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ICES Journal of Marine Science, 57: 1488–1494. 2000 doi:10.1006/jmsc.2000.0912, available online at http://www.idealibrary.com on Population dynamics of two marine polychaetes: the relative role of density dependence, predation, and winter conditions Jaap van der Meer, Jan J. Beukema, and Rob Dekker Van der Meer, J., Beukema, J. J., and Dekker, R. 2000. Population dynamics of two marine polychaetes: the relative role of density dependence, predation, and winter conditions. – ICES Journal of Marine Science, 57: 1488–1494. We modelled the population dynamics of two polychaete species, a prey (Scoloplos armiger) and its predator (Nephtys hombergii) by simple linear and non-linear time–series models using a 29 year data series of population abundance and winter conditions. The log reproductive rate (i.e., the difference in log density between two succeeding years) of the prey population was density dependent and negatively related to the density of the predator. The log reproductive rate of the predator was also density dependent, but was to a greater extent determined by winter temperature, a density-independent factor. Log reproductive rate of the predator was not related to the density of the prey, nor could delayed density dependence be detected. Apparently, the feedback mechanism between predator and prey was weak, perhaps because the predator is a generalist. Predator density can therefore be categorized as a densityindependent factor with respect to the dynamics of the prey species. In neither of the two species was density dependence strong enough to generate complex or chaotic behaviour, as indicated by the weakly negative slope of the reproduction curve at the equilibrium, the apparent lack of a non-linear component in the reproduction curve, and the negative Lyapunov exponent. 2000 International Council for the Exploration of the Sea Key words: chaos, density dependence, Lyapunov exponent, polychaetes, population dynamics, predation. J. van der Meer, J. J. Beukema, and R. Dekker: Netherlands Institute for Sea Research, PO Box 59, 1790 AB Den Burg (Texel), The Netherlands; e-mail: [email protected] Introduction The controversy about the role of density-dependent regulation of population densities has led to one of the most acrimonious and heated debates in ecology (Wolda, 1995; Den Boer and Reddingius, 1996; Bonsall et al., 1998). Though the dispute has not yet been settled, it is now understood that much of it arose from a lack of standardization of terms (Bonsall et al., 1998). Even the terms ‘‘density dependence’’ and ‘‘density-dependent factor’’ have been used ambiguously (Royama, 1992; Wold and Dennis, 1993). At present, the point of interest no longer seems to be a strict choice between either density-dependent regulation or not. Attention has shifted towards quantifying the relative importance of density-dependent and density-independent factors in determining population fluctations, and testifying the hypothesis of regulation per se is no longer thought profitable (Royama, 1992; Wolda, 1995). One way of quantifying the relative importance of densitydependent and density-independent factors is through the use of stochastic population process models that 1054–3139/00/051488+07 $30.00/0 include both types of effects (Royama, 1992). These models are then challenged by real long-term data; the relative role of the two sorts of effects can be determined by choosing the best model (Leirs et al., 1997; Grenfell et al., 1998). A second key topic among population ecologists is the relevance of complex and chaotic dynamics in real populations. May (1974, 1976) showed that very simple deterministic non-linear models of single species in discrete time can reveal chaotic behaviour. Generally, chaotic dynamics in a non-linear process only occur when the level of density dependence is strong (Royama, 1992). However, the earliest applications of non-linear stochastic process models to a number of insect data sets suggested that complex and chaotic behaviour is unlikely to be widespread in real populations (Hassell et al., 1976). This conclusion has been rather influential in discussions on the importance of chaos in population biology, but was questioned by Turchin and Taylor (1992). They argued that the results obtained by Hassell et al. (1976) largely arose from using a single-species model without delayed density dependence. Complex 2000 International Council for the Exploration of the Sea Population dynamics of two marine polychaetes Materials and methods Data S. armiger and N. hombergii belong to the most common macrozoobenthic species of the tidal flats in the Dutch Log biomass density 0 (a) –1 –2 –3 –4 –5 –6 1970 6 Winter temperature dynamics are more likely in higher-dimensional systems (e.g., multi-species system or single-species systems with more than one age class), and analysing such systems in fewer dimensions will hide this complexity and yield erroneous results. Higher-dimensional systems can be rewritten in terms of one-dimensional systems with time delays (Royama, 1981, 1992). Studies based on singlespecies models with time delays indeed pointed to the existence of chaotic dynamics in natural populations (Turchin and Taylor, 1992; Turchin, 1993, 1996). However, other studies could not confirm this result (Perry et al., 1993; Zhou et al., 1997), and the debate over whether chaotic dynamics are relevant to ecological reality continues. A third point of interest is the role predators play in determining the dynamics of a prey species (Berryman, 1992b). Predation may act either as a density-dependent or -independent factor. In terms of the definition provided by Royama (1992, p. 21, ‘‘if the state of existence of an ecological factor, identified in terms of its measure or parameter, is, in turn influenced by the population density of the animal, the factor is said to be density dependent. Otherwise it is said to be densityindependent’’), the criterion is whether the predator’s rate of increase (and hence its population density) is related to prey density. Thus, in the case of a specialist predator the relationship may be tight and may result in (delayed) density dependence. In the case of a generalist predator, whose growth rate is not, or only weakly, related to the density of a single prey species (Hanski et al., 1993; Turchin and Hanski, 1997), predation will more or less act as a density-independent factor. We analysed data that were collected as part of a long-term sampling programme of the zoobenthic community of a tidal-flat area in the Dutch Wadden Sea from 1970. Various non-linear process models were fitted to the time series (29 years) of two polychaete species: Scoloplos armiger (O. F. Müller, 1776) and its main predator Nephtys hombergii (Savigny, 1818). Using the best-fitting model(s), we examined; (i) the relative importance of (delayed) density-dependent and densityindependent factors in determining population growth rate, and, specifically, the role of winter temperature, as the most likely candidate of a density-independent factor (Beukema, 1985); (ii) the role of predation, i.e., what is the effect of prey density on the growth rate of the predator and vice versa; and (iii) the dynamic behaviour of the fitted models: do they exhibit stable or complex behaviour? Lyapunov exponents were estimated to answer this question. 1489 1975 1980 1985 1990 1995 1975 1980 1985 1990 1995 (b) 5 4 3 2 1 1970 Figure 1. Time series of (a) log biomass density (g ash-free dry mass m 2) of Scoloplos armiger (continuous line) and Nephtys hombergii (broken line) at Balgzand and (b) average surface water temperature (C) in winter (December to February) in the Marsdiep tidal channel. Wadden Sea (Beukema, 1976; Dankers and Beukema, 1983). The two species merely live in the upper 10–15 cm of the sediment, where they roam through self-built burrow systems. S. armiger is a detritus feeder, and N. hombergii is a predator of small worms (including S. armiger), crustaceans, and molluscs (Blegvad, 1914; Curtis, 1977; Schubert and Reise, 1986; Bosselmann, 1991). From March to May, females of S. armiger produce egg cocoons that are fixed to the sediment. The larve develop within the cocoon, and upon leaving, the post-larvae immediately burrow in the sediment (Hartmann-Schröder, 1996). Breeding of N. hombergii takes place between April and June, and the larvae are planktonic. Data were collected at Balgzand, a 50 km2 tidal-flat area in the westernmost part of the Wadden Sea. Starting in 1970, 15 randomly chosen permanent plots were sampled annually in late winter (March). At each plot 0.95 m2 was sampled. Cores were taken to a depth of 30 cm and immediately sieved through a 1 mm mesh. More details of the sampling procedure are given by Beukema (1974). Abundance figures used are the annual mean biomass averaged over all 15 plots and expressed in g ash-free dry mass per m2 (Fig. 1a). Biomass data are preferred over numerical 1490 Jaap van der Meer et al. abundance as fragmented worms can be included unambiguously. Mean winter temperature data used are the threemonth average surface water temperatures (December to February) obtained from a daily sampling programme in the Marsdiep, the nearest tidal channel (Fig. 1b). This programme has been running since 1860 (Van der Hoeven, 1982). 2 1 Rt =b0 +b1Nt 1 +b11N2 t +b2Nt1 + 1 2 2 b22N2 t1 +b12Nt Nt1 +t. Turchin and Taylor (1992) considered only a restricted number of values for i (1, 0.5, 0, 0.5, . . ., 3). For the choice i =0, they used logN instead. Hence, the Perry-Zhou model is a special case of the TurchinTaylor model. The first-order Royama model: Rt =Rm exp(a0 a1Xt)+t Models In most stochastic population process models in discrete time, Rt, the log reproductive rate at time t, is related to x, a set of log population densities Xt, Xt1, Xt2, . . ., and to z, a set of exogenous variables Zt, Zt1, Zt2, . . ., by the function f: Rt =Xt+1 Xt =log(Nt+1/Nt)=f(x, z, ), where Nt is the population density at time t, and is a set of constant parameters. Various functions f have been proposed in the past (Berryman, 1992a). Since the results (i.e., the relative importance of x vs. z, and the stability properties of the model) may depend on the model choice, we used a variety of commonly applied models. The simplest stochastic models are the linear models. The second-order linear model is written as: Rt =b0 +b1Xt +b2Xt1 +t, where t is an independent, identically normally distributed number with zero mean. These linear models exhibit only relatively simple dynamics and cannot show chaotic behaviour. Furthermore, there are some conceptual problems, e.g., Rt tends to infinity when Nt tends to zero. We used three types of non-linear models. The first group contains the extensions of the Ricker model, such as suggested by Royama (1992). The second-order version can be written as: can be re-written as a first-order Turchin-Taylor model (first-degree polynomial): Rt =b0 +b1Nt +t, where Rm =b0, a0 = log(b1), a1 = . Thus, as long as b1 <0, the models are equivalent. This is not true for the second-order Royama model, which after re-writing becomes: 2 Rt =b0 +b1Nt 1Nt1 +t. The effects of known exogenous variables Zt, such as mean winter temperature or the density of a prey or predator species, were assumed to be additive: Rt =f(x)+Zt +t. This implies the assumption that any predator species included exhibits a type-I functional response. In other words, the predator term in the prey equation resembles the Nicholson-Bailey model. Normalized (i.e., zero mean and unit variance) data were used for all exogenous variables. A global stability analysis was performed by estimating the stochastic version of the Lyapunov exponent (Ellner and Turchin, 1995; Desharnais et al., 1997). For the first-order models, the exponent is given by: Rt =Rm exp(a0 a1Xt a2Xt1)+t. Second, polynomial models using log-transformed population densities were used, as suggested by Perry et al. (1993) and Zhou et al. (1997). The second-order model, using a second-degree (quadratic) polynomial, is written as: Rt =b0 +b1Xt +b11X2t +b2Xt1 + b22X2t1 +b12XtXt1 +t. Finally, polynomial models allowing a Box–Cox transformation of population densities, N, as suggested by Turchin and Taylor (1992), were used: where F is the (deterministic) function that relates Xt+1 to Xt (Ft =Xt +Rt), and n the number of observations. Exogenous variables in the model (e.g., winter temperature) were kept at the observed average. For the secondorder models the Jacobian-based method was used (Ellner and Turchin, 1995). Statistics Parameters were estimed by least-squares regression. The non-linear Turchin-Taylor model would become a linear model if the non-linear parameters i were known. Population dynamics of two marine polychaetes 1491 Table 1. Goodness-of-fit in terms of the residual sum of squares (RSS) and Mallow’s Cp of various models (see text) relating the log reproductive rate Rt of Scoloplos armiger and Nephtys hombergii to population densities Nt and Nt1. The length of the data series is 29 years, resulting in 27 tri-variate observations Rt, Nt and Nt1. S. armiger Model Constant Linear Royama Perry-Zhou – 2nd degree Turchin – 2nd degree Linear Royama Perry-Zhou – 2nd degree Turchin – 1st degree Turchin – 2nd degree N. hombergii Order df RSS Cp RSS Cp — 1 1 1 1 2 2 2 2 2 26 25 24 24 23 24 23 21 22 19 6.00 4.99 4.99 4.99 4.99 4.74 4.73 4.29 4.44 4.19 5.34 2.24 4.24 4.24 6.24 3.00 4.93 6.69 5.47 10.21 60.74 45.24 45.22 45.22 44.70 44.72 44.66 42.34 44.49 37.13 8.56 2.00 3.99 3.99 5.70 3.71 5.68 8.39 7.59 9.52 Consequently, linear methods could be used for fitting. For this practical reason, Turchin and Taylor (1992) considered only a restricted set of values for i, but this is unnecessarily simplistic (Perry et al., 1993). We used the Golub-Pereyra algorithm for partially linear leastsquares models implemented in S-plus, which estimates only the non-linear parameters by an iterative optimization procedure (Venables and Ripley, 1996). At each iteration step, the linear parameters are estimated by linear methods. We used the same procedure for fitting the Royama models, after a re-expression as presented above. Mallow’s Cp statistic was used for model judgement and selection (Wetherill, 1986): where RSS is the residual sum of squares, ˆ 2 is an estimate of the error variance, n is the number of observations, and p the number of fitted parameters. For least-squares methods, Mallow’s Cp is equivalent to Akaike’s information criterion. Bootstrap pseudo-replicated time-series were generated by simulation (Falck et al., 1995). The model parameters estimated on the basis of the real data were used in the simulations. For each simulated observation, the error was obtained by random sampling with replacement from the observed set of residuals. The initial value(s) of each simulated series were the same as for the real data, and its length was equal to the length of the real series. For each model, we created 500 bootstrap series, and parameters (and, for the non-linear models, the Lyapunov exponent) for each series were estimated in the same way as for the real data. The 95% confidence interval for the true value of each parameter was obtained by Efron’s first percentile method; i.e., the interval is given by the values that exceed 2.5% and 97.5%, respectively, of the generated bootstrap distribution (Manly, 1997). Results A first comparison of the first-order and second-order models revealed that the simplest model, that is the first-order linear model, had the lowest Cp statistic (Table 1). The sample auto-correlation functions and partial auto-correlation functions showed no indication for higher-order terms either (Fig. 2). Inclusion of the exogenous variable winter temperature resulted in a lower Cp statistic for N. hombergii, but not for S. armiger (Table 2, Fig. 3). Taking into account prey density did not improve the fit for the predator N. hombergii. Yet, the log-reproduction rate of the prey species, S. armiger, was related to the density of the predator (Table 2). Second-order models with winter temperature or prey/predator density included (not presented in Table 2) did not show any substantial improvement in goodness-of-fit over the first-order models. For example, the second-order Royama model for N. hombergii, including winter temperature revealed an RSS of 10.43, being only slightly lower than in the first-order model (RSS=11.12). For N. hombergii, the first-order linear model, including the effect of winter temperature, revealed an estimated parameter value b |1 = 0.61 (95% confidence interval: 0.77 to 0.50). This value falls in the range minus one to zero, indicating a stable equilibrium (Royama, 1992). A similar conclusion could be based on the estimated first-order Royama model. The local stability behaviour is indicated by the slope of the reproduction curve at the equilibrium point, which is given by minus the product of b |0 and |. The estimated value of this slope equalled 0.52 (95% confidence interval: 0.74 to 0.41), indicating that the equilibrium is 1492 Jaap van der Meer et al. 1.0 (a) 95 89 9083 75 92 9387 73 71 8474 81 72 77 1 Predicted 0.5 r 98 2 0.0 –0.5 0 –1 –3 0 4 2 6 8 97 91 82 –2 –1.0 86 9476 78 85 88 80 79 96 –2 –4 10 0 2 Observed 1.0 Figure 3. Observed log reproduction rate Rt of Nephtys hombergii versus the one-step-ahead predictions (rs =0.90). Predictions of the first-order model, Rt =b0 +b1Nt +b2Zt +t, where Zt is winter temperature. (b) r 0.5 0.0 The results for S. armiger were similar. The first-order linear model (including predator density P) revealed the estimate b |1 = 0.59 (95% confidence interval: 0.89 to 0.42). The first-order Royama model yielded a slope at the equilibrium point of 0.60 (95% confidence interval: 1.05 to 0.41). The estimated Lyapunov exponent was negative, i.e. |n = 0.90 (95% confidence interval: 2.38 to 0.50). We estimated the local and global stability in the same way for all other first-order and second-order models (although for these models the decrease in the residual sum of squares did not keep pace with the increase in the number of model parameters), and the conclusion was always the same: the equilibrium is globally stable. Apparently, the fluctuations of the two polychaete populations cannot be categorized as chaotic. –0.5 –1.0 2 4 6 Lag 10 8 Figure 2. Sample (a) auto-correlation functions and (b) partial auto-correlation functions of the time series of log biomass density of Scoloplos armiger (continuous lines) and Nephtys hombergii (broken lines) at Balgzand. Lag in years. locally stable (Royama, 1992). The estimated Lyapunov exponent was negative, i.e. |n = 0.76 (95% confidence interval: 1.44 to 0.53), pointing also to global stability (Ellner and Turchin, 1995). Table 2. Goodness-of-fit in terms of the residual sum of squares (RSS) and Mallow’s Cp of various first-order models (see text) relating the log reproductive rate Rt of Scoloplos armiger and Nephtys hombergii to population densities Nt, winter temperatures Zt, and the density of the ‘‘other’’ species Pt. The length of the data series is 29 years, resulting in 28 multi-variance observations Rt, Nt, Zt and Pt. S. armiger Model Constant Linear Royama Perry-Zhou – 2nd degree Turchin – 2nd degree Linear Royama Perry-Zhou – 2nd degree Turchin – 2nd degree Linear Royama Perry-Zhou – 2nd degree Turchin – 2nd degree N. hombergii Covariate df RSS Cp RSS Cp — — — — — Z Z Z Z P P P P 27 26 25 25 24 25 24 24 23 25 24 24 23 6.03 4.99 4.99 4.99 4.99 4.99 4.99 4.99 4.99 3.35 3.35 3.35 3.35 19.02 13.23 15.23 15.23 17.23 15.23 17.22 17.23 19.22 3.00 4.98 4.98 6.98 61.20 45.65 45.64 45.64 45.20 11.81 11.12 11.22 11.11 43.54 43.20 43.02 42.77 106.10 74.52 76.52 76.52 77.56 3.49 4.00 4.21 5.98 71.99 73.24 72.85 74.32 Population dynamics of two marine polychaetes Discussion One problem with using simple non-linear time-series models to describe and characterize the dynamics of populations is that different models may fit equally well, but result in qualitatively different characterizations of the dynamics of the population (Morris, 1990). Berryman (1992a) replied that model choice should not only be based on a goodness-of-fit criterion (or on subjective personal preference), but also on the basis of ecological credibility. A similar view was expressed by Royama (1992), who criticized the generalization of the Ricker model by Turchin and Taylor (1992) to cases with more than one time lag. Royama did not consider this a good model because it is hard to find a possible ecological mechanism for its form. A lack of ecological credibility also holds for the so-called non-linear selfexciting threshold autoregressive (SETAR) models, which have been advanced recently (Grenfell et al., 1998; Stenseth and Chan, 1998). Future population size is modelled as a discontinuous piecewise linear function of the previous population size. This approach allows for a sudden drop from a strongly positive population growth rate to a negative rate when the population size increases at only an infinitely small step. Such response is ecologically rather unlikely, but it fully characterizes the dynamical properties of the (deterministic skeleton of the) model. Berryman (1992a) goes one step further, and takes the radical view that conclusions drawn from incorrect models must themselves be incorrect. Although we sympathize with the plea for ecological credibility, this must be an overstatement. It is widely accepted that all models in ecology are in a way incorrect (Levin, 1981), but the chief question is the extent to which conclusions are robust given imperfect models. However, we were not really confronted with this problem, since none of the fitted models indicated complex dynamics, and therefore we may safely conclude that the occurrence of chaos is rather unlikely in this system. Another point of concern with the use of non-linear time series to characterize the type of dynamics is the risk of over fitting. The response–surface–methodology models as applied by Turchin (1993), for example, had as many as 13 parameters, which is high if there are only as few as 16 to 28 data points. Therefore, Falck et al. (1995) suggest that a better approach for short time series would be to test the hypothesis of linearity first. If that hypothesis cannot be rejected, it does not make sense to search for chaos. We only found marginal differences in goodness-of-fit (in terms of RSS) between linear and non-linear models. The product of the two parameters of the non-linear Royama model that indicates local stability could be estimated with reasonable accuracy, but the parameter estimates separately had much wider confidence intervals. For both species the 95% confidence interval of | (0.58 to +0.19 for N. 1493 hombergii; 1.67 to +2.06 for S. armiger) contained zero, implying that the non-linear component of the model could not reliably be estimated and that the hypothesis of linearity could not be rejected. Generally speaking, predation may lead to a delayed density-dependent response of prey. The feedback mechanism works through high abundance of prey resulting in large numbers of predators, followed by high levels of prey mortality. We found a negative effect of predator abundance on the log reproductive rate of the prey, but no effect of prey abundance on the predator. Delayed density dependence could not be detected either. Apparently, the feedback mechanism was interrupted. This may be because N. hombergii is a generalist predator (Blegvad, 1914; Curtis, 1977; Schubert and Reise, 1986; Bosselmann, 1991), implying that the density of merely one of its prey species could be of minor importance for its total food supply and population growth rate. Note that prey and predator biomass densities were of the same order of magnitude, and energetic considerations make a tight predator–prey coupling also rather unlikely. Thus, the density of N. hombergii can be categorized as a density-independent factor, with respect to the dynamics of S. armiger, one of its prey species. For both species, evidence was obtained for densitydependent regulation, but the impact of winter temperature, a density-independent factor, on the population growth rate of N. hombergii was remarkably strong. One might state that winter temperature also had a considerable impact, albeit indirect and delayed, on the population growth rate of S. armiger. Including winter temperature of the year before had indeed almost the same effect on the population growth rate of S. armiger as the density of N. hombergii. In fitting simple time-series models for a two-species situation, we were able to categorize each effect included as either density-dependent or not. For example, prey density did not affect predator dynamics and, following the definition by Royama (1992), predator density was therefore classified as a density-independent factor with respect to prey dynamics. But suppose that the predator’s rate of change had been influenced both by weather, and prey density. According to Royama’s definition, predator density would then have to be regarded as a density-dependent factor with respect to prey dynamics, despite the fact that predator dynamics were to a large extent determined by weather. 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