Stephan B. Munch, Marc Mangel, and David O. Conover. 2003. Quantifyying natural selection on body size from field data with an application to winter mortality in Menidia menidia. Ecology 84:2168–2177.

This appendix presents analytical solutions to the integral in Eq. 2a for each mortality model assuming allometric growth.  Integration of Eq. 2a with the exponential mortality model requires application of the incomplete gamma function (IG) defined as

(A.1)

where G(a) denotes the gamma function (Abramowitz and Stegun 1965).  Substituting Eqs. 1 and 3a into Eq. 2a, survival under the exponential mortality model is given by

(A.2)

where the natural logarithm of survival is used solely to simplify notation.  After factoring constant terms from the integral, we get

(A.3)

All that remains is to substitute s = m₂x to obtain

(A.4)

which has the form of Eq. A.1 for the incomplete gamma function.  At this point, it can be seen that exponential mortality and allometric growth give

lnS =

(A.5)

for the integral in Eq. 2a.

For allometric mortality, survival is calculated somewhat more simply. Substituting Eqs. 1 and 3b into Eq. 2a, survival is given by

(A.6)

which may be integrated directly to obtain

(A.7)

In both cases, x₁ is given by  where T = t₁ - t₀ is the duration of winter.  If no growth occurs, then survival is calculated from Eq. 2 as

(A.8)

Literature cited

Abramowitz, M., and I. A. Stegun. 1965. Handbook of mathematical functions. Dover, New York, New York, USA.