ECOLOGICAL DETERMINANTS OF STABILITY IN MODEL POPULATIONS

. Detailed studies of the density-dependent interactions of larval and adult life stages of Drosophila melanogaster suggest that the stability of populations is dependent on environmental variables. Models of Drosophila population dynamics predict that the levels of food supplied to the adults and larvae will have important effects on population stability: stability is enhanced when adults are given low amounts of food and larvae are given high amounts of food. Experimental results presented here are consistent with these general predictions. These observations have implications for the management of biological populations and present an opportunity for studying the evolution of population stability.


INTRODUCTION
A long-standing problem in population biology has been understanding the factors in the environment and properties of organisms that affect the stability of population dynamics. Fluctuations in population size around an equilibrium may be due to the manner in which life history characters respond to population density, or they may reflect fluctuations in qualities of the environment that affect the numbers of organisms that can be supported. In this paper we describe how detailed life history components of Drosophila melanogaster are affected by density and how they, in turn, affect population stability.
The ability of biological populations to undergo dramatic and regular changes in population size was demonstrated by Nicholson (1957) (May 1974). These theoretical results stimulated several studies of natural and laboratory populations (Hassell et al. 1976, Thomas et al. 1980, Mueller and Ayala 1981. Although these studies used a wide variety of techniques, they shared the common goal of attempting to derive estimates of the dynamical characteristics of biological populations. The results of these studies were similar: most populations showed asymptotically stable dynamics. I Manuscript received 12 March 1993;revised 20 June 1993;accepted 30 June 1993. A more recent analysis of data from natural populations suggests that complex dynamical behavior may be more common than suggested earlier (Turchin and Taylor 1992). Turchin  We consider the factors that affect the stability of a model population, Drosophila melanogaster, by studying the population dynamics of Drosophila in the welldefined environment of the laboratory. Although we are ultimately interested in the behavior of populations in their natural environment, the factors that affect population stability are complex. We feel that little progress will be made in understanding these factors unless experiments are carried out under carefully controlled conditions not possible in field environments.

Population dynamic model
The effects of crowding on the life history of Drosophila are well known (Mueller 1985(Mueller , 1988. Crowding will restrict food availability for larvae, and decrease survival and adult size (Bakker 1961, Nunney 1983, Mueller et al. 1991. Crowding also increases the concentration of waste products, which will reduce larval viability (Botella et al. 1985) and adult size (L. D. Mueller, unpublished data). For populations of Drosophila kept on a discrete cycle of reproduction (e.g., reproduction takes place on a single day), effects of adult crowding on adult survival can safely be ignored.
However, female fecundity is affected immediately by crowding in two ways. Larval crowding causes larvae to pupate at a smaller size, and these small adults lay fewer eggs than large adults, all other things being equal.
Adult crowding will also decrease female fecundity. It March 1994 POPULATION STABILITY 431 was thought for some time that this effect was primarily mediated through increased behavioral interactions at higher densities (Mueller 1985). However, as we show in this paper, food availability (which decreases with increasing adult numbers) dramatically affects female fecundity.
In populations of Drosophila that are kept on fully discrete generations the number of eggs at time t + 1, n, +, can be described by n,= -G(N,)F(n, ) W(n,) Vn,, (1) 2 where V is the probability of an egg becoming a first instar larva, W is the density-dependent function describing the viability of first instar larvae, F is the mean fecundity of adult females and reflects the effects of food limitation (during the larval life stage) on female size and hence fecundity, and G(N,) describes the effects of adult density, N, on female fecundity (Mueller 1988).
It is worth noting that the effects of adult crowding are assumed to be independent of the larval crowding effects. In fact, to our knowledge this assumption has never been tested, but testing could in principle be done by conducting experiments similar to the ones done in this paper with adults of different sizes.

Components of this model have been tested and
shown to account adequately for the effects of crowding on viability (Nunney 1983, Mueller et al. 199 la) and reduction of adult size (Mueller et al. 199 la), which ultimately reduces female fecundity (Mueller 1987).
The stability properties of Eq. 1 have also been examined (Mueller 1988). This previous study showed that population stability was especially sensitive to changes in parameters of the function G(N,). Stability was enhanced in this theoretical study when female fecundity declined rapidly with increasing adult density. If this rate of decline was only modest, then cyclic and chaotic population dynamics were observed.
The functions used in Eq. 1 are summarized below.
The viability function is, W(n,) =J+(y) dy, (Mueller 1988) where 0(y) is the standard normal density function, y is the amount of food consumed by the larval life stage, and x = (m Vn,K I -1)r 1. In this model m is the minimum amount of food a larva must consume to successfully pupate, K is the total amount of food available for larvae, and o' is the variance in food consumption. The effects of larval crowding on female fecundity are described by, ln[F(n,)] = c(, + clln(g), where s is the mean adult size of females that have experienced larval crowding (n,) and is equal to = W(n1)-f s[K(oy + 1)V In,'](P(y) dy, (Mueller 1988) and, s(k) = a, + a,{ -exp [-a2(k -m)] (Mueller et al. 1991a), a,, a, and a are empirically determined constants. Lastly, the effects of adult density on female fecundity are modeled by a hyperbolic function, where f is the maximum fecundity achieved at low adult densities and a measures the sensitivity of female fecundity to increasing density. Rodriguez (1989) used an exponential model for this function and obtained a good empirical fit to his data. Mechanistic models of egg-laying and intraspecific interactions can be developed that lead to the hyperbolic model (Pearl 1932) or the exponential model (Rodriguez 1989). We do not attempt to resolve the issue of which of these two functions is best. Nor do we feel the qualitative predictions made here will be sensitive to changes in this particular function, although this is certainly an interesting area for further research. In the numerical iteration of Eq.  (Rodriguez 1988, 1989. In this study we have examined the relationship between fecundity and density at two different food levels. Adults used in these experiments had been raised for two generations in a common environment. Adults that were 1-2 d old were placed in 30-mL vials at one of six adult densities (2, 4, 8, 16, 32, or   are given high food they continue to lay a large number of eggs even when the population is crowded (Fig. 2).
As the population approaches carrying capacity the adult population will easily lay many more eggs than can possibly survive, especially when larval resources are low, and hence the adult population crashes in the next generation. However, since well-fed females lay up to 80 eggs per day the population recovers quickly from the crash and the cycle repeats.
It is worth emphasizing that even the detailed model used here involves numerous assumptions that will not be true even in the carefully controlled laboratory environment. For instance, the model described here con- tions. Nevertheless, these models have provided qualitative predictions concerning population stability and food levels that would not have been possible with other, more simple models. These qualitative predictions must be tested experimentally. However, it is probably unrealistic to expect accurate quantitative agreement between this theory and our experiments given the number of assumptions that have been required to achieve this theory.

Experimental populations
The HH population shows a rapid increase in its carrying capacity and large but irregular fluctuations after the first few generations (Fig. 4). The HL populations show a slow but regular increase in population size until about the seventh generation; after this time the numbers fluctuate about the equilibrium less wildly than the HH populations. The differences between these populations can be attributed to the large differences in number of eggs laid. In the HL populations the number of eggs produced was very low for each female, and hence changes in the total population size were never very spectacular. Finally, the LH populations give the appearance of a regular two-point cycle: large adult numbers invariably followed by very small population size. In these populations the adults are well fed, and hence the females laid large numbers of eggs, but since the amount of food that is available for the larvae is limited, there are few surviving adults after a large pulse of eggs. Small numbers of adults are capable of increasing their numbers rapidly, however, so the population recovers from its "crashes" with an overexplosion of adults, and the cycle continues.
One qualitative prediction that follows from the models considered previously is that the carrying ca- Further, the fact that the magnitude of these correlations decreases over time is more consistent with the cycle being generated from endogenous factors (density dependence) than an environmental factor that fluctuates cyclically (Turchin and Taylor 1992). Given the controls that are placed on the laboratory environment, the likelihood that uncontrolled factors would oscillate in a regular fashion is further reduced.
The autospectrum (Fig. 5) shows that there is a high-

Nicholson's blowflies revisited
A natural question is whether the population dynamics of other species will be affected in a similar fashion with changing food levels. In one experiment performed by Nicholson, the adult ration of ground liver was changed from ad lib to 1 g/d. The pronounced cycles that were present when the adults were given unlimited food were significantly attenuated by restricting the adult food ration. While many other fac- For the HH population the first data point was not used, for the HL populations only the data from generation 7-12 were used, and for the LH populations all the data were used. The eigenvalue was estimated by first fitting a quadratic equation to the N, vs. N. , data. The first derivative of this quadratic is then taken and evaluated at the empirically determined equilibrium population size. The confidence intervals are calculated from the five independent population eigenvalues. It has been shown that eigenvalues estimated from adult-to-adult transitions may be biased (Prout and McChesney 1985). However, numerical studies (Mueller 1986) (Rosenzweig 1972).
Several previous studies with D. melanogaster have generally concluded that population dynamics are asymptotically stable (Thomas et al. 1980, Mueller andAyala 1981). However, one feature of all these studies is that they were carried out under a standard set of conditions that most closely resembles the HL type of environment used in this study. Hence, the prior studies would not be expected to reveal the types of dynamics shown by the LH populations, since the LH environment is quite different from the standard "laboratory" environment.

Evolution of population stability
To what extent is population stability a phenotype that is under the influence of natural selection? Experimental work on this topic has been almost nonexistent until recently (Stokes at al. 1988). An analysis of a long- Others (Thomas et al. 1980) have argued that group selection would prevent populations from evolving demographic properties that result in severe oscillations or chaotic population dynamics. Theoretical arguments based on individual selection produced a variety of predictions concerning the evolution of population stability that depended critically on certain assumptions and models used in the analysis (Heckel and Roughgarden 1980, Turelli and Petry 1980, Mueller and Ayala 1981, Hansen 1992. In models where tradeoffs in certain parameter values were allowed, the out-come of evolution depended critically on the type of trade-offs permitted (Mueller andAyala 1981, Gatto 1993). These studies were initially motivated by the pervasive observation of populations with stable dynamics. Although the occurrence of populations with cyclic or chaotic dynamics may be more common than previously thought (Turchin and Taylor 1992), the evolution of population stability is still an unsettled area. The ability to generate cyclic dynamics in Drosophila offers the possibility of doing more detailed studies of the evolution of this phenomenon. Such studies would benefit greatly from the many features of Drosophila that make them such a useful model system in evolutionary biology (Rose 1984).