH(d-1)/d) fs(βH1/d/V)
is dependent on the average exploitation of the resource (fe), the interference between the individuals (fι), and the local exploitation of the individuals themselves (fs). These components are normally seen only as a function of N, but they are also dependent on life history traits like mass specific metabolism (β), body mass (w), foraging speed (V), and home range (H); and on the spatial dimensionality (d) of the geometry of ecological interactions.
Exploitation occurs through the reduction of the resource density by the consumption of resources by the population. As such it is a function of the overall energy that is used by the population [ fe(Nwβ) ].
Regulation by interference is as function of the frequency of encounters between individuals. This frequency is proportional to the overlap between home ranges, times the frequency of foraging within the home range. The home range overlap [ NH ] is simply the average home range (H) divided by the per capita availability of space (1/N). And the frequency of foraging [ V/H1/d ] is the foraging speed (V) divided by the length of the foraging tracks that is proportional to the dth root of the d-dimensional home range (H1/d). Regulation by interference may thus be given as fι(NVH(d-1)/d).
Local resource exploitation by foraging self-inhibition is not in itself density regulation, but the component is needed for the natural selection of a realistic density regulation. It is a function of the frequency of foraging [ V/H1/d ], as it is the time interval between foraging tracks that determines the regrowth, or dispersal, of new resources in areas that have been exploited by the individual. Yet, self-inhibition is relative to the average frequency of re-harvesting from all the individuals in the population when foraging tracks are infinitely long and never reused by the individual itself. This frequency is expected to be proportional, among others, with the overall biotic pace (β), and local resource exploitation may thus be expressed as fs(βH1/d/V).