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Using stochastic population process models to predict the impact of climate change Jaap van der Meer , J.J. Beukema, Rob Dekker NIOZ Royal Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Den Burg, The Netherlands abstract article info Article history: Received 28 October 2011 Received in revised form 20 August 2012 Accepted 31 August 2012 Available online 8 September 2012 Keywords: Wadden Sea Population dynamics Non-linear time series models Global change More than ten years ago a paper was published in which stochastic population process models were tted to time series of two marine polychaete species in the western Wadden Sea, The Netherlands (Van der Meer et al., 2000). For the predator species, model ts pointed to a strong effect of average sea surface winter temper- ature on the population dynamics, and one-year ahead model forecasts correlated well with true observa- tions (r = 0.90). During the last decade a pronounced warming of the area occurred. Average winter temperature increased with 0.9 °C. Here we show that despite the high goodness-of-t whilst using the orig- inal dataset, predictive capability of the models for the recent warm period was poor. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Climate change may have a crucial inuence on population dy- namics. Physiologically-structured population models are probably the most appropriate tool for obtaining a mechanistic understanding of the role of climate-related environmental factors in interaction with endogeneous feedback processes. Yet, these models require de- tailed information on food availability, feeding processes, reserve dynamics, growth, reproduction and survival rates in relation to size and reserves, and so on. For many species such detailed informa- tion is lacking. If only long-term abundance data are available, time series analysis offers an alternative approach (Bjørnstad and Grenfell, 2001). Over the last decades a growing interest arose in the use of linear and non-linear time series models to describe the ef- fects of climatic factors and ultimately predict the impact of climate change. Most studies followed the approach advocated by Royama (Berryman, 1999; Royama, 1992) and used diagnostic tools such as autocorrelation and partial autocorrelation functions in combination with building stochastic population process models. These models relate the per capita rate of population change log(N t +1 /N t ), where N t is the population density at time t, to the population densities at previ- ous time steps, N t , N t -1 , N t -2 , and to environmental (climatic) noise Z t by some linear or non-linear function. Examples are Soay sheep responding to the occurrence of gales, winter severity and vegetation production (Berryman and Lima, 2006; Grenfell et al., 1998; Stenseth et al., 2004), wolves affected by winter temperature and snow accumu- lation (Ellis and Post, 2004), Alpine ibex responding to the maximum depth of snow (Lima and Berryman, 2006), reindeer responding to the rate of snow melting and winter severity (Tyler et al., 2008), rodents inuenced by vegetation production, spring and summer rainfall and spring temperature (Andreo et al., 2009), and wolverines that have to cope with declining snowpack depth (Brodie and Post, 2010). Whilst the population process models applied in these studies de- scribed the observed time series reasonably well, it remains to be seen whether the models are capable of correctly predicting the population trajectory under future and perhaps more severe climate change. Only Jacobsen et al. (2004) and Lima and Berryman (2006), studying the Al- pine ibex, split their data set in two parts. The training set covered 20 years and the test set 19 years. Berryman and Lima showed that sev- eral of the models that performed well during the rst period in terms of the usual selection criteria such as the coefcient of determination R 2 or Akaike's Information Criterion AIC, had poor predictive capabili- ties. They concluded that the only real way to judge and compare eco- logical models with any degree of condence is in their ability to predict independent observations (Lima and Berryman, 2006). Yet, they did not further discuss what is actually meant with independent observations. A general problem with statistical models, to which these stochastic population process models belong, is to clearly dene the domain for which they are actually valid. A statistical model may be perfectly capable to predict future and independent observations if environmental conditions do not systematically change, but at the same time may fail to correctly predict population changes when the environment runs out of the previous domain. Ten years ago, we tted several linear and non-linear stochastic pop- ulation process models on 29-year time series (19701998) of the abundance of two polychaete species in the western Wadden Sea (Van der Meer et al., 2000). The predatory species Nephthys hombergii, which happens to be a generalist predator, did not respond to the abundance of the prey species Scoloplos armiger, but was strongly inuenced by winter temperature. In contrast, the per capita rate of change of the prey species was negatively affected by the predator Journal of Sea Research 82 (2013) 117121 Corresponding author. Tel.: +31 222 369357. E-mail address: [email protected] (J. van der Meer). 1385-1101/$ see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.seares.2012.08.011 Contents lists available at SciVerse ScienceDirect Journal of Sea Research journal homepage: www.elsevier.com/locate/seares

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Page 1: Using stochastic population process models to predict the impact of climate change

Journal of Sea Research 82 (2013) 117–121

Contents lists available at SciVerse ScienceDirect

Journal of Sea Research

j ourna l homepage: www.e lsev ie r .com/ locate /seares

Using stochastic population process models to predict the impact ofclimate change

Jaap van der Meer ⁎, J.J. Beukema, Rob DekkerNIOZ Royal Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Den Burg, The Netherlands

⁎ Corresponding author. Tel.: +31 222 369357.E-mail address: [email protected] (J. van de

1385-1101/$ – see front matter © 2012 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.seares.2012.08.011

a b s t r a c t

a r t i c l e i n f o

Article history:Received 28 October 2011Received in revised form 20 August 2012Accepted 31 August 2012Available online 8 September 2012

Keywords:Wadden SeaPopulation dynamicsNon-linear time series modelsGlobal change

More than ten years ago a paper was published in which stochastic population process models were fitted totime series of two marine polychaete species in the western Wadden Sea, The Netherlands (Van der Meer etal., 2000). For the predator species, model fits pointed to a strong effect of average sea surface winter temper-ature on the population dynamics, and one-year ahead model forecasts correlated well with true observa-tions (r=0.90). During the last decade a pronounced warming of the area occurred. Average wintertemperature increased with 0.9 °C. Here we show that despite the high goodness-of-fit whilst using the orig-inal dataset, predictive capability of the models for the recent warm period was poor.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Climate change may have a crucial influence on population dy-namics. Physiologically-structured population models are probablythe most appropriate tool for obtaining a mechanistic understandingof the role of climate-related environmental factors in interactionwith endogeneous feedback processes. Yet, these models require de-tailed information on food availability, feeding processes, reservedynamics, growth, reproduction and survival rates in relation tosize and reserves, and so on. For many species such detailed informa-tion is lacking. If only long-term abundance data are available, timeseries analysis offers an alternative approach (Bjørnstad andGrenfell, 2001). Over the last decades a growing interest arose inthe use of linear and non-linear time seriesmodels to describe the ef-fects of climatic factors and ultimately predict the impact of climatechange. Most studies followed the approach advocated by Royama(Berryman, 1999; Royama, 1992) and used diagnostic tools such asautocorrelation and partial autocorrelation functions in combinationwith building stochastic population process models. These modelsrelate the per capita rate of population change log(Nt+1/Nt), where Nt

is the population density at time t, to the population densities at previ-ous time steps, Nt, Nt−1, Nt−2,… and to environmental (climatic) noiseZt by some linear or non-linear function. Examples are Soay sheepresponding to the occurrence of gales, winter severity and vegetationproduction (Berryman and Lima, 2006; Grenfell et al., 1998; Stensethet al., 2004), wolves affected bywinter temperature and snow accumu-lation (Ellis and Post, 2004), Alpine ibex responding to the maximumdepth of snow (Lima and Berryman, 2006), reindeer responding to therate of snow melting and winter severity (Tyler et al., 2008), rodents

r Meer).

rights reserved.

influenced by vegetation production, spring and summer rainfall andspring temperature (Andreo et al., 2009), and wolverines that have tocope with declining snowpack depth (Brodie and Post, 2010).

Whilst the population process models applied in these studies de-scribed the observed time series reasonably well, it remains to be seenwhether the models are capable of correctly predicting the populationtrajectory under future and perhaps more severe climate change. OnlyJacobsen et al. (2004) and Lima and Berryman (2006), studying the Al-pine ibex, split their data set in two parts. The training set covered20 years and the test set 19 years. Berryman and Lima showed that sev-eral of the models that performed well during the first period in termsof the usual selection criteria such as the coefficient of determinationR2 or Akaike's Information Criterion AIC, had poor predictive capabili-ties. They concluded that the only real way to judge and compare eco-logical models with any degree of confidence is in their ability topredict independent observations (Lima and Berryman, 2006). Yet,they did not further discuss what is actually meant with independentobservations. A general problem with statistical models, to whichthese stochastic population process models belong, is to clearly definethe domain for which they are actually valid. A statistical model maybe perfectly capable to predict future and independent observations ifenvironmental conditions do not systematically change, but at thesame time may fail to correctly predict population changes when theenvironment runs out of the previous domain.

Ten years ago,wefitted several linear andnon-linear stochastic pop-ulation process models on 29-year time series (1970–1998) of theabundance of two polychaete species in the western Wadden Sea(Van der Meer et al., 2000). The predatory species Nephthys hombergii,which happens to be a generalist predator, did not respond to theabundance of the prey species Scoloplos armiger, but was stronglyinfluenced by winter temperature. In contrast, the per capita rate ofchange of the prey species was negatively affected by the predator

Page 2: Using stochastic population process models to predict the impact of climate change

Table 1Parameter estimates ±SE and residual sum of squares (RSS) of the linear model Rt=b0+b1Xt+b2Zt+ t, where Rt is the log reproductive rate, Xt the log population density,and Zt a normalized environmental variable; 25 degrees of freedom. See text for furtherdetails.

Species b0 b1 b2 RSS

N. hombergii −1.110±0.209 −0.613±0.093 1.129±0.136 12.29S. armiger −0.719±0.209 −0.530±0.134 −0.284±0.083 3.73

118 J. van der Meer et al. / Journal of Sea Research 82 (2013) 117–121

abundance. For both species first-order linear models performed best.Particularly for the predator species, the fit was good. The correlationbetween the one-year ahead forecasts, using the observed winter tem-perature, and the true observations equalled 0.90. During the last11 years, that is after themodel results were published, winter sea sur-face temperature increased considerably in the western Wadden Sea,reflecting not so much a change in the zonal winter winds over theNorth Atlantic Ocean as indicated by North Atlantic Oscillation (NAO)index, but merely a real large-scale warming trend (Van Aken, 2008).It seems that climate has run out the domain as occurring during the pe-riod 1970–1998. Here we use the data from this relatively warm period1999–2011 to test the stochastic population process models that werefitted on the 1970–1998 data. This way we explore the predictive capa-bility of ecological time series models.

2. Materials and methods

Datawere collected at Balgzand, a 50‐km2 tidal-flat area in thewest-ernmost part of the Wadden Sea. Starting in 1969 and up to 2011, 12randomly chosen transects of 1‐km long and 3 randomly chosen quad-rats with a size of 30 by 30 m were sampled annually in late winter(March). At each transect 50 cores with a surface area of 0.019 m2

were taken covering a total of 0.95 m2. In each quadrat approximatelya similar surface area was sampled. Cores were taken to a depth of30 cm and immediately sieved through a 1 mm mesh. Animals weresorted and counted alive in the laboratory, dried during two to threedays at 60 °C in a ventilated stove, and incinerated. Ash mass wassubtracted from dry mass to obtain ash-free dry mass. More details ofthe sampling procedure are given in Beukema (1974). Abundance fig-ures used are the annual mean biomass of N. hombergii and S. armigeraveraged over all 15 sites and expressed in ash-free dry mass per m2.Biomass data are preferred over numerical abundance as fragmentedworms can be included unambiguously.

Mean winter temperature data used are the three-month averagesurface water temperatures (December to February) obtained from adaily sampling programme in the Marsdiep, the nearest tidal channel(Van Aken, 2008).

Data from the period 1970–1998 have been used in a previouspublication by Van der Meer et al. (2000), but since then a few datacorrections have taken place, resulting in minor changes in parameterestimates and goodness-of-fit measures.

3. Models

First-order stochastic population processmodels in discrete time areused (Berryman, 1999; Royama, 1992). The log reproductive rate Rt isrelated to log population density Xt= logNt and to an exogeneous vari-able Zt by a function f:

Rt ¼ Xtþ1−Xt ¼ logNtþ1

Nt

� �¼ f Xt ; Zt ; ζð Þ þ �t

where ζ is a parameter vector, and t is an independent, identically nor-mally distributed variable with zero mean. Based on the resultsobtained by Van der Meer et al. (2000), first-order linear models areused, with winter temperature as the exogenous variable Zt for N.hombergii and the density of N. hombergii as the exogenous variable Ztfor S. armiger. The first-order linear model can be written as

Rt ¼ b0 þ b1Xt þ b2Zt þ �t

Both the terms b2Zt and twould be classified as vertical perturbationeffects by Royama (1992).Van der Meer et al. (2000) using the1970–1998 data, showed that more complex and higher-ordernon-linear models, such as the models proposed by Royama (1992) or

by Ellner and Turchin (1995), did not result in improved fits. Suchmodels are therefore no longer considered here.

4. Statistics, forecasts and predictions

Parameters of the first-order linear models are estimated byleast-squares regression using the 1970–1998 data. One-year aheadforecasts of the log reproductive rate Rt are made using the estimatedparameters based on the 1970–1998 period, which may be called thetraining dataset. These forecasts require knowledge of the logpopulation density Xt and the exogeneous variable Zt. Forecasts withinthe training data set are compared with forecasts within the test dataset, i.e. the years 1999–2011.

Two types of long-term predictions aremade for the 1999–2011 pe-riod, using in both cases the estimated parameters based on the1970–1998 period. The first type of prediction only uses the exactknowledge on the annual temperatures during the period 1999–2011.Hence knowledge of the population densities during the period1999–2011 is not used. Predictions for the log reproductive rate Rt ofthe prey species S. armiger are thus based on the predicted values Xtfor the predator species N. hombergii. The second type of predictionalso uses knowledge of the other species' density during the period1999–2011. Hence, predictions for the log reproductive rate Rt of theprey species S. armiger are now based on the observed values Xt forthe predator species N. hombergii. For the predator species, there is nodifference between the two types of predictions, since its reproductiverate is not a function of the prey species.

For each model 1000 bootstrap pseudo-replicated time series forthe testing period are generated by simulation, using the estimatedset of parameters and sampling with replacement from the observedset of residuals from the training period. The 95% prediction intervalsare obtained by Efron's first percentile method (Manly, 1997).

5. Results

The estimate for themodel parameter b1 falls in the interval rangingfrom−1 to 0, both for N. hombergii and S. armiger (Table 1). Hence theequilibrium point is globally stable and the series will, apart from fluc-tuations caused by the environmental variable Zt and random noise,asymptotically converge to the equilibrium point (Royama, 1992). Forthe predatory worm N. hombergii the environmental variable averagewinter temperature had a negative effect on the log reproductive rate.The predator population therefore increases at low densities andwarm winters and decreases at high densities and cold winters(Fig. 1). The prey population S. armiger responds negatively to the pred-ator population, since the parameter b2 is negative.

Averagewinter temperature data showed higher values over the test-ing period (1999–2011) than over the training period (1970–1998). Thedifference was equal to 0.9 °C, which is equivalent to 0.57 SD units(Fig. 2). As a result the model, estimated using the data from the trainingperiod, predicts for the predator N. hombergii an increase in the popula-tion equilibrium equal to−b2

b1ΔZt , where ΔZt is the change in the normal-

ized average winter temperature of 0.57 SD units. In back-transformednumbers it means an increase in predators with a factor of 2.8, which isconsiderable. The predicted population trajectory for N. hombergii washowever consequently above the observed series (Fig. 3). Since the

Page 3: Using stochastic population process models to predict the impact of climate change

−6 −5 −4 −3 −2 −1 0

01

23

45

6

Log biomass density

Win

ter

tem

pera

ture

in d

egre

es C

elsi

us

Fig. 1. Three-month average surface water temperatures (December to February) in °Cin the Marsdiep versus log biomass density in g per m2 of N. hombergii for the period1970–1998. Symbols show the log reproductive rate, where a triangle pointing up in-dicates a population increase and one pointing down a decrease. Symbol size indicatesthe change in population size.

1970 1980 1990 2000 2010

−6

−4

−2

02

4

Year

Log

biom

ass

dens

ity

Fig. 3. Predicted (thin grey solid line) and observed (black solid line) log biomass den-sity Xt for the predator N. hombergii versus time. Predictions on the basis of populationdensity and winter temperature. Grey dashed lines refer to the 95% bootstrap interval.Thick grey solid line gives the prediction in the absence of perturbation effects, butwith a systematic change in temperature. It shows a rapid transition from the initialvalue and from the equilibrium in the training period when the average normalizedwinter temperature Zt equalled zero to the equilibrium in the testing period when Zt ¼0:57 SD.

Mean density

Fre

quen

cy

−1.7 −1.6 −1.5 −1.4 −1.3 −1.2

050

100

150

200

Fig. 4. Bootstrap frequency distribution of the average log biomass density Xt for the pred-atorN. hombergii during the testing period. Grey circle indicated the observed average logbiomass density.

119J. van der Meer et al. / Journal of Sea Research 82 (2013) 117–121

observed average biomass density during the testing period 1999–2011was situated in the tail of the bootstrap distribution (Pb0.01), the obser-vation is classified as significantly different from the prediction (Fig. 4). Ifthe observed population is below the predicted equilibrium, one expectson average apopulation increase. Indeed,most forecasts of the populationchange R in the testing period showed a positive sign, implying a popula-tion increase. The actual population change was, however, in most ofthese cases considerably lower than the forecast (Figs. 5 and 6).

Forecasts for the prey species S. armiger do not systematically differbetween the training and the testing period (Figs. 7 and 8). Predictionsare also in accordance with observations when based upon observeddensities of the predator species (Fig. 9). When, however, predictedpredator densities are used to predict the prey density trajectory duringthe testing period, then predictions fall apart from observations (Fig. 9).

6. Discussion

The goodness-of-fit of the 1970–1998 models was high, particularlyfor N. hombergii. The correlation between the one-year ahead forecastscorrelated well (r=0.90) with the true observations (Fig. 5). Theresidual standard error equalled 0.702, which can be interpreted as anaverage difference by a factor of 2.03 between forecast andobservation. For S. armiger the correlation was r=0.5 and the averagedifference is a factor of 1.48. These differences should be interpreted

1970 1980 1990 2000 2010

−2

−1

01

2

Year

Sta

ndar

dize

d w

inte

r te

mpe

ratu

re

Fig. 2. Three-month average surface water temperatures (December to February) in °Cin the Marsdiep. The average temperature during the second period (1999–2011) is0.9 °C, which equals 0.57 in SD units, higher than during the first period (1970–1998).

−4 −3 −2 −1 0 1 2

−3

−2

−1

01

2

Observed R

For

ecas

ted

R

69

70

71

7273

74

7576

77

78

79

80

81

82

83

84

85

86

87

88

89

90

9192

93

94

95

96

97

9899

0001

02

03

04

05

06

07

0809

10

Fig. 5. Forecasted versus observed log reproductive rate Rt for the predator N. hombergii.Forecasts on the basis of population density and winter temperature. Grey numbersrefer to the training data set.

Page 4: Using stochastic population process models to predict the impact of climate change

1970 1980 1990 2000 2010

−6

−5

−4

−3

−2

−1

01

Year

Log

biom

ass

dens

ity

Fig. 6. Forecasted and observed changes in log biomass density Xt for the predatorN. hombergii versus time. Forecasts on the basis of population density and winter temper-ature. Grey arrows refer to the forecast within the training data set.

1970 1980 1990 2000 2010

−2.

5−

2.0

−1.

5−

1.0

−0.

50.

0

Year

log

Bio

mas

s de

nsity

Fig. 8. Forecasted and observed changes in log biomass density Xt for the prey S. armigerversus time. Forecasts are based on observed population densities and predator densities.Grey arrows refer to the forecast within the training data set.

120 J. van der Meer et al. / Journal of Sea Research 82 (2013) 117–121

in comparison with the observation that the maximum and minimumaverage densities are by a factor 480 for N. hombergii and by a factorof 17 for S. armiger. Although it is not always clear what thesegoodness-of-fit values are in other studies, we get the impression thatmost fits are not that good. The ibex studies reported coefficients of de-termination between 0.78 and 0.93 (Jacobsen et al., 2004; Lima andBerryman, 2006).

Despite these promising results using the training set, one-yearahead forecasts and long-term predictions using the testing set weresignificantly different from the observed trajectory. The predicted in-crease in predator abundance and decrease in prey abundance did nottake place. One might argue that with the choice of another time seriesmodel, a better job would have been done. Perhaps, but we examined awide range of models that had been used before and our model selec-tion procedure was more or less standard. Mallow's CP was used asthe selection criterion, which is for least squares methods equivalentto the widely used Akaike's Information Criterion (AIC). The linearmodels revealed a lower CP than all the non-linear models that we ex-amined and several authors have argued that it does not make muchsense to use the more complex non-linear models if non-linearity can-not be shown unambiguously (Falck et al., 1995). Nevertheless, we ap-plied several non-linear models, for example the first-order Royamamodel Rt=Rm−exp(a0−a1Xt)+ t, on the testing set but the resultshardly changed. Forecasts and predictions for the testing set remainedpoor.

−0.5 0.0 0.5 1.0

−0.

4−

0.2

0.0

0.2

0.4

0.6

Observed R

For

ecas

ted

R

69

70

71 72

73

74

75

76

77

78

79

80 81

8283

84

85

8687

88

89

90

91

92

9394

95

96 97

98

99

00

01

02

03

04

0506

070809

10

Fig. 7. Forecasted versus observed log reproductive rate Rt for the prey S. armiger. Fore-casts are based on observed population densities and predator densities. Grey numbersrefer to the training data set.

Amore serious point of concern, however, becomes apparent from acareful look at the residuals of the forecasts for N. hombergii (Fig. 10). Itis indeed true that most years within the testing period showed amuchlower log reproductive rate R than forecasted, but a declining trend inthe residuals seems to have started already in the mid 1980s. Onemight thus argue that the original model, despite the high correlationof r=0.90 between model and observations, was not appropriate be-cause of this non-random temporal pattern in the residuals, and thatthe failure of themodel to predict future population abundance as a re-sult of temperature change may not come as a surprise. However, thisargument may also be turned around, because this residual pattern isonce again a warning signal that unknown factors may change overtime and can ruin the predictive power even of models that at firstsight show a very good fit. If we would have used data up till 1985 asthe training set, the correlation between model and observationswould have been even higher, r=0.94, and the pattern in residualswould not have been present.

Hence the poor predictability may have been caused by interactionswith another factor, which levels have changed over the last decades.One might for example think of the introduction of invasive species,such as the exotic polychaeteMarenzelleria viridis. Yet, its biomass den-sity has increased only from 1996 onwards to values much higher thanthose of N. hombergii or S. armiger. This all remains a speculation and amore complete understanding of the complex interactions within thebenthic community will only be achieved when knowledge of the

1970 1980 1990 2000 2010

−6

−5

−4

−3

−2

−1

01

Year

log

Bio

mas

s de

nsity

Fig. 9. Predicted (grey solid lines) and observed (black solid line) log biomass densityXt for the prey S. armiger versus time. Predictions are based upon the population den-sities and either on the observed (thick grey solid line) or on the predicted (thin greysolid line) predator densities. Grey dashed lines refer to the 95% bootstrap interval.

Page 5: Using stochastic population process models to predict the impact of climate change

1970 1980 1990 2000 2010

−1.

5−

1.0

−0.

50.

00.

51.

01.

5

Year

Res

idua

l

Fig. 10. Residuals, that is observed R minus forecasted R for N. hombergii versus time.Grey symbols refer to the training set.

121J. van der Meer et al. / Journal of Sea Research 82 (2013) 117–121

species' physiology and ecology allows the building of mechanistic andindividual-based models. This will take some time and for the timebeing predictions on the impact of climate change on population dy-namics on the basis of time series models should be treated withsome skepticism.

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