Eddy-Induced Variability in Larval Settlement of Nearshore Marine Species
Mitarai, S., Siegel, D.A., Warner, R.R., Winters, K.B, Kendall, B.E., Gaines, S.D.
Will be submitted to MEPS
Abstract
Quantitative descriptions of larval dispersal are critical for the predictive understanding
of many nearshore marine populations. For sessile species with a planktonic life stage,
larval dispersal is the predominant means that connects spatially segregated
populations. Larval dispersal is driven by mean currents, wind-driven Ekman circulation
and coastal eddy motions as modified by the larval development time course and larval
movements. Eddy motions predominate in the variability of ocean circulation, and have
been considered to be a major source of stochasticity in settlement and recruitment
events. In contemporary marine ecology, however, the vast majority of models
oversimplify the processes of larval dispersal by simply neglecting eddy motions and
often describe it as a simple diffusion process. In this study, we clarify intrinsic
stochasticity arising from coastal eddy motions in larval dispersal by introducing simple
scaling theory that counts the number of arriving eddies to habitats for a single
spawning season. This scaling theory suggests that coastal eddy motions set a strong
source of uncertainties in settlement patterns and population connectivity of coastal
marine species for a single spawning season even when extreme abundance of larval
production is available, and provides a means to quantify eddy-induced stochasticity in
larval settlement patterns. We demonstrate that the scaling theory can accurately
account for eddy-induced stochasticity in simulated dispersal patterns obtained from
idealized coastal circulation simulations of the California Current. These results give
new insights into the nature of larval dispersal and its potential for regulating population
dynamics of nearshore marine species.
1 Introduction
1
Quantitative descriptions of larval dispersal are critical for the predictive understanding
of many nearshore marine populations. Many of the marine species that comprise
nearshore populations are relatively sessile as adults, with a dispersive planktonic larval
stage. The planktonic larval stage is often the predominant means that connect spatially
separated habitats for sessile species. Hence a predictive knowledge of larval dispersal,
including source and destination locations and the degree of their connectivity, is key
information for the study of nearshore marine population dynamics (Roughgarden et al.,
1988; Kinlan and Gaines, 2003; Largier, 2003; Sale et al., 2005; Warner and Cowen,
2002). Due to the small size of marine larvae, larval transport by coastal circulation
processes is likely the dominant process driving larval dispersal (e.g., Jackson and
Strathmann, 1981; Siegel et al., 2003), although larval behavior (such as vertical
migratory behavior and late-developmental period swimming) can also have some
influence on integrated larval transport (Leis, 2006; Leis et al., 2007; Gerlach et al.,
2007; Paris and Cowen, 2004; Siegel et al., in press).
Larval settlement and recruitment observations provide many hints of the stochastic
nature of larval dispersal for many fish and invertebrate populations on intra-seasonal to
inter-annual time scales (Caffey, 1985; Caselle and Warner, 1996; Farrell et al., 1991;
Hamilton et al., 2006; Swearer et al., 1999; Myers, 2001). One major source of
stochasticity is changes in the circulation of the coastal ocean driven by mean currents,
wind-driven Ekman circulation and coastal eddy motions. Among these physical drivers,
coastal eddy motions predominate in the variability of coastal systems, creating
seemingly chaotic flow patterns (e.g., Poulain and Niiler, 1989; Swenson and Niiler,
1996; Signell and Geyer, 1991; Strub et al., 1991; Haidvogel et al., 1991; Richardson,
1993; Bernstein et al., 1977; Dong and McWilliams, 2007). These eddy motions have
recently been considered to be a major source of stochasticity in larval settlement
patterns, and this can have an influence on stock recruitment dynamics (e.g.
Roughgarden et al., 1998; Siegel et al., 2003, in press; Cowen et al., 2006; PfeifferHerbert et al., 2007; Mitarai et al., 2008).
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The vast majority of models in contemporary marine ecology fails to account for the
chaotic nature of coastal eddy motions. For example, a conventional diffusion modeling
approach (e.g, Largier, 2003; Jackson and Strathmann, 1981) describes larval dispersal
as a smooth and homogeneous diffusive process (often as a Gaussian process), and
eddy-induced stochasticity in larval dispersal is simply ignored. By definition, a diffusion
modeling approach describes time-averaged dispersal patterns of water parcels,
smoothing out turbulent eddies (e.g., Pope, 2000; Tennekes and Lumley, 1972). In
another example, an unstructured larval pool has been often assumed in many
ecological studies, where larvae are produced in the nearshore, enter a common pool
offshore, and are then returned to nearshore habitats often by downwelling favorable
winds (Bakun and Parrish, 1982; Farrell et al., 1991; Parrish et al., 1981; Roughgarden
et al., 1991). While the larval pool modeling approach can account for pulses of
settlement, homogeneous mixing makes all sites potential sources of larvae for all other
sites, and thus may connect populations that are in reality unconnected. A larval pool
has to be spatially structured by ocean circulation.
Numerical simulations of coastal circulation processes are a powerful tool to assess
population connectivity via advection of water parcels (e.g., James et al., 2002; Siegel
et al., 2003, in press; Cowen et al., 2006; Pfeiffer-Herbert et al., 2007; Aiken et al.,
2007; Mitarai et al., 2008). However, very few of these models have been utilized to
assess the underlying mechanism of eddy-induced stochasticity in population
connectivity and larval settlement patterns. Do coastal eddy motions set spatio-temporal
variations in larval settlement and recruitment patterns? Under what circumstances can
conventional diffusion-based models be approximately valid? More importantly, how
does eddy-induced stochasticity in larval dispersal affect the understanding and
predicting of ecological processes and the design of fishery management? Many of
these questions are yet to be answered because we still lack a clear picture of the
intrinsic stochasticity arising from coastal eddy motions.
The goal of this study is to clarify intrinsic stochasticity in larval settlement patterns and
population connectivity arising from chaotic coastal eddy motions. We introduce simple
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scaling theory that accounts for this eddy-induced stochasticity in larval dispersal by
counting the number of coastal eddies arriving to coastal habitats. Predictions by the
proposed scaling theory suggest that coastal eddy motions set a strong source of
uncertainties in larval settlement patterns and population connectivity when viewed on
annual time scales, even when extreme abundance of larval production is available. We
test this scaling theory by using simulated larval dispersal patterns obtained from
idealized coastal circulation simulations of the California Current along the central
California coastline (Mitarai et al. 2008). We show that the proposed scaling theory
successfully accounts for eddy-induced stochasticity in the simulated dispersal patterns.
Based on the results, we discuss important consequences of scaled eddy-induced
stochasticity in stock dynamics and community structure of nearshore marine
populations. We illustrate fundamental sources of stochasticity in larval dispersal and
marine population dynamics by using simple scaling theory.
We first present scaling theory for eddy-induced stochasticity in larval dispersal in
Section 2. The proposed scaling theory is then examined by using simulated larval
dispersal patterns obtained from the idealized coastal circulation simulations of Mitarai
et al (2008) in Section 3. Finally, we discuss the importance of coastal eddy motions in
marine population dynamics and a spatial fishery management in section 4.
2 Scaling Theory for Eddy-induced Stochasticity
2.1 Eddy-induced stochasticity in larval dispersal
Larval dispersal of coastal marine species takes place in the turbulent coastal ocean,
where coastal eddies are ubiquitous and drive the stirring and mixing of water parcels
(e.g., Fig. 1 a). Coastal eddy motions evolve rapidly in response to winds, while being
advected with the mean currents and wind-driven Ekman circulation. Studies show that
ocean surface drifters are advected by these turbulent currents (Winant et al., 1999;
Dever et al., 1998; Swenson and Niiler, 1996; Poulain and Niiler, 1989; Ohlmann et al.,
2001). Surface drifters are advected by the ocean currents, mostly along coastal eddies
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due to geostrophic currents, while transported by mean currents and wind-driven
Ekman circulation, as illustrated in Fig. 1 b (e.g., see also Ohlmann et al., 2001). Hence
eddies act to sweep larvae together into streaks, and transport them as coherent
packets (see also Roughgarden et al., 1988, Mitarai et al., 2008, Siegel et al., in press).
Population connectivity among nearshore habitats, therefore, has to be strongly
controlled by the departure and arrival of these “larval packets.” This is very different
from a diffusion modeling approach, where larval dispersal is described as a
homogeneous and small-scale diffusion process. While a diffusion modeling approach
always results in smooth and homogeneous connectivity among coastal populations,
larval packet transport can lead to sparse and heterogeneous population connectivity
(Mitarai et al., 2008; Siegel et al., in press).
An unstructured larval pool has been often assumed in many ecological studies, where
larvae are produced in the nearshore, enter a common pool offshore. Coastal eddies,
however, will make a larval pool heterogeneous in space. Larval production occurs
within a few kilometer of the shoreline, at least in the California Current (Refs; need help
from Bob here), while the size of coastal eddies is typically much larger and the center
of eddies is located 10’s to 100’s km offshore (as explained below). Larvae are then
seeded and transported in the peripheral of eddies due to Geostrophic currents, leaving
the center of eddies rather unoccupied (Fig. 1 b). This makes a larval pool spatially
heterogeneous structured by coastal eddy motions. Settlement pulses observation
driven by downwelling favorable winds (Bakun and Parrish, 1982; Farrell et al., 1991;
Parrish et al., 1981; Roughgarden et al., 1991) may be caused by, among other things,
this heterogeneous larval pool formed by coastal eddies. Settlements pulses can occur
when downwelling favorable winds bring coherent larval packets formed by eddies from
an offshore larval pool towards a shore. Coastal eddy motions can be, therefore, a
strong source of uncertainties in determining larval settlement patterns.
The exact patterns in larval dispersal processes are expected to be different for each
independent spawning season. This occurs because the statistics of coastal circulations
are 1) actually different in different years (what we will call real inter-annual variability)
5
or 2) the flows are statistically identical but the spawning seasons samples a different
collection of eddy motions resulting in different dispersal patterns (here defined as
apparent inter-annual variability). We examine apparent inter-annual variations in
dispersal patterns, due to chaotic coastal eddy motions, by introducing simple scaling
theory to describe them below.
2.2 Scaling eddy-induced stochasticity in settlement events
The scaling of the number of settlement pulses will be key to understanding eddyinduced stochasticity in larval dispersal processes. The fluid dynamics of ocean
circulation systems suggests that larvae are swept into packets by coastal eddies, while
transported by mean currents and wind-driven Ekman circulation, as described above.
Larval settlement patterns along a shore can be then modeled as a superposition of
arriving pulses of settlements (packets) formed by coastal eddies. The resulting
settlement patterns and degree of eddy-induced stochasticity will depend on the total
number of settlement pulses arriving for the domain, N_{ev}, and the spatial extent for
each settlement pulse, δ_{ev}, which is normalized by the domain size. A large number
of events, each providing occurring over relatively large spatial scales, will result in a
smooth settlement pattern; whereas fewer, smaller sized events will result in a patchy
pattern of settlement (e.g. Siegel et al., in press).
The number of coastal eddies arriving in the domain can be estimated as the ratios of
the domain size, L, to the eddy size, l (the number of eddies that occupy the domain)
and the duration of the spawning season, T, to the eddy residence time, τ (the number
of eddies arriving at a site per season) and the fraction of arriving eddies that contain
settling larvae, f_{sv} , i.e.,
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The survivability fraction, f_{sv}, should be f_{sv} = 1 if all coastal eddies contain
particles (larvae), and f_{sv} < 1 otherwise. Since the goal of this study is to assess
stochasticity induced by coastal eddies, and not due to biological variability, we assume
f_{sv} = 1 for the rest of this paper. The relative spatial scale, δ_{ev}, is set solely by the
eddies in the flow field, and estimated as δ_{ev} = l/L.
The time and length scales of coastal eddy motions can be estimated by using available
oceanographic information. The first-mode baroclinic Rossby radius of deformation is a
natural scale in the ocean associated with boundary phenomena such as boundary
currents, fronts, and eddies (Gill, 1982). As an example, let us estimate the number of
arriving eddies to the central California coast using the baroclinic Rossby radius of
deformation. Along the US west coast, the first-mode baroclinic Rossby radius of
deformation is approximately 20 to 30 km (Chelton et al., 1998). Hence the portion of
the coastline covered by each coastal eddy can be estimated as l = 2r = 40 – 60 km,
where r is the first-mode baroclinic Rossby radius of deformation. The mean current
speed of California Current is about u ≈ 4 cm/s (e.g., Swenson and Niiler, 1996; Poulain
and Niiler, 1989; Siegel et al., 2003). The eddy residence time can then be scaled by τ =
l/u = 12 – 17 days. The central California coast extends approximately 250 km
(approximately between Pt. Conception and Monterey). Hence each eddy covers 20%
of the coastline, i.e., δ_{ev} = 50 / 250 = 0.2. For a single spawning season (e.g., 90
days), the total number of arriving packets can be scaled by N_{ev} = 30, given the
parameters L = 250 km, l = 50 km, T = 90 d, τ = 15 d and f_{sv} = 1.
One way to assess the spatial variability of the number of settling larvae among sites is
to use a coefficient of variation statistic (CV_s). If the probability that a given larval
packet lands on a particular site is δ_{ev} and each event is independent, then the
expected number of packets arriving at a site is δ_{ev} N_{ev} and the expected value
of the CV_s can be approximated by using binomial sampling theory as
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If N_{ev} or δ_{ev} increases, the settlement pattern will become less stochastic. Given
the parameters L = 250 km, l = 50 km, T = 90 d, τ = 15 d and f_{sv} = 1, for example,
Eqs. (1) and (2) yield <CV_s> ≈ 0.37. Thus, eddy-induced stochasticity causes about
40% of variation in settlement events among sites, even without biological variability in
larval production (f_{sv} = 1). Adding biological source of variability makes spatial
settlement patterns for a single spawning season even more stochastic.
The scales of coastal eddies are regional. While the first-mode baroclinic Rossby radius
of deformation is approximately 20 to 30 km along the US west coast, the radius quickly
increases towards the equator, e.g., south of Baja California (Chelton et al., 1998). The
presence of topographic features such as irregular coastlines will have a strong
influence on the surrounding circulation. Small topographic features, for example, may
create smaller coastal eddies. Given a larger (or smaller) eddy size, the eddy-induced
stochasticity may vary substantially. However, scaling theory, Eqs. (1) and (2), suggests
that the eddy-induced stochasticity is rather insensitive to the choice of eddy size. For a
case with a larger eddy, for example, the number of arriving eddies is reduced [Eq. (1)],
which will inflate eddy-induced stochasticity [Eq. (2)]. However, the domain covered by
an eddy is increased, which reduces spatial heterogeneity. These two counter-acting
effects nearly cancel out, leaving eddy-induced stochasticity rather unchanged. For
example, the scaling theory yields <CV_s> ≈ 0.44, given the parameters L = 250 km, l =
100 km, T = 90 d, τ = l / 0.04 ≈ 30 d and f_{sv} = 1. Stochastic settlement due to coastal
eddies are expected, not only in the California Current, but also other coastal oceans.
2.3 Eddy-induced stochasticity in population connectivity
Connectivity between given nearshore sites is often quantified using a “connectivity
matrix” (e.g., James et al., 2002; Largier, 2003; Cowen, 2002; Cowen et al., 2006;
Siegel et al., in press). Definitions for the connectivity matrix differ widely. A connectivity
matrix can be defined by 1) accounting for water-parcel transport from a source site j to
a destination site i within spawning and larval development time-course windows as
8
modified by larval behavior characteristics; 2) the spawned larvae from site j for the
season that successfully settle at site i; or 3) those spawned larvae from site j that
successfully recruit to the adult stage at site i. The first definition excludes any biotic
sources of stochasticity other than larval life history characteristics. The second and
third superimpose the biotic spatial processes of production and post-settlement
interactions onto the first abiotic definition. Cowen et al. (2006), for example, used the
third definition in their illustration of a connectivity matrix. In this study, we refer to the
first, abiotic, definition as the connectivity matrix, so that we can separate the effect of
eddy-induced stochasticity from other biological sources of variability.
The scaling relationship can be used to develop a “packet model” for population
connectivity to aid in spatial population dynamics modeling. The idea is to portray
population connectivity as the departure and arrival of N_{ev} independent, equallysized, settlement packets. The source and destination locations of each packet are
determined by random sampling of the long-term averaged Gaussian dispersal kernels
(e.g., see Siegel et al, 2003; Largier, 2003). Destination locations for the N_{ev} packets
(X) are selected randomly from within the domain and their source locations (Y) are also
determined randomly selecting from the Gaussian distributions [Eq. (3)], which accounts
for the downcoast displacement, i.e.,
where x and y are sample space variables for X and Y, respectively; f_x and f_y are the
probability density functions for X and Y, respectively; and µ and σ, respectively,
represent the mean and the standard deviation of the dispersal distance of successfully
settling larvae. Here values of Gaussian parameters µ and σ can be estimated from
oceanographic information for given larval development time courses (see Siegel et al.,
2003). Connectivity matrices are then modeled based upon the number of packets
between a given source and destination. Mathematically,
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where X_n and Y_n indicate the source and destination locations for the n-th larval
packet; Θ_x (a, b) is the boxcar function that is equal to 1 for a < x < b and 0 otherwise,
representing a destination (source) area covered by each eddy. As he number of
packets increases (approaches infinity, precisely), modeled connectivity matrices
become identical to Gaussian dispersal kernels, i.e.,
which gives a smooth and homogeneous connectivity among populations (diffusion
models). We numerically demonstrate when modeled connectivity [Eq. (4)] can be
reasonably approximated by smooth Gaussian connectivity [Eq. (5)] below.
The packet model suggests that coastal eddy motions set a strong source of
stochasticity in population connectivity when viewed on annual time scales. In other
words, the number of larval packets formed by eddies is not large enough to achieve
smooth and homogeneous population connectivity, even when extreme abundance of
larval production is available. Figures 2 a − c show the connectivity matrices obtained
from three different realizations of the packet model predictions (predicted using a
different random seed in determining the source and destination locations of packets
mimicking chaotic coastal eddy motions), given the parameters L = 250 km, l = 50 km, T
= 30 d, τ = 15 d, f_{sv} = 1, µ = -68 km and σ = 71 km. Here the values of µ and σ are
determined by using the regression formula by Siegel et al. (2003), assuming 20 days of
pelagic larval durations in the California Current. The realized connectivity matrices
show that some sites receive less settling larvae, while others receive pulses of
settlements from a wide array of source locations. There are “hot spots” that indicate
strong connections between particular source and destination locations. The
connectivity matrices are not only heterogeneous in space, but also intermittent in time.
Different realizations produce distinctly different patterns in settlement pulses and
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different connections among nearshore sites; these are still spatially heterogeneous, but
the locations and intensities of the hot spots change (cf. Figs. 2 a - c). These
realizations illustrate that coastal eddy motions alter larval settlement patterns from one
spawning season to the next, due to the chaotic nature of turbulence.
The realized connectivity matrices are very different from a diffusion model prediction
(cf. Figs. 2 a – c and d). Graphically, the packet model for population connectivity can
be considered as randomly placing N_{ev} of eddy-size “patches” in the smooth,
Gaussian connectivity matrix. When averaged over many independent spawning
seasons, the domain is filled with many successful settlement events and the signature
of spatial heterogeneity in population connectivity becomes smoother (Siegel et al.,
2003, in press). Figure 3 shows the packet model prediction for 1 and then averaged
over 5, 10 and 100 independent spawning seasons. Comparison of Fig. 2 d and Fig 3. d
shows that the resulting connectivity patterns become smoother approaching the
Gaussian patterns of connectivity predicted by a diffusion model. “Hot spots” in the
connectivity matrices are still discernible after 10 spawning seasons (Fig. 3 c), although
connectivity patterns are smoother. The connectivity matrices become very similar to
the diffusion model prediction when averaged over 100 spawning seasons (Fig. 3 d).
The scaling analysis, Eqs. (1) and (2), clearly describes this change in connectivity. By
substituting Eq. (1) to Eq. (2), we obtain
Larval settlement (and connectivity) patterns will become smoother as T , δ_{ev} , or
f_{sv} increase or as τ decreases. Given the parameters L = 250 km, l = 50 km, τ = 15
d, f_{sv} = 1 and T = 90, 450, 900 and 9000 d, Eq. (6) yields <CV_s> ≈ 0.37, 0.21, 0.12
and 0.04. These values show a good agreement with CV_s values computed from the
realized connectivity matrices in Fig. 2 (see the caption of Fig. 2).
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Note that the connectivity described here is independent of adult population
demographics. Adding variability in larval production will only make the connectivity
diagrams presented here more stochastic, and requires even longer time until spatial
heterogeneity in the simulated population connectivity is smoothed out. These results
suggest that population connectivity should be stochastic even when extreme
abundance of larval production is available, and chaotic coastal eddy motions, thus, set
avoidable uncertainties in population connectivity for a single spawning season. Also
note that the packet model assumes that the probability that a given larval packet lands
on a particular site is equal [Eq. (3)]. Complex coastal topography, however, will modify
the probability distributions for the packet landing location, and may create preferable
destination sites over seasons if persistent topographic eddies are formed (e.g.,
Graham and Largier, 1997). This topographic effect may make population connectivity
even more heterogeneous in space, but probably less intermittent in time. While it would
be possible to include topographic effects in scaling theory (with much more
complexity), it is beyond the scope of this study. Our primary focus is to clarify eddyinduced stochasticity in larval dispersal processes, separated from other sources of
stochasticity. The proposed scaling theory shows that ocean stirring makes larval
connections among nearshore sites a stochastic process that is both spatially
heterogeneous and temporally intermittent.
3 A Test of Scaling Theory
In the previous section, we introduced scaling theory that accounts for stochasticity in
population connectivity and settlement events driven by coastal eddy motions. The
natural next step would be to examine the proposed scaling theory (the packet model).
It would be challenging to test scaling theory by using realistic circulation simulations
(e.g., James et al., 2002; Cowen et al., 2006; Pfeiffer-Herbert et al., 2007; Aiken et al.,
2007) and in-situ settlement and recruitment observation data (Caffey, 1985; Caselle
and Warner, 1996; Farrell et al., 1991; Hamilton et al., 2006; Swearer et al., 1999;
Myers, 2001) because of the difficulty to separate eddy-induced stochasticity from other
sources of biological and topographic variability. In this study we test the scaling theory
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against idealized simulations of coastal circulation processes modeled after the
California Current (Mitarai et al., 2008). The coastal circulation-simulations of Mitarai et
al. (2008) assume homogeneous environment in the along-shore direction and separate
eddy-induced stochasticity from other sources of variability. This numerical
configuration, thus, allows us to test the packet model effectively.
3.1 Idealized coastal circulation-simulations
Model forcing and domain configuration of Mitarai et al. (2008) are patterned after
typical flow conditions of the California Current at line 70 (off shore of Pt. Sur, California)
of the California Oceanic Cooperative Fisheries Investigations (CalCOFI) (Lynn and
Simpson, 1987; Chelton et al., 1998). The domain is modeled to be homogeneous in
the along-shore direction with a domain size of 256 km in the along-shore direction and
288 km in the cross-shore direction. The domain is discretized horizontally by a 2-kmresolution grid (128 grid points in the along-shore direction and 144 grid locations in the
cross-shore direction). Twenty vertical (depth) levels are considered, with enhanced
resolution near the top and bottom boundaries. The bathymetry has a steep continental
slope modeled after CalCOFI line 70. No bathymetric variations are considered in the
along-shore direction, and periodic boundary conditions are used at the northern and
southern boundaries. This scenario reasonably represents the US west coast while still
providing a numerical system capable of addressing the fundamental processes of
larval dispersal driven by eddy motions.
The simulated flow fields are mainly driven by stochastic wind stress applied to the top
surface. The wind field is assumed to vary on spatial scales much larger than the alongshore scale of the simulated domain while its magnitude decreases towards the shore
(Pickett and Paduan, 2003; Capet et al., 2004). Each component of the wind vector is
modeled as a statistically-stationary Gaussian random process, given the statistics
estimated from hourly buoy wind data of the National Data Buoy Center (stations 46028,
46012 and 46042) for summer and winter and the spatial wind observations of Pickett
and Paduan (2003). We simulate two distinctive flow regimes, i.e., a strong upwelling
13
condition in summer (July) and a weak upwelling condition in winter (January). Here we
simulate 28 different realizations of flow fields for each condition by changing the
random number seed in wind forcing, while the other parameters and configuration are
unchanged. Using these realizations of simulated flow fields, we examine apparent
inter-annual variations in dispersal patterns.
3.2 Modeling of larval dispersal in the simulated flow fields
A large number of Lagrangian particles are released and tracked in the simulated flow
fields to simulate dispersal of larvae. Lagrangian particles are passively advected by
coastal circulation processes in horizontal directions while they are capable of changing
their vertical locations, mimicking ontogenetic development of vertically migrating
behaviors. Values of planktonic larval duration (PLD) can range from a month to several
months for typical reef fish (Victor, 1986; Wellington and Victor, 1989). Only larvae
transported to nearshore suitable habitats during their settlement competency time
windows (at the end of the PLD) are counted as successful settlers.
Nearshore habitats where particles are released and settle are defined here as all
waters shallower than 100 m in depth (within 10 km from the coast). The exaggerated
offshore extent of suitable habitat (at least for the California coast) was selected to
account for active swimming towards suitable habitat in the last stages of larval
development. After the simulation flow fields reach statistically stationary conditions,
1,000 Lagrangian particles are released daily for 90 days in the upper 10 m of the water
column, uniformly distributed in nearshore waters. The total number of released
particles within the domain is 90,000 for each realization. We examine two different
scenarios for vertical positioning. For the first scenario, Lagrangian particles are
released near the sea surface (within the upper 10 m) and stay at this depth while they
are passively transported horizontally. For the second scenario, particles are released
near the top surface, and shift their vertical locations to 30-m deeper after 5 days from
their release, keeping their depth after that. Most fish larvae are found within the upper
50 m or so (Leis, 1991; Cowen, 2002). These two behaviors are expected to give a
14
general idea of the importance of larval vertical positioning, coupled with coastal eddy
motions, in determining population connectivity.
Successful settlement is defined as an event that occurs when the Lagrangian particles
are found within the nearshore habitat region within competency time windows. We
examine four sets of competency time windows, i.e., 10 – 20, 20 – 40, 30 – 60 and 40 –
80 days. As stated before, the pelagic larval durations (PLDs) of many benthic reef fish
are about a month or two and the duration of the competency time window tends to be
greater in fishes with larger PLD (Victor, 1991). Lagrangian particles are transported by
coastal circulation until the end of the competency time windows. During the time
period, Lagrangian particles are allowed to record successful settlement more than
once, in order to account for individual differences in development within a larval cohort.
Particles settling to (and from) outside of the domain are also accounted for by taking
advantage of the along-shore periodicity counting the number of periodic domains the
particle has traversed.
3.3 Larval packet transport in coastal circulation-simulations
Coastal circulation processes and eddy water parcel motions can be estimated using
the temporal evolution of the sea level distribution because contours of sea level
provide geostrophic streamlines for flow at the sea surface. Figure 4 shows model
depictions of sea level contours and the trajectories of Lagrangian particles under the
strong upwelling conditions characteristic of summer. In the figure, low sea level
features (blue regions) correspond to cyclones and support counter-clockwise
geostrophic currents. Anti-cyclones, high sea level features (yellow region), create
clockwise circulations. The sea level contours represent coastal eddy motions well.
Coastal eddy motions evolve rapidly in response to statistically forced winds, while
being advected with the mean equatorward currents. Comparison of each panel of Fig.
4 shows that eddy patterns change greatly over the period. Lagrangian particles are
advected by the simulated currents, mostly along lines of constant sea level, and these
patterns evolve rapidly as the sea level patterns change (Fig. 4). Although Lagrangian
15
particles are uniformly released from nearshore waters, their density quickly becomes
non-uniform as the particles are advected by the coastal eddy motions. Eddies act to
sweep larvae together into streaks, and transport them as coherent packets, as
illustrated in the previous section (cf. Figs. 1 and 4).
Eddy-induced stochasticity in larval settlements can be seen by examining a time series
showing the temporal and spatial patterns of successful settlements (Fig. 5). Figures 5 a
– d show time series of the density and along-shore destination locations of particles
that successfully settle in the domain for the default case. Four different realizations of
the simulations are shown, using different number seeds in the wind forcing, which
results in the circulation patterns being different for each of the realizations,
corresponding to four independent spawning seasons or years. Successful settlements
occur in infrequent pulses, because settling larvae are accumulated by coastal eddies
and delivered to nearshore habitats as a coherent group (Fig. 4). Different realizations
produce different patterns in settlement pulses that are still spatially heterogeneous, but
the locations and intensities of the hot spots change (cf. Figs. 5 a – d).
Simulated settlement patterns are stochastic regardless of the spawning season, PLDs
and vertically migrating behavior. In the winter off California, upwelling-favorable winds
are diminished along with the strength of the currents (Parrish et al., 1981; Pickett and
Paduan, 2003), and reduced mean offshore surface currents keep more released larvae
within the region of suitable habitat, which may diminish eddy-induced stochasticity. The
simulated dispersal patterns, however, show that the same larval release schedule
applied to a typical winter flow field still shows a high level of stochasticity (Fig. 5 e).
Ontogenetic descents move larvae from higher speed surface flows to deeper, slower
flows, which is expected to reduce eddy-induced stochasticity. When we model vertical
migrating ontogenetic behavior as a descent from the surface to 30 m after 5 days from
release, the arrival locations and times of settlements pulses are modified at certain
times, but the highly heterogeneous nature of the settlement remains (cf. Figs. 5 a and
h). Dispersal patterns can change greatly depending on the PLD because larvae are
transported further in time and distance as their PLD increases. Figures 5 f and g show
16
arrival time series obtained with two different competency time windows, i.e., 10 – 20
and 30 – 60 days, in exactly the same flow fields as in Fig. 5 a. While simulated
dispersal patterns are distinctly different under these differing competency time
windows, they all remain stochastic.
3.4 Spatio-temporal scales emerging from the simulated dispersal
We can measure the temporal and spatial scales of arriving settlement pulses using a
variogram (Rossi et al., 1992). Time scales are calculated using the variogram of the
arrival density with arrival location held constant, while length scales are calculated
using the variogram range of the arrival density with arrival location held constant.
Arrivals before the first settlement is observed (e.g., day 20 in Fig. 5 a) and after the last
settlement occurs (e.g., day 130 in Fig. 5 a) are not used in the computation. The
variogram range is defined as the minimum spatial lag at which the variogram values
reach the variance of the arrival densities. The obtained time and length scales are
summarized in Table 1. The temporal and spatial scales of settlement pulses were
rather consistent regardless of the seasonal upwelling conditions, PLDs and vertically
migrating behavior, although some variation occurs. The arrival time scales of
settlement pulses range from 11 to 16 days. The arrival length scales of settlement
pulses measure 35 to 57 km. These values are similar to the eddy time (τ = 12 – 17 d)
and length scales (l = 40 – 60 km) estimated by the scaling theory for California Current.
Nearshore eddies appear to set the spatio-temporal scales of successful larval
settlement events in the simulated flow fields from this analysis, although we cannot
really prove it from this analysis.
3.5 Scaling theory vs the circulation simulations
In order to assess connectivity via advection of water parcels in the simulated flow
fields, nearshore waters are delineated into multiple equal-area sites with 4-km spacing
in the along-shore direction (64 sites in the domain). Sites are identified by their alongshore locations as i (or j) = 1, 2, ..., 64. Connectivity matrices are obtained based upon
17
the number of settling particles between a given source and destination for each
realization of the simulations, i.e.,
where S^n_{ji} indicates the number of successfully setting particles that are transported
from site j to site i during the n-th spawning season (here, realization of the simulations),
and N_j is the number of particles released from site j. This matrix defines coastal
connectivity independent of adult population demographics.
The connectivity matrices obtained from the simulated flow fields are neither smooth nor
homogeneous for a single spawning season, similarly to the predictions by the packet
model (cf. Figs. 2 and 6). Figure 6 shows the resulting connectivity obtained from four
different realizations of the simulations (simulated using a different random initial seed
in the wind forcing), corresponding to the arrival time series shown in Figs. 5 a – d.
Connectivity matrices obtained from the coastal circulation simulations show a good
qualitative agreement with the packet model (cf. Figs. 2 and 6), that is, spatial
heterogeneity in connectivity predicted by the packet model shows similar patterns to
the results obtained from the coastal circulation simulations. Coastal eddy motions alter
connectivity patterns from one time period (spawning season) to the next. As averaged
over many independent spawning seasons, the spatial heterogeneity in connectivity
decreases both in the coastal circulation simulations and the packet model predictions
(cf. Figs. 3 and 7). Equally, both connectivity matrices are smoothed out, and can be
approximated by a diffusion model. Thus, the packet model shows a good qualitative
agreement with the circulation simulations.
The packet model also accurately quantifies eddy-induced stochasticity in the simulated
settlement events. Figure 8 compares estimates of <CV_s> obtained from the
circulation simulations and with those predicted by scaling theory, as a function of the
number of spawning seasons. The prediction from scaling theory [Eq. (6)] shows good
18
agreement with the results obtained from circulation simulations. For the case with
shorter larval release duration, there are less settlement pulses for a single spawning
season (smaller N_{ev}). Accordingly, <CV_s> is higher, and it decays more slowly than
in the default case when averaged over many independent spawning seasons. The
packet model reasonably approximates the decay of <CV_s> for each case. The good
agreement between the scaling theory and the circulation simulations suggests that
scales of spatial settlement variations, induced by ocean circulation, can be estimated
by counting the number of coastal eddies and their spatial extent in the target area.
Averaged over many independent spawning seasons, more eddies bring more larvae to
nearshore habitats, and spatial settlement variations are smoothed out. Coastal eddy
motions set the spatial and temporal scales of larval settlement patterns at least in the
simulated flow fields.
3.7 Caveats of scaling theory
Stochasticity of settlement events is reduced when settling larvae stay within nearshore
waters during their planktonic duration. For short PLDs or under the weak upwelling
conditions (as in winter), a considerable number of settling particles may stay within
nearshore waters during their entire PLD. For example, for the case with the short PLD
(around two weeks) under the weak upwelling condition of winter, approximately 25% of
the settling particles stay in nearshore waters throughout their PLDs (Fig. 9 a), and their
cross-shore dispersal range (within which 99% of the settling particles are contained) is
limited to 50 km from the coast at most (Fig. 9 b). Dispersal patterns of settling larvae
that are not advected away by currents or eddies from nearshore waters will not be
much affected by coastal eddy motions. Reflecting that these settlers that stay within
nearshore waters, eddy-induced stochasticity is reduced for shorter PLDs and under the
weak upwelling condition of winter (Fig. 8). For many of the cases considered in the
dispersal simulations (and for many benthic species), however, not many larvae remain
nearshore, and coastal eddy motions are important. Typically, more than 90% of settlers
are advected out of nearshore waters before settling (Fig. 9 a). Under the strong
19
upwelling condition in summer, in particular, almost all settlers do not stay within
nearshore waters, and dispersal range exceed 100 km.
5 Discussion
The proposed scaling theory (the packet model), counting the number of arriving eddies
to nearshore habitats, suggests that coastal eddy motions should set a strong source of
uncertainties in larval dispersal patterns, even when extreme abundance of larval
production is available. Larvae are swept into streaks (or packets) by coastal eddies,
while transported by mean currents and wind-driven Ekman circulation (Fig. 1). The
number of larval packets formed by coastal eddies is not large enough to achieve
smooth and homogeneous larval settlement patterns when viewed on annual time
scales (Fig. 3). Therefore, even without any biological or topographic sources of
variability, population connectivity patterns are expected to be heterogeneous in space,
creating only a few strong connections among sites (Fig. 2). This scaling analysis is
good news and bad news for the study of marine ecology. While the packet model
provides a means to quantify this eddy-induced stochasticity in larval settlement
patterns (Fig. 8), it also implies that coastal eddy motions may make larval settlement
and population connectivity patterns almost unpredictable, due to the chaotic nature of
turbulence (Figs. 2 a – c). However, this eddy-induced stochasticity in larval dispersal is
important only if it alters the predictions and/or understanding of nearshore marine
population dynamics. We discuss the importance of these findings below.
One case where coastal eddy motions can play an important role in marine population
dynamics is in the case of interspecific interactions where competition among larvae
may be important for determining post-settlement recruitment rates. Larvae from
different spawning periods can “catch” different eddies, resulting in different dispersal
patterns on a year-to-year or generation-to-generation basis. Hence, there is the
possibility that larvae from an inferior competitor will occasionally land in locations that
are free of the superior competitor’s larvae – if it happens often enough, the two species
can coexist. This can be considered as a spatial variant of the storage effect where rare
20
recruitment events can lead to coexistence of interacting species (e.g., Warner and
Chesson, 1985, Berkley et al. in prep).
Another important effect of coastal turbulence could be in the condition to be satisfied
for retention of local production and upstream invasion. Byers and Pringle (2006)
derived the condition for the upstream-transport/retention condition in an advective
environment, in terms of the Gaussian parameters µ and σ in Eq. (3), assuming a
smooth, Gaussian larval dispersal and no intra-specific and interspecific competition at
recruitment. The packet model, however, suggests that larval should be transported as
coherent packets, not as a diffusion process, and dispersal patterns can be
approximated by Gaussian distributions only when averaged over many independent
spawning seasons (Fig. 4). Hence the upstream-transport/retention condition may be
altered by accounting stochastic larval transport and also intra- and inter-specific
interactions at recruitment.
Chaotic eddy transport of larvae may be an important mechanism that makes species
coexistence in an advective environment possible. Imagine two identical species that
disperse in an advective environment with initial conditions in which one species is
distributed upstream from the other (cf. Byers et al, 2006). If the two species are
demographically identical (e.g., the same mortality, same fecundity, same competency
time windows and post-settlement density dependence factors), and competing for
limited resources (or space), the upstream species has a great advantage because it
can send more larvae to settling sites downstream due to the mean advection in the
system. Downstream species may be able to invade upstream because their larvae can
be occasionally transported upstream as a coherent packet, not as a weak diffusion
processes, into areas where upstream species happen to be rare. The number of
successful settlements at a site will show substantial variation year to year, even without
any other sources of uncertainties (Fig. 2). The “packet” transport of larvae by coastal
eddy motions, coupled with life history, may structure nearshore marine populations
because it adds strong inter-annual variation in settlements.
21
The stochastic nature of population connectivity influences many other ecological
processes. For example, the pulsed nature of larval transport, even at long distances
from a source, can act to alleviate the Allee effects that limit the success of longdistance colonization (Gascoigne and Lipcius, 2004). Further, occasional large, isolated
pulses of larval settlement predicted here should enhance the contribution of the
storage effect on species persistence and coexistence (Warner and Chesson, 1985).
Similarly, the long lives and high fecundity that are characteristic of most marine
organisms may in fact be evolutionary responses to the inherent uncertainty of
recruitment. Different life histories (e.g., various vertical positioning behavior, spawning
timing and PLDs) may be employed in order to take advantage of chaotic coastal eddy
motions. While the present study identifies important features of complex nearshore
systems, more work is required to understand the interactions among variable coastal
circulation, organism life cycles, and the management of these important ecosystems
and the fisheries they support.
The cohort transport of fish larvae will also make the management of nearshore
fisheries difficult. Fishery managers must assess stocks on annual time scales, and
here we show that at that population connectivity is inherently stochastic. Fisheries
scientists and marine ecologists have long tried to explain the sources of recruitment
variability so that recruitment could be predicted knowing only environmental factors
(Hjort, 1914; Chavez et al., 2003; Cowen et al., 2006; Walters and Martell, 2004; Myers,
2001). But, because larval dispersal is not a deterministic diffusion process, local rates
of larval settlement will be largely decoupled from local stocks; thus stochastic
population connectivity may provide an unexplored source of noise in observed stockrecruitment relationships (Walters and Martell, 2004; Myers, 2001). The proper design
of a network of marine protected areas (MPAs) requires accounting for the spatial
distribution of reproductive adults and the connectivity among populations as well as the
spatial distribution of fishing effort. In most MPA modeling studies, larval dispersal is
described as a smooth and homogeneous diffusion process, given certain regional
oceanographic information (e.g., Largier, 2003; Siegel et al., 2003). Our work indicates
that the connectivity among MPAs will not be persistent year to year, reflecting chaotic
22
coastal eddy motions. The connectivity among MPAs may be very different depending
on species characteristics as well (e.g., spawning timing and ontogenetic behaviors).
Fishery managers must cope with this unavoidable unpredictability in nearshore
populations and the fisheries they support.
Acknowledgments
The authors acknowledge a series of enlightening discussions with Chris Costello, Ray
Hilborn, Steve Polasky, Libe Washburn, Sally McIntyre, Jenn Caselle, Brian Kinlan, Tim
Chaffey, James Watson, Heather Berkley, Charles Dong, Jim McWilliams and Jamey
Pringle. This work is a contribution of the “Flow, Fish and Fishing” biocomplexity project
work and is supported by the National Science Foundation (NSF grant # 0308440).
Figure 1. (a) Sea-surface temperature adjacent to the central California coastal revealed by the
Advanced Very High Resolution Radiometer (AVHRR). Images are colored from dark blue (~ 5 C), though
light blue to green to yellow, ending in red (~ 15 c) to indicate a progression from cold to warm water.
Dates of photographs are May 18, 1982, April 6, 1983, and May 21, 1984. Notice that the California
Current consists of many coastal eddies that predominates in variability in the circulation system. (b)
Schematic diagram of larval transport in California Current system adjacent to central California. Coastal
eddy motions indicated with orange circles (clock-wise eddy motions) and light blue circles (counter-clockwise eddy motions) evolve rapidly in response to winds, while being advected with the mean currents
(blue arrow) and wind-driven Ekman circulation (red arrow). These eddies act to sweep larvae (green
circles) together into streaks, and transport them as coherent packets. [This diagram is Fig. 7 of
23
Roughgarden et al. (1988) with additional cartoons on top of it. I will make a new diagram getting
some pictures from Erik Fields for the panel a and using Adobe illustrator for the panel b.]
24
Figure 2. a) - c) Connectivity matrices obtained from three different realizations (using different random
number seed in determining source and destinations of packets) of the packet model [Eqs. (1), (3) and
(4)], given the parameters L = 250 km, l = 50 km, T = 30 d, τ = 15 d, f_{sv} = 1, µ = -68 km and σ = 71 km.
d) Connectivity matrix predicted by a diffusion model [Eqs. (3) and (5)], given the parameters µ = -68 km
and σ = 71 km. The source and destination locations are given by along-shore location. Smaller values
indicate locations that are downstream. Values less than 0 or greater than the domain size (250 km)
indicate source locations that are outside of the domain. The dashed line in the connectivity matrices
represent self settlement; i.e., the source and destination locations of larval packets are identical. Points
above the self-settlement line indicate larval transport from upstream to downstream. Color scales for the
diffusion model show the probability densities for the source (or destination) locations of successfully
settling larvae to (from) a given site. Each row (or column) of the connectivity predicted by the diffusion
model has a Gaussian profile given by Eq. (3). Color scales for the packet model are adjusted so that
predicted connectivity becomes identical to the diffusion model as averaged over infinite number of
spawning seasons (realizations), as given by Eqs. (3) and (5).
Figure 3. Connectivity matrices obtained from the packet model [Eqs. (1), (3) and (4)] for 1 season and
averaged over 5, 10 and 100 independent spawning seasons. Here, the model parameters are set to L =
250 km, l = 50 km, τ = 15 d and T = 90, 450, 900 and 9000 d. Mean dispersal scales are set to µ = -68 km
and σ = 71 km, corresponding to the condition used in Fig. 2. The coefficients of variation for the number
of the settlement among sites (variation for summation of each column of the matrix) are 0.38, 0.18, 0.10
and 0.02 (from left to right) for this particular realization. For the detailed descriptions of connectivity
matrices, see the caption of Fig. 2.
Figure 4. Model depictions of sea level (color contours in cm) and the trajectories of Lagrangian particles
for days 45, 60, 75 and 90. Here, only 2000 randomly chosen particles are shown. The circles show the
location of the particles while the white trails behind each show their previous 3-day trajectories. Red
circles indicate successfully settling particles (i.e., found within 10 km from the coast during the
competency time window.) The competency time window is set to 20 to 40 days. The vertical dashed red
line indicates the boundary for the nearshore habitat from which particles (larvae) are released and where
settlement can occur. Low sea level features correspond to cyclones and support counter-clockwise
geostrophic currents. Anti-cyclones, high sea level features, create clockwise circulations. The flow field is
modeled to represent conditions found in the central coast of California (CalCOFI line 70) during a typical
July (high upwelling conditions). The simulated domain is 256 km in the along-shore direction and 288 km
in the cross-shore direction (only the inner 200 km are shown in the cross-shore direction). Particles are
released near the top surface, and keep their depth while they are passively transported by coastal
circulation in horizontal directions. There is a mean southward flow in the domain.
Figure 5. Arrival time series showing the density (color) and arrival to along-shore locations (vertical
axis) of successfully settling particles from all source locations. Left panels: arrival time series obtained
from four different realizations of the simulations (using different random number seeds in the wind
forcing) for the base case (strong upwelling condition of summer). In this case, Lagrangian particles
remain near the top surface while being passively transported in horizontal directions, and the
competency time window is set from 20 to 40 days. Settling particles from outside of the domain are
simulated by taking advantage of the along-shore periodic flow fields. Right panels: arrival time series e)
under the weak upwelling condition in winter, f) when the competency time window is set from 10 to 20
days and g) from 30 to 60, and h) for vertically-migrating larvae (migrating to 30 m after 5 days, mimicking
an ontogenetic vertical migration) days. The flow fields are identical for the panels a, f – h. The three
vertical lines in the time series indicate the day when the first larvae are able to settle; when larval
releases stop; and when settlement ends, respectively. The densities (colors) are given by the number of
settlement events per 4 km per day normalized by the total number of settlement events for each case.
Table 1. Length (km) and time (days) of the arriving pulses of settlements (see Fig. 5) calculated using a
variogram range (Rossi et al., 1992) under different conditions depending on seasonal upwelling
conditions, planktonic larval durations (PLDs) and vertically migrating behavior. Two scenarios are tested
for vertically migrating behaviors: i) staying near the top surface and ii) vertically migrating to 30 m after 5
days. Values are shown for a strong upwelling condition (summer) and a weak upwelling condition
(winter). The time scales are calculated using the variogram range of the arrival density with arrival
location held constant, while the length scales are calculated using the variogram range of the arrival
density with arrival location held constant.
Figure 6. Connectivity matrices obtained from four different realizations of the simulations (using different
random number seeds in the wind forcing) under the strong upwelling condition in summer. Lagrangian
particles are transported passively in horizontal directions while they stay near the top surface. The
competency time window is set to 20 to 40 days. Nearshore waters are delineated into multiple equalarea sites with 4-km spacing in the along-shore direction (64 sites in the domain). Connectivity (color) is
given by the number of successful settlers advected from a source site j to a destination site i divided by
the total number of particles released from site j [Eq. (7)]. The source and destination site locations are
shown with along-shore location here. Smaller values indicate site locations that are downstream. Values
less than 0 or greater than the domain size (256 km) indicate source site locations that are outside of the
domain. Settling particles from outside of the domain are simulated by taking advantage of the alongshore periodic flow fields. The dashed line in the connectivity matrices represent self settlement; i.e., the
source and destination locations of settling particles are identical.
Figure 7. Connectivity matrices obtained from the coastal circulation-simulations for a) a single spawning
season, b) averaged over 5 spawning seasons, c) averaged over 10 spawning seasons and d) averaged
over 20 spawning seasons under a strong upwelling conditions of summer. Connectivity matrices for a
number of seasons are generated by combining the same number of randomly-chosen realizations of the
simulations. The coefficients of variation for the number of the settlement among sites (variation for
summation of each column of the matrix) decreases as 0.58, 0.23, 0.13 and 0.08. For detailed
descriptions of connectivity matrices, see the caption of Fig. 6.
Figure 8. Comparison of the coefficient of variation for the number of settlements among sites (CV_s)
obtained from the circulation simulations (symbols) and predicted by scaling theory (packet model), Eqs.
(6) as a function of the number of spawning seasons (or realizations). The expected CV_s value,
<CV_s>, is plotted as a function of the number of spawning seasons. As averaged over many
independent spawning seasons, spatial heterogeneity in settlement patterns decrease (see Figs. 3 and
7), and accordingly <CV_s> decreases. We examine a) the default case with 90-day larval release
duration for a single spawning season and b) the reduced 30-day release case. Accordingly, T is set to a)
T = 90, 180, ..., 2250 d and b) T = 30, 60, ..., 750 d for the scaling theory version. The other parameters
are the same between the two cases: L = 256 km, l = 50 km, τ = 15 d and f_{sv} = 1. For the circulation
simulations, the competency time window is set to 20 to 40 days. Two seasonal upwelling conditions are
tested: a strong upwelling condition of summer (black circles) and a weak upwelling condition of winter
(open circles). Two different scenarios are examined for ontogenetic vertical migration: staying near the
top surface (circles) and migrate 30 m after 5 days from release (triangles). Simulated settlement patterns
for a number of spawning seasons are generated by combining the same number of randomly-chosen
realizations of the simulations. The expected value is computed using 20 different combinations. The
horizontal line indicates CV_s = 0.13, corresponding to the rather smooth connectivity matrix shown in
Fig. 3 panel c.
Figure 9. Top panels: fraction of the total number settling particles that remained in nearshore
waters (i.e., within 10 km from the coast) in the total number of settling particles obtained from
the circulation simulations as a function of the competency time window under two seasonal
upwelling conditions: a) a strong upwelling condition of summer and b) a weak upwelling
condition of winter. Two scenarios are tested for vertical migrating behavior: staying near the top
surface (black bars) and vertically migrating to 30 m after 5 days (white bars). Bottom panels:
cross-shore dispersal range of the settlers (the distance from the coast within which 99% of the
settling particles are contained) as a function of the competency time window under c) a strong
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