How Well Does the Edmonson-Paloheimo Model Approximate Instantaneous Birth Rates?

An equation independently derived by Edmondson and Paloheimo for estimating instantaneous birth rates without a knowledge of mortality rates has been extensively employed by zooplankton ecologists. This expression is exactly true only under the unlikely conditions that the age distribution is stable and that egg mortality is equal to that averaged across the entire population. Here I compare estimates of instantaneous birth rates ( h )using the Edmondson-Paloheimo equation to those determined with a general model which accounts for age structure instability and egg mortality in two Daphnic~ pulr~x populations. Despite the fact that the assumptions of the Edmondson-Paloheimo model are rarely met, it mimics the seasonal pattern of b quite well, with few exceptions. An analysis of variance suggests that the model will provide an adequate approximation of the true population birth rate when a variance in b of 0.003 can be tolerated. A much more critical factor in the analysis of instantaneous rates in plankton populations is the development of accurate sampling techniques. Key n,ord.s: birth rutes; Clrrdoc~eru; Daphnia pulex; dr,nzogrcrphy; egg rc~tio; rotifcrs; ;ooplanXton. A great deal can often be inferred about the factors h,,, = INN, + C,) In(N,) regulating population densities from measures of two n population parameters, the instantaneous birth and mortality rates, h and m. The estimation of h has been --In(l + E ) (2) D thought to be particularly simple for planktonic cladocerans and rotifers whose adulta bear live young (Edmondson 1968). The attractiveness of both of these which are easily enumerated in preserved samples and expressions lies in their independence from m which whose egg development rates are simple functions of is often the final product of interest. Given an additemperature. Edmondson (1960) first introduced the tional estimate of the actual rate of population growth, egg ratio technique as a means of estimating instanr , the population mortality rate can then be detertaneous birth rates with preserved samples of plankmined by difference using the relation tonic rotifers. In theory the method can be extended m = h r . (3) to any population characterized by continuous growth. Under the special conditions of zero population Since few populations are either free from egg morgrowth, zero egg mortality and a stable age distributality o r stable in size or age distribution, Eqs. 1 and tion, the age distribution of eggs will be even, and the 2 would seem to be inappropriate. However, the use finite birth rate of the population is of Eq. 2 has been further encouraged by a derivation p = EID by Paloheimo (1974) which, despite its incorporation of egg mortality, turns out to be independent of morwhere E = C,IN, is the average number of eggs per tality and identical to Eq. 2. This convenient absence individual in the population (the egg ratio), C, and N, of egg mortality from Eq. 2 occurs only when the morbeing the total number of eggs and individuals in the tality rate of eggs is exactly equal to that averaged population at time t , and D is the duration of egg across the entire population ( m ) . Furthermore, the indevelopment (days). Under these constraints, p is redependence of Eq. 2 from r is still a consequence of lated to the instantaneous birth rate by assuming a stable population age distribution. Threlkeld (1979) has suggested that the accuracy of birth rate estimates could be improved by examining the age distribution of eggs. Although this approach would seem to be an advance over the Edmondson-Palohei(1) mo model, it is not yet clear whether the additional An alternate expression for h which makes no aswork is warranted. sumptions about the age distribution of eggs, but A large part of the problem is that, with few excepwhich still assumes zero population growth and zero tions (DeMott 1980. Keen and Nassar 1981), previous egg mortality, is investigations involving the egg ratio method have not attempted to estimate the variance of the rates in Eq. Manuscript received 25 September 1980; revised 5 ~~~~h 3. In the absence of such information it is not only 1981; accepted 27 April 1981. impossible to compare the adequacy of different tech1 Febr~~u l -y 1982 ZOOPLANKTON BIRTH RATES 13 niques, but a comparison of instantaneous rates between dates o r sites is risky at best. An analysis of variance for b and r becomes particularly critical when /?I is estimated by difference. Many studies show dramatic fluctuations in in over very short time intervals (Clark and Carter 1974, Kwik and Carter 1975, Goldman et al. 1979). and without variance estimates these cannot be identified a s real o r artifactual. Negative estimates of in, which often arise from Eq. 3, are usually attributed to the hatching of uncensused resting eggs, although they are just as likely to be a consequence of poor estimates of r resulting from inadequate sampling or biased determinations of h resulting from faulty assumptions of the EdmondsonPaloheimo model. Despite these problems, zooplankton ecologists continue to rely heavily on Eq. 2 (see Bottrell et al. 1976, Goldman et al. 1979. Kerfoot and Peterson 1979, Lynch 1979 for several recent applications), and it seems likely that many will continue to d o so in the interest of time. Thus. it is of interest to know the extent to which deviations from the assumptions inherent in the Edmondson-Paloheimo equation are reflected in Door estimates of birth rates in field studies. Such an analysis has not been possible in the past because the magnitude by which the assumptions have been violated has been difficult to quantify. Computer simulations are of only limited value in examining this problem (Seitz 1979) because the choice of input parameters is necessarily subjective. The application of a sequential sampling technique for estimating sizespecific mortality rates in cladoceran populations (M. Lynch, pc~r~oncll oh,cert~atiotl) now allows the collection of the necessary information. This paper provides an empirical test of the efficacy of the Edmondson-Paloheimo model by comparing estimates of b by Eq. 2 with those obtained for two Daphnicl p~rlex populations for which instabilities in age structure and differential egg mortality are accounted for. Since a generalized model for estimating instantaneous birth rates from egg ratio data has not been previously published I will start by deriving such an expression. Throughout this paper I will assume a homogeneous population, i.e., one in which all individuals are exposed to the same physical and biological factors. I also assume that egg development times and size-specific rates of growth, egg laying, and mortality are constant over the interval ( t L), t). (Since the structure of the population may change over [t D, t] , this does not imply constancy of these rates as population parameters.) For populations for which L) is on the order of a few days, as it is in most of the planktonic organisms to which the egg ratio model has been applied, such approximations do not seem unreasonable. However, where these assumptions are significantly violated, no simple expression for estimating b by an egg ratio method can be derived, and alternate empirical approaches must be sought. The ideal egg ratio model makes no assumptions about the internal structure of the population, but considers the individual contribution of each class of individuals to the total population birth rate. Although the classification scheme is arbitrary, I will derive the model in the context of a population of an arbitrary number of size classes ( i t , ) since most organisms may be more easily classifed by size than by age. The total number of eggs derived from size class x which hatch during time ( t , t + dt) is equal to the total number of eggs laid during time ( t L), t L) + dt ) by individuals of that size class which survive to hatching (most planktonic organisms, especially cladocerans, do not change size classes while carrying a clutch), b,. N,., ,tit = tit, I,.N,, , , e r = l J e i " ~ (4) where h,. = the instantaneous birth rate of size class x (day-'), N,,,, = the number of individuals in size class s at time t (number per square metre), r,. = (In &,,, In N,,,-7)/~ is the growth rate of size class x (day-') I , = the instantaneous rate of egg laying by size class x (day-'), and tn, = the instantaneous mortality rate of size class x (day-'). The size-specific rate of egg laying, I,. , can be re-expressed in terms of the mean size-specific clutch size E,{ . as follows. The total number of eggs carried by size class x at time r is