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 e^{1 – 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 e^{1 – 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.

Two examples

Lack’s fitness trade-off between current reproduction and future survival has been documented in a number of species (e.g., Lack, 1954; Boyce and Perrins, 1987; Tinbergen and Daan, 1990; Daan et al., 1990). It is illustrated in Fig. 2 for the great tit and the kestrel. Both species have a natural brood size with a higher fitness than could be obtained from manipulated broods that were either larger or smaller than the natural brood size.

Fig. 2 Left: Relative fitness (total reproductive value) as a function of brood size for great tit. Data from Tinbergen and Daan (1990). Right: Relative fitness (total reproductive value) as a function of brood size for kestrel. Data from Daan et al. (1990). Filled circles are natural brood sizes, and open circles manipulated brood sizes, showing that both species have brood sizes close to the fitness optimum.

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References

• Boyce, M.S., and C.M. Perrins 1987. Optimizing great tit clutch size in a fluctuating environment. Ecology 68:142--153.
• Charlesworth, B. 1994. Evolution in age-structured populations. 2nd ed. Cambridge University Press, Cambridge.
• Daan, S., C.Dikstra and J.M. Tinbergen 1990. Family planning in the kestrel (Falco tinnunculus): The ultimate control of covariation of laying data and clutch size. Behavior 114:83--116.
• Lack, D. 1947. The significance of clutch size. Ibis 89:302--352.
• Lack, D. 1954. The natural regulation of animal numbers. Oxford University Press, Oxford.
• Schaffer, W.M. 1983. The application of optimal control theory to the general life history problem. The American Naturalist 121:418--431.
• Tinbergen, J.M., and S.Daan 1990. Family planning in the great tit (εm Parus major): Optimal clutch size as integration of parent and offspring fitness. Behavior 114:161--190.