A THEORY WHICH PREDICTS STABLE EQUILIBRIUM FOR INVERSION POLYMORPHISMS IN THE GRASSHOPPER MORABA SCURRA

In a recent analysis of inversion polymorphisms in the grasshopper, Moraba scurra, Lewontin and White (1960) found that each of the populations sampled occupied a "saddle" position rather than a peak on the adaptive surface. They examined reasons for the apparent equilibrium condition at such an unstable point and tentatively hypothesized that the equilibrium is maintained by a combination of gene frequency-dependent selection and yearly fluctuations in the physical factors of the environment. In the construction of their adaptive topographies, Lewontin and White assumed panmixia, an assumption they noted may not be strictly correct because, according to White (1957), these insects "live and die within a few metres of the point where they hatched from egg." It is possible that the apparently anomalous results obtained by Lewontin and White are related to the assumption of panmixia. Consequently, we have recomputed the adaptive surfaces, taking inbreeding into account. The procedure followed was basically that used in the original computations except that expected genotypic frequencies were calculated on the basis of Wright's equilibrium rule. It was assumed that the tendency for close neighbors to mate and for relatives to be close neighbors results in some equilibrium level of inbreeding which can be represented by F, the correlation between uniting gametes (Wright, 1922). The adaptive surfaces for the Wombat B population (1956, 1957, and 1959 data combined) assuming F = 0.00, 0.05, 0.10, and 0.25 are given in figs. 1-4. The "saddle" found consistently by Lewontin and White appears for the combined 1956, 1957, and 1959 Wombat data when F is assumed to be zero. However, for F = 0.05 the saddle is replaced by an adaptive peak which coincides closely with the observed population point. The apparently stable equilibrium which exists in the several populations is easier to explain on the basis of this topography than the topography for F = 0. Nevertheless, the adaptive peak is little more than a hillock and it seems possible that the adaptive gradients are too slight to prevent populations with such a topography from drifting to fixation. When higher levels of inbreeding are assumed, the adaptive peak becomes increasingly prominent. For F = 0.25 the peak is the outstanding feature of the landscape and the entire adaptive surface assumes an aspect that seems compatible with stable equilibrium (fig. 4). However, assumption of this level of inbreeding causes the adaptive peak to shift toward the center of the adaptive surface to a position where it no longer coincides precisely with the observed population point. Values of F in the range of 0.10 to 0.15 (fig. 3) appear to fit the data best because they produce topographies featuring a prominent adaptive peak which is also coincident with the population point. Such values of F seem not unreasonable for these sedentary insects. Each of the populations (Royalla, Hall, Williamsdale) reported by Lewontin and White were reanalyzed in similar fashion and it was found that the population points in all cases lie near adaptive peaks. These results indicate that the present composition of the populations is what one would expect from information about the 1This study was supported in part by a grant from the National Institutes of Health, GM 10476. The use of a digital computer made available by the computing center, University of California, Davis, is gratefully acknowledged.