Understanding demography in an advective environment: modelling Calanus finmarchicus in the Norwegian Sea

Authors


Douglas C. Speirs, Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, Richmond Street, Glasgow G1 1XT, Scotland, UK. E-mail: dougie@stams.strath.ac.uk

Summary

  • 1Attempts to understand the demography of natural populations from time-series can be hampered by the fact that changes due to births and deaths may be confounded with those due to movement in and out of the sampling area.
  • 2We illustrate the problem using a stage-structured time-series of the marine copepod Calanus finmarchicus sampled in the vicinity of a fixed location but where the demography is shown to be inconsistent with the assumption of a closed population.
  • 3By combining a realistic simulation of the hydrodynamic environment with a model of phenology we infer the time and location at which the stages observed in each sample were recruited as eggs. This yields a spatial and temporal map of the recruitment history required to produce the observed densities.
  • 4Using an empirical relationship between C. finmarchicus egg production and the abundance of phytoplanktonic food, the spatio-temporal patterns in chlorophyll a can be inferred. The distributions during the spring bloom are spatially heterogeneous, and we estimate that the phytoplankton patches are of the order of 30 km across. This result is robust to substantial variations in the assumed stage-dependent mortalities.
  • 5We conclude that information on advective transport can be used to make testable predictions about the scale of spatial heterogeneities. These, in turn, imply the appropriate spatial scale over which time-series might be replicated in order to obtain more information about unknown processes such as mortality.

Introduction

Demographic data on natural populations frequently come in the form of time-series obtained by repeatedly sampling within restricted areas. At its most basic level, making sense of such data means being able to do book-keeping on the fundamental processes of birth, death, immigration and emigration. In spatially homogeneous systems immigration and emigration from the sampling area will balance even when their magnitudes are large, and so the migration terms can be safely ignored. For spatially heterogeneous populations, however, the observed dynamics will potentially be affected by asymmetrical migration terms which therefore need to be quantified.

One broad class of organism whose populations are notoriously spatially heterogeneous and where the basis for movement is particularly simple, are the zooplankton. Planktonic organisms are by definition incapable of directly controlling their position in relation to water currents. As a consequence, in advective environments − that is, environments where transport involves a deterministic translation in space such as those produced by oceanic or river currents − individuals can be dispersed over large distances. This combination of patchiness in space and advective transport means that it is difficult to distinguish changes in population density arising from births and deaths from those due to a spatially heterogeneous population advecting past the observation point, unless the vital rates are independently known.

Although the difficulty outlined above is widely recognized (Aksnes & Blindheim 1996; Aksnes et al. 1997), it is rarely directly addressed in practice. Most commonly it is assumed that the population drifting past a fixed sampling location is spatially homogeneous over distances at least as large as those produced by advective transport during the period of observation. This is justified only if transport is negligible, or if the period of interest is very short. However, in the open ocean currents can yield displacements of the order of 10 km day−1. Thus, unless we can assume spatial homogeneity over very large distances, any meaningful demographic analysis is limited to processes occurring over a few days at best, a period much shorter than the typical lifespan of most meso-zooplankton species.

One of the major reasons why advection is frequently ignored in demographic analysis is that often little or nothing is known about physical transport itself. In recent years, however, coupled physical–biological models, in which spatial population models are driven by realistic hydrodynamics, have become more common (e.g. Lynch et al. 1998; Gurney et al. 2001). The explicit representation of the movement in such models provides the necessary linkage between the observed local dynamics and spatial heterogeneity in the wider environment. Unfortunately, this does not guarantee that we can make sense of data because as a rule we know nothing about the initial condition of the population in space, or about temporally and spatially varying factors, such as food availability, that can drive the observed demographic heterogeneity. Thus, even when transport is quantified, when only single time-series available we are usually in a position where the factors determining the population dynamics are under-constrained.

In this paper we argue that, given a few key assumptions, it is none the less often possible to make testable inferences from such time-series on the basis of coupled physical–biological models. Suppose, first, that our time-series takes the form of stage-structured data (or that censused individuals can be aged). If advection rather than diffusion dominates physical transport, then all individuals observed at a given time must have experienced the same environment and hence must have been born at the same time and location. If we either know or assume mortality and development rates, then running a structured population model which includes physical transport backwards in time from the observations yields a recruitment history that extends in both space and time. If transport is unimportant then this recruitment history may vary in time but must be similar at all locations. If, however, spatial heterogeneity in recruitment is inferred this implies a significant role for transport.

In order to illustrate the above assertions we will focus on the demography of the herbivorous marine copepod Calanus finmarchicus sampled at a fixed location in the Norwegian Sea. The following sections therefore begin by detailing the biology of this copepod, and by outlining the sampling programme and the resultant stage-structured data. It is shown that the observed time-series are inconsistent with closed-population demography. We then introduce a hydrodynamic simulation which can be used to locate the time and place at which individuals observed at a given stage were recruited into the population as eggs. It turns out that this implies a spatial patchiness to the phytoplankton on which C. finmarchicus feeds which is robust to a wide variation in mortality rates. These results imply that the apparent inconsistencies in the observed time-series of C. finmarchicus at this site can be understood in the context of hydrodynamic transport. Moving beyond this particular species, we conclude that for populations where physical transport is important and can be quantified, the scale of demographic spatial heterogeneity identified using our approach can be used to develop improved sampling programmes.

Calanus finmarchicus

Secondary production by zooplankton is a key linkage in most open-sea foodwebs − pumping carbon fixed by primary producers (phytoplankton) upwards into trophic groups of more immediate commercial and aesthetic concern such as pelagic fish, marine mammals and seabirds. Throughout the northern part of the Atlantic Ocean and the Norwegian sea the key player in this process is the boreal copepod C. finmarchicus (Gunnerus). At the northern end of its range (north of Iceland) it co-occurs with the Arctic species C.glacialis Jashnov and C.hyperboreus Kroyer, while in its southerly limit (north-east Atlantic, South Norwegian Sea, North Sea) it is found along with C.helgolandicus (Claus). However, throughout its central range it is by far the most abundant meso-zooplankton species and is thus responsible for much of the of secondary production.

C. finmarchicus passes through six naupliar and five copepodite stages (which we label as N1→N6 and Cl→C5, respectively) during its passage from egg to reproductively mature adult. During the spring and summer months the population is found mainly in the upper part of the water column and contains all life-history stages. During the winter, mature adults, eggs, nauplii and early copepodite stages are essentially absent; the population then consisting almost exclusively of late copepodite stages, which are found mainly at depths in excess of 500 m. Here they seek refuge from predation and reduce their physiological costs by entering a state akin to the diapause exhibited by terrestrial insects.

Although C.finmarchicus is ecologically important in shelf regions such as the North Sea, it does not overwinter there in substantial numbers, but is believed to re-invade in spring after overwintering in deeper water (Heath et al. 1999). A prerequisite to an understanding of the long-term dynamics of C. finmarchicus in shelf-waters is thus an understanding of its dynamics in deep ocean basins. Although it is known that these are determined by a subtle interaction of biology and oceanographic processes (Bryant, Hainbucher & Heath 1998), efforts to identify the key behavioural and physical mechanisms which determine year-on-year changes in abundance have been greatly hampered by the shortage of spatio-temporal density data against which basin-scale models can be tested.

As part of an effort to redress this lack, the EU TASC (Trans-Atlantic Study of Calanus) programme funded an intensive effort to obtain high-resolution stage-resolved time-series of C.finmarchicus abundance at a station in the central basin of the Norwegian sea. Although the sampling effort and costs involved in these measurements limited the operation to a little over three months (centring on the anticipated time of the spring phytoplankton bloom) the resulting time-series show a number of subtle features, which are not understood readily unless physical as well as biological processes are considered.

Observations at Ocean Weathership M

From 22 March 1997 until 9 June 1997, staff from the Institute for Marine Research (Bergen), the Plymouth Marine Laboratory and the Alfred Wegener Institute (Bremerhaven) carried out daily plankton sampling from the MV Polarfront in the vicinity of Ocean Weathership Station, which we refer to henceforth as OWSM (Irigoien et al. 1998; Niehoff et al. 1999; Heath et al. 2000; Hirche, Brey & Niehoff 2001). The times and locations of the samples are shown in Fig. 1, which also illustrates OSWM's situation (66 N, 2E) in the deep central basin of the Norwegian Sea, at roughly the same latitude as Iceland and close to the Norwegian continental shelf. Stage-resolved abundances of C. finmarchicus were obtained from samples collected with a 63 µm mesh net fitted to a WP2 net with 50 cm mouth area and hauled vertically from 100 m depth to the surface. The resulting data, which we illustrate in Fig. 2, have been described previously by Niehoff et al. (1999), Hirche et al. (2001) and Ohman & Hirche (2001).

Figure 1.

The main picture shows the locations of the samples taken in the vicinity of Ocean Weathership M. The numbers indicate the days elapsed from 22 March 1997. Samples were taken repeatedly at locations A (on days 8, 15 and 64), B (23, 67, 70–72, 79) and C (21, 58–61, 63, 73, 75–77, 80). The inset shows the geographical situation of Station M.

Figure 2.

Stage-resolved observations of the abundance of Calanus finmarchicus in the surface waters (above 100 m) at Ocean Weathership M during the spring and early summer of 1997.

Ohman & Hirche (2001) have analysed the short-term processes determining local egg mortality during these observations on the assumption that the demographics at OWSM can be regarded as closed over time-scales of the order of days. However, Fig. 2 suggests that the longer-term dynamics cannot be understood on the same basis. If an assumption of ‘closedness’ was accurately justified, we would expect to observe major features of the time-series in the abundance of one stage being reflected in its successor after a delay comparable with the stage transit time. In none of the time-series can we discern any features which plausibly bear such an interpretation. All the stages up to Cl, for example, show marked peak densities that are approximately synchronous at about day 135. Another evident inconsistency is that the abundances of N3s generally exceed those of the combined NI–N2 stage despite having a substantially shorter through-stage development time. Assuming a closed population would lead us to the absurdity of negative mortality rates.

To reinforce this central point we used a spatially explicit demographic model − which will be described in detail in a later section − to infer the time-series of egg deposition rates needed to produce the observed abundances for each developmental stage if the water sampled at OWSM remained stationary at OWSM for the entire observation period. If the assumption of closedness were correct, these inferred series would be identical, but we see from Fig. 3 that the required recruitment rates differ greatly both in absolute value and temporal pattern.

Figure 3.

The time-series of population egg deposition rates which would be needed to produce the abundance time-series observed for each developmental stage assuming a closed and homogeneous population.

The discrepancies in temporal pattern between successive stages are especially revealing. Consider, for example, the recruitment rate series for stages N3, N4–6 and Cl. Each has a characteristic peak, which occurs at day 122 for N3s, day 115 for N4–6 and day 108 for Cls. This temporal separation follows from the fact that the recruitment-rate peaks reflect observed abundance peaks at OWSM which all occur about day 135 (see Fig. 2). Because the developmental delay from egg to Cl is approximately 15 days longer than that from egg to N3, these simultaneous changes in observed abundance must be a consequence of recruitment events separated in time by about 15 days.

We thus conclude that the C. finmarchicus demography observed at OWSM cannot be understood without taking account of the physical environment − particularly near-surface water movements in the surrounding area.

The hydrodynamic environment

In order to reconsider the observations shown in Fig. 2 in a spatial context we need to describe the physical and hydrodynamic conditions in the region. To do this, we used a simulation originally developed for the TASC project using the Hamburg Shelf-Ocean Model (HAMSOM) − a three-dimensional baroclinic level-type model which solves the primitive equations of motion on an Arakawa C-grid (Backhaus & Hainbucher 1987).

The flow-fields and temperatures for the area of interest (Fig. 1 inset) were taken from a fine-resolution model covering the region from (56 Ν, 30 W) to (72 N, 20 E) at a resolution of 7·5′ (latitude) × 15′ (longitude) with 10 depth layers. This was nested within a coarse resolution model of the whole Atlantic north of 45 N, which was forced with 3-month averaged wind-speeds to provide sea–surface elevation boundary conditions for the fine-scale model (Harms et al. 2000).

The fine-scale model was forced with daily wind-stress, air-pressure, temperature, relative humidity and cloud cover from the ECMWF database and by freshwater inputs for the North Sea, Norway and Iceland. Daily average flow-fields and in-situ temperatures were output from this simulation for the year in which the data shown in Fig. 2 were obtained (1997).

In Fig. 4 we show a typical set of drifter tracks computed from these simulated flow-fields assuming that the drifter is located a constant 20 m below the surface. Each track starts on Julian day 80 of 1997 and continues until Julian day 160 the same year. The tracks start from the points denoted by the filled circles, which have been chosen so that each track passes under one of the sampling locations in Fig. 1 at the same time that the sample was taken. Note that similarities in the shapes of the various tracks indicate some spatial coherence in the velocities.

Figure 4.

Simulated tracks of three drifters driven by the HAMSOM 1997 simulation flow-fields at a depth of 20 m. The tracks start from the locations shown by the filled circles on Julian day 80 and finish on day 160. The crosses indicate the locations where the tracks coincide in space and time with samples shown in Fig. 1, and are labelled with the Julian day when this occurs.

To compute these tracks we note that the lat/long position (θN, θE) of a parcel of water being advected north at speed VN and east at speed VE, changes at a rate:

image(eqn 1)

where α ≡ 2π/360 and R is the radius of the earth. We solve these differential equations using a standard Runge–Kutta fourth-order algorithm with a time step of 1 h, driven by flow-fields linearly interpolated in time and space from the daily average HAMSOM fields. For each track we start the integration at the location and time at which a sample was taken, and integrate forwards to Julian day 160 and backwards to Julian day 80.

As an example, consider the track sampled at Julian day 124. The surface temperature in the area at that time is of order 6 °C, which would make the time taken to develop from an egg to copepodite stage C5 approximately 60 days (Gurney et al. 2001). Hence we see that an individual observed as, for instance, a C4 on that day was deposited as an egg round about Julian day 80 some 150 km further south. Thus, the stage-resolved observations of Fig. 2 are influenced by physical and hydrodynamic conditions over an area much larger than that actually sampled.

Spatio-temporal origins of the observations

egg release

Gurney et al. (2001) give a model for the development of juvenile stages of C. finmarchicus derived from the work of Corkett, McLaren & Sevigny (1986) and Miller & Tande (1993). This model postulates that all developmental stages show the same temperature dependence of development rate, so we can define a development index q, running from 0 (a newly deposited egg) to 1 (a C5 about to become an adult), with intermediate values at the end of each stage (Table 1).

Table 1.  Percentage development (equivalent to q × 100, where q is the development index) at the transition between each stage and its successor
StageEN1N2N3N4N5N6ClC2C3C4C5
q × l003·58·914·319·725·130·535·944·053·162·171·8100

The rate of change of q, g(Τ), depends only on water temperature T, and is given by:

image(eqn 2)

Thus, if we consider a water parcel which arrives at OWSM on day d and assume (as a strategic simplification) that it is a closed system, then an individual with development index q on the day of observation must have been recruited at a time τd(q) defined by:

image(eqn 3)

where Td(x) represents the in-situ temperature of the water-parcel at time x.

Because morphological stage i covers a development index range qi−1qi, individuals observed in that stage on day d must have been recruited over the period τd(qi−1)→τd(q). For illustrative purposes we define a ‘typical’ recruitment time for individuals observed in stage i at OWSM on day d, as:

image(eqn 4)

This represents the recruitment time for an individual with a development index on day d which lies in the centre of the stage i interval.

To calculate the recruitment time and position for a typical stage i individual observed on day d, we start a fixed-depth water parcel 20 m below OWSM on day d and use eqn 1 to track it backwards in time. We also characterize the parcel by a development index, which we initialize to q̂i on day d and then drive its diminution according to eqn 2 with the temperature appropriate to the local in-situ temperature in the water parcel. The time when the development index reaches zero is inline image and the position of the water parcel at that time is the equivalent recruitment location.

In Fig. 5a we show all the ‘typical’ recruitment points which occur on Julian day 81. The individuals concerned are sampled in almost every state from eggs to C5s at times between days 83 and 139 according to the spatial position at which they are recruited. In Fig. 5b we show the analogous spatial relation between recruitment position and arrival time/stage for eggs released on day 119.

Figure 5.

The spatial origins of observed stage abundances. The points show the position of eggs deposited on days 81 (a) and 119 (b) which reach sampling locations at different times, and thus as different developmental stages. Each point is marked in bold with the Julian day at which the parcel of water concerned is sampled, and (in square brackets) with the developmental stage which the survivors of that cohort have then reached.

egg production

In the last section, we noted that the eggs that form the cohort whose survivors are observed in developmental stage i on day d are released into the relevant water-parcel between days τd(qi−1) and τd(qi). If we now denote the rate of deposition of eggs in the water-parcel at time τ by Rd(τ) and the proportion of eggs deposited on day τ which survive until day d as Sd(τ) then we can see at once that the observed abundance of stage i individuals on day d is:

image(eqn 5)

To calculate the survival Sd(τ) we need to know the per-capita mortality rate of the developing individuals. As part of the TASC programme Eiane et al. (2002) estimated stage-dependent mortality rates for C. finmarchicus at two different Norwegian fjords that differed in predation regime. Because OWSM is in deep water we assume that the mortality rates measured by Eiane et al. (2002) at Lurefjorden (where tactile predators predominated and predation rates overall were lower) were more appropriate (Table 2).

Table 2.  Stage-dependent mortality rates (day−1) used in the model. The values for eggs–C5 are from Eiane et al. (2002)
StageEN1/N2N3N4–N6C1C2C3C4C5A
µ (day−1)0·1820·3360·1490·0260·0150·0000·0100·0100·0150·050

As these rates depend strongly on stage, we can most conveniently calculate Sd(τ) by noting that:

image(eqn 6)

We then characterize each water-parcel we track backwards from the sampling location by a survival S, which is initialized to unity and integrated backwards according to eqn 6 until the parcel reaches the recruitment point – when S = Sd (τ).

Our main purpose is to use eqn 5 to infer the recruitment rate Rd. Unfortunately, its structure precludes us from achieving this is any simple way. We thus choose instead to define a new quantity, R̂d(inline image), which represents the constant recruitment rate which, if applied across the time-interval τd(qi−1)→τd(qi) would produce the same stage i abundance on day d as that actually observed. Equation 5 implies that:

image(eqn 7)

To compute θ we use a series of water-parcels launched with development increments within the stage i range and tracked back to their release points to determine a series a values of Sd(τ) for τd(qi−1)→τd(qi) < τ < τd(qi). We then determine θ using the trapezoidal approximation to the integral in eqn 7.

In Fig. 6 we show spatial patterns of egg-depositon rate computed from the observed time-series of stage-abundances. To increase spatial coverage, we have consolidated the calculated points into 5-day groups before interpolating across the group to produce the false-colour maps of population total egg production rate shown in the figure.

Figure 6.

The spatial distribution of egg production rates at 10-day intervals. The points denote the inferred positions at recruitment to the egg stage for individuals that were later observed at different stages − eggs (open circles), N1/2 (open squares), N3 (open triangles), N4/6 (black circles), Cl (black squares), C2 (black triangles), C3 (red circles), C4 (red squares), C5 (red triangles).

At Julian day 80 egg production is low everywhere (mostly < 5000 eggs m−2 day−1) By Julian day 90 raised egg production rates (−20 000 eggs m−2 day−1) are observed in a patch about 40 km square centred 50 km south and 10 km east of OWSM. By day 100 the major centre of egg production has moved about 20 km west and intensified to ≈ 80 000 eggs m−2 day−1. By day 110, however, egg production rates are high almost everywhere. By Julian day 130 egg production rates have dropped back almost to pre-bloom levels. For days beyond 135 (not shown) population egg production rates are at pre-bloom levels throughout the region of inference.

adult abundance

C. finmarchicus females who have overwintered at depth do not rise to the surface and begin reproduction until (roughly) the time at which the observations discussed in this paper begin (Julian day 80, i.e. mid-March). At the temperature prevailing in the neighbourhood of OWSM during the period of observation, the egg to adult development time exceeds 60 days. Hence, we can conclude that all adults observed at OWSM before day 140 must be survivors of the overwintering cohort.

Under our simplifying assumption that each water-parcel is closed we can further assert that all adults observed on day d must be the survivors of those initially present in the water-parcel in the early spring. Hence we can see that, if the observed abundance of adults on day d is NA(d) then the abundance in the same water-parcel at time τ, NA(τ), must be:

NA(τ) = NA(d) exp[µA(d − τ)],(eqn 8)

where µA is the adult per-capita mortality rate (Table 2).

Applying this prescription, we can calculate a time-series of adult abundance values back along the track of each water-parcel. If we do this for each sample and consolidate the results into 5-day groups, we can interpolate between the inferred abundance values to produce the semisynoptic distributions shown in Fig. 7. At Julian day 80 the abundance is uniformly high (≈ 20 000 per m2) − exactly as we would expect if the overwintering population had recently risen to the surface. As time progresses the spatial average abundance inevitably decreases, as the action of mortality is not yet being counteracted by any recruitment into the adult class. However, we also notice that the distribution of adults becomes patchy, with a relatively well-defined scale of the order of 10–20 km.

Figure 7.

The spatial distribution of adult abundance at 10-day intervals. The points denote the inferred positions at recruitment to the egg stage for individuals later observed at different stages, using the same symbols as for Fig. 4.

per-capita productivity

We can determine the adult per-capita egg-production rate by dividing the population total egg-production rate (Fig. 6) by the appropriate local adult abundance (Fig. 7). We produced Fig. 8 by calculating this ratio for every point at which we had inferred values of population egg production, consolidating the resulting productivity estimates into 5-day groups and then interpolating over as large an area as the resulting inferences allowed.

Figure 8.

The spatial distribution of per capita fecundity at 10-day intervals. The points denote the inferred positions at recruitment to the egg stage for individuals later observed at different stages, using the same symbols as for Fig. 4.

We see from this figure that per-capita fecundity is uniformly low (typically less than 2–3 eggs female−1 day−1) until Julian day 100. At this time the average fecundity increases markedly to over 10 eggs female−1 day−1. By day 110 peak production has increased further, particularly in the region south-west of OWSM, but there is also evidence of relatively low production near OWSM. At day 120 females over almost the whole area for which we can infer per-capita productivity are producing over 10 eggs per day. By day 130, although patches of high productivity remain, the average per-capita production is just beginning to fall away, and for times later than day 130 (not shown) per-capita production falls almost to pre-bloom levels.

chlorophyll

Ship-board laboratory measurements at OWSM (Niehoff et al. 1999) suggest that prior to and during the phytoplankton bloom the per-capita fecundity of adult C. finmarchicus females, β, depends predominantly on chlorophyll abundance, C (µg l−1). We fitted a Michaelis–Menten saturating function to this data using Gauss–Newton optimization to obtain:

image(eqn 9)

which accounts for some 61% of the variance in per-capita egg production prior to Julian day 147. After this time fecundity seems to drop disproportionately, probably due to senescence of the females.

In Fig. 8 we show the spatial distribution of per-capita fecundity; thus we can calculate the local chlorophyll abundance from:

image(eqn 10)

Figure 9 shows the results of this calculation. Although the relation between adult fecundity and chlorophyll is non-linear it is monotonic, so we expect the inferred chlorophyll distribution to be qualitatively similar to the adult fecundity distribution from which it is derived.

Figure 9.

The spatial distribution of chlorophyll a at 10-day intervals. The points denote the inferred positions at recruitment to the egg stage for individuals observed later at different stages, using the same symbols as for Fig. 4.

Before Julian day 100 the inferred chlorophyll density is below 0·1 mg m−3 almost everywhere. At Julian day 100 an irregular distribution of patches of enhanced chlorophyll density appears, with peak abundances of over 3 mg m−3. Between Julian days 110 and 120 both the peak chlorophyll density and the overall average rise, with the peak being around 5 mg m−3 by Julian day 120. By Julian day 130 the bloom is beginning to die away, and for later times (not shown) the average chlorophyll abundance is little different from pre-bloom levels.

Characteristic scales

Figures 6–9 lead us to suspect that the inferred values of chlorophyll, and hence of recruitment, are consistent locally but are more variable between widely separated points. It is therefore important to determine both whether the claim of patchiness is statistically valid and to quantify the scale of patchiness.

We use the semivariogram (e.g. Cressie 1993) as our measure of spatial dependence. For each pair of points located at si and sj and separated by a distance, or lag, h, the semivariance, γ(h), measures half the average squared difference of the data values, yi and yj,

image(eqn 11)

If there is no spatial dependence then the semivariance for each pair of points is estimating the same thing − the total variance of the data – and a plot of the semivariance against lag will show no trend. If, on the other hand, there is a patchy distribution the semivariance will be low for low lags and only asymptote to the total variance for large lags.

In order to obtain a synoptic picture of the spatial dependence spanning the main bloom period from Julian days 100–140, we began by taking the ln(x + l) transformation of the chlorophyll densities inferred over this period. These values were then divided into 5-day blocks, and the semivariances were calculated from the pairs of points in each block. Treating each 5-day period separately was necessary because the area over which we predict values changes in size and location with time. Because the spatially averaged levels of chlorophyll also change over the period we scaled the data in each 5-day block by the standard deviation of all the values in that block, thereby ensuring that the asymptotic semivariance is always unity. In practise, plots of semivariances calculated in this way are extremely noisy and have a high density of points from all the pairwise estimates, thereby making trends difficult to detect. Thus, instead of plotting the empirical semivariances we capture the trend using Friedman's ‘super smoother’ (Freidman 1984). The result, shown by the long-dashed line in Fig. 10, shows that the zero-lag semivariance is about 0·5 (the ‘nugget’ variance, indicating that 50% of the variance is inherent variability or measurement error) and asymptotes to unity (the scaled ‘sill’).

Figure 10.

Semi-variogram showing the scale of patchiness in the inferred chlorophyll distributions. The smooth through the data (long-dashed line) gives a non-parametric estimate of how the semivariance changes with distance. The solid line shows the best fit exponential model. The short-dashed lines show the 0·05 and 0·95 percentiles on the smooths through semivariograms obtained from randomly reassigned chlorophyll values and locations.

We repeated the process of obtaining a smoothed semivariogram for 100 trials in which the data were assigned randomly to the locations. As expected from random data the semivariograms showed no trend as the lag increased, and the 5% and 95% percentiles provide us with confidence intervals for spatially independent data. Because the original smooth lies consistently below the 5% percentile for low lags, we can conclude that the observed spatial dependence is statistically significant.

The smoothed semivariogram corresponds tolerably well to the ‘authorized’ theoretical model:

γ(h) = 1 + (c0 − 1) exp(−h/h0)(eqn 12)

where c0 is the nugget and h0 is the characteristic distance of the approach to the sill. The best fit exponential model is shown by the dashed red line in Fig. 10. From eqn 12 we can define the effective range, re, as the distance at which the semivariance has reached 95% of the sill value,

re ≡h0 × (ln(l − c0) − ln(0·05)).(eqn 13)

Using Gauss–Newton optimization to obtain the least squares fit of the exponential model yields a characteristic length scale of 5 km and an effective range of 15 km. The effective range corresponds approximately to the radius of a patch, thus we infer that the patches are roughly 30 km in diameter.

Robustness

As a start towards addressing the wider issue of robustness we focus here on what is probably the least certain aspect of our model − the stage-dependent mortalities. Although these mortalities were adopted from a field study in a fjord, OWSM lies in deep water and so they remain a potentially major source of error.

We repeated the calculations leading to the best-fit effective range (eqn 13) on 100 randomizations in which the log-transformed values of the mortalities are drawn from a normal distribution with mean given by the log of the values in Table 2 and with a coefficient of variation of 20% (although Eiane et al. 2002 give confidence intervals for their estimates we do not use those as we are trying to simulate conceivable mortality rates rather than those pertaining at Lurefjorden). The C2 stages had zero observed mortality, and so for the randomizations we assumed a mean of 0·01 per day − the same value as for the next two copepodite stages.

The resulting frequency distribution of the effective ranges is given in Fig. 11, and shows that despite the very wide variation in mortality rates the effective ranges remain typically in the 6–18 km range, well within plausible levels for chlorophyll patchiness. The small number of instances with somewhat larger (> 20 km) effective ranges all had large adult mortalities, implying that there is some sensitivity of our results to adult mortality rates.

Figure 11.

Frequency histogram showing the robustness of the predicted patch scale to random perturbations in stage-dependent mortality. The effective ranges are from the best-fit exponential models to the semivariograms of egg production rate (eggs female−1 day−1). The log-transformed values of the mortalities used in the randomizations are drawn from a normal distribution with mean given by the log of the values in Table 2 and with a coefficient of variation of 20%.

Discussion

One of the fundamental difficulties in estimating population parameters from time-series is that observed changes in population density arise through differences between birth and immigration, on one hand, and deaths and emigration on the other hand. This makes many demographic problems under-determined, even in non-spatial situations where migration can be ignored (Nisbet & Wood 1996). With spatial heterogeneity and transport the situation is clearly even worse. Our study of the OWSM C. finmarchicus time-series demonstrates that plausible levels of environmental heterogeneity and even modest transport will make it impossible to ignore migration other than at very short time-scales. OWSM is located in the Norwegian Sea gyre and is therefore generally considered a more retentive area in which the effects of transport might be expected to be relatively small. Indeed, an earlier study comparing times series at various locations in the north-east Atlantic found that OWSM had the most self-contained demography of all the sites examined (Heath et al. 2000).

The characteristic pattern of the OWSM time-series −the near simultaneous peaks of all the developmental stages up to Cl − is inexplicable in terms of closed-population demography. We have shown that it results from the passage of bodies of water that have experienced an extended period of high recruitment due first to high densities of adults in the presence of low chlorophyll, and later due to lower numbers of adults that are more fecund as they pass through a chlorophyll patch (away from OWSM). The high recruitment did not start early enough to produce a simultaneous peak in the later stages C2–C5, but packets of water that were further away from OWSM and experienced the same bloom pass through OWSM later, leading to a simultaneous accumulation in the densities of the later stages.

As well as producing a spatially explicit demographic model that is now completely consistent with the observed population changes at OWSM, our approach makes testable inferences about the scale of environmental heterogeneity required to achieve this. The inferred chlorophyll fields are, in principal, directly testable from satellite information on sea colour. Regrettably, there are no data available for the start of 1997 when the demographic observations were made; the Coastal Zone Color Scanner (CZCS) on-board NASA's Nimbus 7 satellite operated from 1978 to 1986, while the Sea-viewing Wide-Field-of-view Sensor (SeaWiFS) on the SeaStar satellite was not launched until August 1997. At a more qualitative level, however, if the inferred chlorophyll distributions were uniform we would have to conclude that transport is not an important process in the demography. If, on the other hand, the inferred values were extremely heterogeneous, with all points apparently unrelated to their neighbours, the implausible degree of chlorophyll patchiness thereby implied would falsify our approach. The values that we actually detected were of the order of 30 km, which is entirely consistent with observations on phytoplankton patches which suggest that meso-scale patches are in the range of 15–50 km (Parsons, Takahashi & Hargrave 1984; references therein).

Clearly, the results are predicated on a set of more-or-less assailable assumptions, such as the accuracy of the hydrodynamic model providing the transport and temperature, and the accuracy of the development model. Because the development model has been parameterized from extensive experimental data it seems unlikely that it would be a major source of uncertainty. By contrast, it is easy to imagine that errors in the transport determined by the hydrodynamic model may have considerable implications, not least for the scale of the inferred chlorophyll patches. This suggests that future studies aimed at understanding the effects of advection may gain from more attention on quantifying flow-fields in the field in order to validate or optimize hydrodynamic models. An additional problem is that in all our simulations movement was determined by velocities at 20 m depth, and in so doing ignored diel vertical movements. While some degree of position control can be achieved by vertical migration (Eiane, Asknes & Ohman 1998), the vertical gradients in current speeds over the top 100 m or so in the deep water around OWSM are small. Perhaps more importantly, we ignored turbulent diffusion, a phenomenon that will be increasingly important over longer time-scales. Although a quantitative effect on the inferred heterogeneity of chlorophyll is probable, it seems likely that the qualitative conclusions relating to demographic features observed around OWSM will remain intact.

Our results are entirely self-consistent, in the sense that a plausible chlorophyll field combined with plausible transport and a wide range of assumed mortalities can entirely recreate the observed data in a way not possible by treating the population as closed. An unfortunate consequence of this is that it demonstrates the impossibility of obtaining truly reliable estimates of mortality rates from data in open-water sites. However, the results point a way forward for the design of sampling programmes aimed at measuring mortality rates in advective environments. Ideally, one would wish for complete spatio-temporal coverage over an area large enough to cover the distances moved by individuals over their lifespan. This is not practical, but it is sensible to ask what might be done with realistic resources. One possibility is to trade temporal resolution for spatial coverage. For the OWSM data, our analysis implies that over a period of 80 days the dynamics at a point are influenced by what is happening at locations of up to some 200 km away. This suggests that if we select a grid of sampling locations separated by the inferred patch scale we can visit (within the limits imposed by sea conditions) those locations in sequential order and have a good chance of resampling patches several times along their route. This is no different in principle to the sampling programme actually undertaken in the vicinity of OWSM, except that rather than aiming at a fixed location the sampling is dispersed deliberately over a scale that harnesses what we know about transport. Lagrangian sampling achieved by following drogued drifters is a methodology that will result similarly in resampling the same body of water, but will lose information on spatial patterns. Moreover, mortality estimates obtained from Lagrangian sampling are conditional on local spatial homogeneity so that diffusive population exchange being small. By contrast, with a grid of sampling stations we introduce a the possibility of obtaining reliable mortality estimates for open water organisms by directly fitting a spatially explicit population model incorporating both advective and diffusive transport.

Acknowledgements

This work was supported by the NERC under grants NER/T/S/1999/00072 and NER/T/S/2002/00142 and by the EU funded TASC project. Barbara Niehoff kindly provided the locations at which the OWSM samples were taken. We would also like to thank two anonymous referees for their constructive comments.

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