A test of simple models of population growth using data from very small populations of Drosophila melanogaster |
V. Sheeba and Amitabh Joshi*
Evolutionary Biology Laboratory, Evolutionary and Organismal Biology Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560 064, India
Three simple discrete-time models of density-dependent population growth were fit to data from eight small laboratory populations of Drosophila melanogaster subjected to either stabilizing or destabilizing food regimes. The exponential logistic model gave the best fits to data from both types of populations. In contrast, the linear logistic and hyperbolic models did not yield good fits, especially in the case of the destabilized populations. Parameter estimates from the exponential logistic model were in close agreement with those from previous studies using very large Drosophila populations subjected to the same type of stabilizing and destabilizing food regimes. Our results suggest that the simple exponential logistic model can reasonably well capture the major features of the dynamics of Drosophila populations even for very small population sizes where one may expect demographic stochasticity to play a major role in determining the population size.
ALTHOUGH the
importance of demographic stochasticity in population ecology has been appreciated for a
long time by theoreticians^{1–8}, most empirical work on population dynamics
has been structured around deterministic models^{9–13}. Many of the
commonly-used models of population growth are extremely simple and deterministic^{14–16},
yet empirical results have typically shown reasonable agreement with predictions of the
deterministic models^{10–13,17}. However, such studies have typically
*For correspondence. (e-mail: ajoshi@jncasr.ac.in)
used populations large enough to render the effects of demographic stochasticity on their dynamics unimportant. In recent years, the dynamics of small populations has been receiving considerable attention in ecology, especially because of conservation concerns^{18–22}. Most attention, however, has been focused on the impact of demographic stochasticity on the likelihood of extinction of small populations^{23–27}. On the other hand, the effect of demographic stochasticity on the dynamics of small populations, has rarely been studied empirically^{28}. It is, therefore, of considerable interest to assess whether deterministic models of population growth and dynamics can adequately capture the essential features of the dynamics of very small populations, or whether we need to explicitly incorporate demographic stochasticity into our models of population dynamics in order to make them applicable to smaller populations. In the study reported here, we fitted three simple discrete-time models of population growth to data from eight small (average size N = 74.69, st. dev. = 57.23) laboratory populations of Drosophila melanogaster to examine how well these simple deterministic models fit data from populations that may be expected to show relatively large amounts of demographic stochasticity.
We studied the dynamics of eight small populations of D. melanogaster, each of which was maintained in a single 8-dram vial (25 mm d ´ 95 mm h). Two sets of four populations each were derived from a large (N ~ 2000 adults) and outbreeding ancestral laboratory population (JB-1), derived from the UU populations of Mueller (UU and JB maintenance described, respectively, in refs 29, 30), that had been maintained in the laboratory at low larval densities for many years. At the beginning of the experiment, sets of 8 males and 8 females were placed into each of 8 vials and allowed to lay eggs for 24 h. The adults were then discarded, and the vials used to initiate the experimental small populations, which were maintained for a further 10 generations in the laboratory, as described below. Four of the populations were subjected to a food regime which combined low levels of food (3 ml food medium per
Figure 1. Eleven generations of census data from the four stabilized (HL) and four destabilized (LH) populations.
Table 1. The goodness of fit of the linear logistic, exponential logistic and hyperbolic models of population growth to data from the eight HL and LH-populations. Entries are the values of the coeffi-cients of determination (R^{2}) for each non-linear regression
vial) during the larval phase with high levels of food during the adult phase (i.e. regular medium supplemented with a generous dab of live yeast paste on the side of the vial): this type of regime is known to induce large fluctuations in population size in Drosophila cultures^{17}. These populations are henceforth referred to as LH-populations (Low food to larvae, High food to adults). The other four populations were subjected to a food regime combining high levels of food (10 ml food medium per vial) during the larval phase with low levels of food to the adults (i.e. no supplementary live yeast): this type of regime is known to give rise to relatively stable dynamics, with relatively smaller random fluctuations in population size as compared to the LH-regime^{17}. These populations are hereafter referred to as HL-populations.
In each generation, the number of adult flies present in each population (vial) was counted on the 21st day after egg-lay. The flies were then placed in a fresh vial containing either 3 ml (LH-populations) or 10 ml (HL-populations) of banana food and allowed to lay eggs for exactly 24 h, after which the adults were discarded. The larvae developed and pupated in these vials. Due to high larval density in these vials, eclosion was typically staggered over several days. Consequently, from day 8 through day 18 after egg-lay, any eclosing flies in these vials were collected daily into fresh vials with ~ 5 ml food in them. Eclosing flies were added daily into these adult collection vials and every other day all adults collected from a specific population till that time were shifted to a fresh vial containing ~ 5 ml food. On the 18th day after egg-lay, the egg vials were discarded and all eclosed adults of each population transferred to fresh vials containing ~ 5 ml of food with (LH-populations) or without (HL-populations) a supplement of live yeast paste added to the wall of the vial.
We fitted three simple discrete-time models of population growth to the census data of each of the eight HL and LH-populations by least-squares non-linear regression using the Levenburg–Marquardt procedure, implemented by Table Curve 2D^{31}. The models fitted were
(i) Linear logistic: N_{t}+1 = N_{t} [1 + r{(K – N_{t})/K}],
(ii) Exponential logistic: N_{t}+1 = N_{t} exp [r(K – N_{t})/K],
(iii) Hyperbolic: N_{t}+1 = N_{t}[a_{1}/(1 + a_{2}N_{t})],
where r and K represent the per capita intrinsic rate of increase and the carrying capacity of the population, respectively. These models were chosen as they are commonly used in ecological studies and their parame-
Figure 2. Plots of N_{t}+1 versus N_{t} for one representative population from each of the two maintenance regimes (HL/LH). The points represent the observed values and the line is the best-fitting curve based upon the exponential logistic equation N_{t}+1 = N_{t} exp [r(K – N_{t})/K].
ters lend themselves to relatively clear biological interpretation. We also estimated the coefficients of variation (CV) of population size over time as a crude measure of the relative magnitude of the fluctuations in population size in the different HL and LH-populations. The use of the CV of population size over time as a measure of the relative degree of stability was motivated by a pragmatic definition of stability in ecology as being the state in which a population undergoes relatively small fluctuations in size^{32,33}.
Both the HL and LH-populations showed fairly large fluctuations in population size over time, with the LH-populations exhibiting behaviour close to two-point cycles, as predicted by a detailed model of Drosophila population growth incorporating the effects of pre-adult and adult density on various components of fitness such as survivorship, body size and female fecundity^{34} (Figure 1). The HL-populations exhibited relatively smaller fluctuations in size: the mean CV of population size in the LH-populations (0.8766) was significantly greater (t-test with 3 df; P < 0.05) than that of the HL-populations (0.6044). These results support previous observations made on large populations of Drosophila (N ~ several thousand flies) subjected to LH and HL type of maintenance regimes^{17,35}.
Of the three models of population growth tested, only the exponential logistic model gave consistent fits with at least moderate R^{2} values (Table 1). The fit of the linear logistic and hyperbolic models was consistently poorer than that of the exponential logistic model for both HL and LH-populations, and the two former models essentially failed to explain even a small fraction of the variation in the case of the LH-populations (Table 1). Moreover, the fit of the exponential logistic model was better in the case of the LH-populations (Table 1, Figure 2): the mean R^{2} for the LH-populations (0.6514) was significantly greater (t-test with 3 df; P < 0.05) than that for the HL-populations (0.2869). This is not altogether surprising, given that the large fluctuations in the size of the LH populations are deterministically driven by nature of the maintenance regime^{17,34}. The combination of severe larval competition for scarce food resources, together with yeast being provided to the adults, results in a tendency for the population to crash due to heavy pre-adult mortality when the number of eggs is large. The recovery of the population from such a crash, however, is rapid because the yeasting of the adult food renders female fecundity relatively unresponsive to adult density^{17,34}.
The parameter estimates (Table 2) from fitting the exponential logistic model to the HL and LH-population data also support the predictions about the dynamics of populations under LH and HL type maintenance regimes. The Drosophila population growth model of Mueller^{34}, incorporating the effects of pre-adult and adult density on various components of fitness such as survivorship, body size and female fecundity, predicted that LH type of regimes would induce periodic fluctuations while the HL type of maintenance regime would result in populations exhibiting asymptotic stability. These predictions have since been verified using large populations of Drosophila^{17,35}. In our HL-populations, all estimates of r satisfy 1 < r < 2 (Table 2), a condition in which the exponential logistic model exhibits an oscillatory approach to a stable equilibrium^{14} (Figure 3). Of the four estimates of r for the LH-populations
Figure 3. Observed and predicted population size trajectories over time in one representative population from each of the two maintenance regimes (HL/LH). The predictions were based on r and K values estimated from the best-fitting exponential
logistic curve.
Table 2. Estimates of the parameters r (intrinsic rate of increase) and K (carrying capacity) for the eight HL and LH-populations.
Estimates are from the best fitting exponential logistic model
(Table 2), three exceed r = 3, a situation resulting in chaotic fluctuations in population size^{14}, while one population shows r = 2.294, in which case the population is expected to show stable two-point cycles^{14} (Figure 3). The consequences of these differing values of r for population dynamics are best visualized through plots of N_{t}+1 versus N_{t} for different values of r (Figure 2); the steeper the hump in these density-dependent functions, the greater the instability of the resultant dynamics^{15}. Our estimates of r, the parameter essentially determining the dynamic behaviour of the exponential logistic model, are in close agreement with estimates of r from large Drosophila populations subjected to HL and LH type of maintenance regimes^{36}.
Our results clearly suggest that the important gross features of the dynamics of Drosophila populations in the laboratory can be captured reasonably successfully by the simple exponential logistic model even for populations that are extremely small in size, especially if the fluctuations in population size are, at least in part, driven by deterministic forces, as in our LH-populations. In an earlier study on 25 small laboratory populations of Drosophila, Rodriguez^{28} found that a fairly detailed multiple-life stage density-dependent model was not able to successfully predict even the gross features of the observed dynamics. In that study, parameters of the model were independently estimated from data other than time series of population size^{28}. It should also be noted that the conditions of his study^{28} were similar to our HL regime, in which the exponential logistic model did not give very good fits (Table 1). Certainly, fitting models with a greater number of parameters may provide a better fit to data from HL type populations, but interpretation of the parameters in ecologically meaningful terms then becomes difficult. In light of an increasing interest in studying the dynamics of small and destabilized populations, however, we think that our finding that the exponential logistic model gives a reasonably good fit to data from destabilized small populations of Drosophila is useful in that it suggests that such laboratory populations can be fruitfully used to investigate various issues pertaining to the stability and dynamics of small populations and metapopulations. While there are many sources of stochasticity, other than size, that small natural populations may face, we feel that being able to use a laboratory system to test predictions from population dynamics theory regarding the behaviour of small populations is likely to lead to a better understanding of what might or might not be possible in natural settings.
ACKNOWLEDGEMENTS. We thank M. Rajamani and Vishal Gohil for assistance in the laboratory, two anonymous referees for helpful comments on the manuscript, and Laurence D. Mueller and Vijay K. Sharma for some very thought-provoking observations on fitting population dynamic models to Drosophila data. This work was supported by a grant from the Jawaharlal Nehru Centre for Advanced Scientific Research to A. J.
Received 27 January 1998; revised accepted 2 September 1998