Malthusian Relativityι**=7/3ψ
Life history evolution - traits & transitions

Lack's clutch size

Density dependent interactive competition selects a balanced Lack’s clutch size

Fig. 1 Left: Fitness as a function of lifetime reproduction given a trade-off between current reproduction and future survival, with Lack’s clutch size indicated by the circle. Middle: Exploitative competition selects for a continued decline in the trade-off between current reproduction and future survival, and this causes an evolutionary increase in lifetime reproduction as determined by Lack’s clutch size. Right: Density dependent interference competition selects for an intermediate reproductive rate, with the evolutionary equilibrium for Lack’s clutch size indicated by the filled circle.

Just as the quality-quantity trade-off makes the reproductive rate trade-off against body mass, there is an energetic trade-off between current reproduction and the future survival of either offspring or parents or both. This was noted by Lack (1948) when he proposed that optimal reproduction is where most offspring survives. Including also the trade-off to adult survival (Schaffer, 1983; Charlesworth, 1994), the optimum of intermediate reproduction became known as Lack’s clutch size.

Lack’s clutch size is based on a physiological selection with constant relative fitnesses, with a trade-off between survival (p) and lifetime reproduction (R)

p = p*L e1 – R/R*L

where p*L is the probability to survive and reproduce, and R*L lifetime reproduction, at Lack’s optimum (*L). With fitness being r = ln(p R), we obtain the fitness profile

r = lnp*L + lnR + 1 – R/R*L , the selection gradient

∂ r / ∂ R = 1/R – c , and Lack’s clutch size

R = R*L ,

from ∂ r / ∂ R =0, with p = p*L e1 – R/R*L reducing to p*L. The physiology is thus selecting for a partial optimum (Fig. 1, left), with fitness declining if the reproductive rate is above or below the optimum.

With Lack’s clutch size evolving by the partial selection of the physiology, we expect the trade-off to adjust to the other life history traits of the organism; with a large set of Lack optima capturing the potential evolution of the reproductive rate. The physiological fitness optimisation behind each of these optima, is also operating on the whole set of Lack optima with a selection gradient

∂ r / ∂ ln R*L = 1 , => R*L → ∞

that selects for a continued increase in the reproductive rate (Fig. 1, middle). This selection for unbalanced reproduction is our base case expectation in the absence of interactive competition. And with a lifetime reproduction that is inversely related to body mass [ R ∝ 1 / w ] it reflects, among others, the selection of the quality-quantity trade-off for the absence of mass [ ∂ r / ∂ ln w = -1, => w → 0 ].

With interactive competition, we find that it is the population dynamic feed-back of the density dependent competitive interactions that is selecting the optimal life history from Lack’s set of physiological optima. For multicellular animals with stable energetic states, this implies the selection of the intermediate body mass (w**) that generates an intra-specific resource bias of unit [ ψι** = 1 ]. And with the selected allometries for multicellular animals including mass invariant survival and lifetime reproduction, we find the feed-back of interactive competition to select an intermediate and body mass invariant Lack optimum

R*L ∝ w0 and p*L ∝ w0

as illustrated by the right plot in Fig. 1.


  • Charlesworth, B. 1994. Evolution in age-structured populations. 2nd ed. Cambridge University Press, Cambridge.
  • Lack, D. 1947. The significance of clutch size. Ibis 89:302--352.
  • Schaffer, W.M. 1983. The application of optimal control theory to the general life history problem. The American Naturalist 121:418--431.