Appendix A: Methodological details
A. Empirical data collection
Detailed methodology of fish collection, otolith extraction, preparation and analyses for the
samples used in this manuscript can be found in Shima and Swearer (2009). Briefly, we
collected recently settled common triplefin (Forsterygion lapillum) from standardized artificial
collectors (Ammann 2004) deployed in triplicate to rocky reef sites (n=3 sites in Wellington
harbour, n=3 sites along Wellington south coast), and sampled at approximately weekly intervals
from December 2003 to March 2004.
We extracted and analysed sagittal otoliths from a sub-sample of collected fish. One
otolith from each fish was embedded in cyanoacrylate and polished along the sagittal plane with
3 µm diamond lapping film to expose daily growth increments across the postrostral axis. We
used image analysis software (Image Pro Plus v.5.0) to quantify daily otolith increments and
visible indications of major life-history transitions (hatching, settlement). The remaining otolith
from each fish was cleaned in a buffered 15% H2O2 solution, embedded in inert resin, and
polished along the sagittal plane to within 10 µm of the otolith core using 9 µm diamond lapping
film. We used LA-ICPMS to quantify trace element concentrations (relative to Ca) for Li, B,
Mg, P, S, Mn, Cu, Zn, Sr, Ba, and Pb with an 80 µm wide transect running from the core to the
edge of the otolith along the post-rostral axis.
For each fish, we decomposed the time-series of each trace-element concentration (from
hatching to settlement) to 13 statistical descriptors (“extracted features”, sensu Wang et al 2006)
and used these to cluster individuals (using a K-means algorithm) with similar trace-element
patterns across their larval development. We used consensus among the cubic clustering
criterion (CCC), pseudo F statistic, and pseudo t2 to determine the most parsimonious number of
clusters in our data set (this analysis suggested 2 distinct clusters; Shima and Swearer 2009)
Patterns of variation in trace element concentrations and extracted features were indicative of
development in nearshore/coastal waters for one cluster, and in offshore waters for the other.
B. Metapopulation model
We used a simple metapopulation model with two identical patches to demonstrate the potential
for dispersal-induced variation among individuals to influence population density. Each patch
contains a subpopulation of adults that produce dispersing larvae. Larvae that return to settle in
their natal patch are “non-dispersers” and those that move to the other patch are “dispersers”.
We model a “direct cost” of dispersal (e.g., Burgess et al. 2012, and references therein)
by reducing the proportion of “dispersers” that successfully settle. This assumption of a direct
cost to dispersal is indirectly supported by our previous observations for the common triplefin,
that (i) putative dispersal pathways are independent of putative natal origins (Shima and Swearer
2009), and (ii) putative “non-dispersers” settle in greater numbers than putative “dispersers”
(Shima and Swearer 2010). Taken together, these results suggest that, all else being equal,
dispersers have reduced larval survivorship relative to non-dispersers. We vary the strength of
this direct cost in our model to explore its effect on population density.
Settlers enter the adult stage after a maturation delay (e.g., Kirkpatrick 1984), which
represents time required to grow and develop to reproductive maturity. We model an “indirect
cost” of dispersal (e.g., Burgess et al. 2012, and references therein) by assuming that dispersal
increases the maturation delay relative to non-dispersal. This rests on the observation that
dispersers have lower growth rate and the assumption that maturation occurs at a fixed size. As a
consequence, survival to the adult stage is reduced for dispersers. This assumption is supported
by the empirical results of this paper (Fig 1), and by our previous observations that suggest (i)
putative dispersers have reduced short-term survival rates relative to non-dispersers, and (ii)
post-settlement survival to adulthood is disproportionately comprised of putative non-dispersers
(Shima and Swearer 2010). Finally, we assume that post-settlement survival through the
maturation delay is density dependent with respect to adult densities. This assumption is based
upon a number of empirical studies that demonstrate post-settlement density-dependent survival
in small reef fish (reviewed in Hixon 2011), and indirectly supported by a negative correlation
between densities of juvenile and adult common triplefin (Mensink and Shima 2014).
We denote adult density in patch i at time t as Nai(t), where i=1 or 2. The rate of change
of adults in patch i is the difference between maturation of both settler types (at rates Mni(t) and
Mdi(t) for non-dispersers and dispersers, respectively) and adult mortality:
𝑑𝑁𝑎𝑖 (𝑡)
𝑑𝑡
= 𝑀𝑛𝑖 (𝑡) + 𝑀𝑑𝑖 (𝑡) − 𝛿𝑎 𝑁𝑎𝑖 (𝑡)
(1)
where a is the adult per capita mortality rate. At equilibrium,
∗
∗
∗
𝑀𝑛𝑖
+ 𝑀𝑑𝑖
− 𝛿𝑎 𝑁𝑎𝑖
=0
(2)
Maturation rate is equal to recruitment rate multiplied by survival through the maturation delay,
i.e., from settlement to reproductive maturity. We assume the mortality rate during this period
depends on adult density, such that
∗
∗
∗ −(𝛿𝑗 +𝛼𝑁𝑎𝑖 )𝜏𝑛
𝑀𝑛𝑖
= 𝑅𝑛𝑖
𝑒
(3)
where Rni* is recruitment of non-dispersers to patch i, j is the background juvenile mortality
rate,  represents the increase in mortality with adult density, and n is the maturation delay for
non-disperses. Similarly,
∗
∗
∗ −(𝛿𝑗 +𝛼𝑁𝑎𝑖 )𝜏𝑑
𝑀𝑑𝑖
= 𝑅𝑑𝑖
𝑒
(4)
Our assumption that slower growth of dispersers results in a longer maturation delay (d>n)
implies that, for a given adult density (Nai*), dispersers have lower survival than non-dispersers.
Furthermore, the duration of the maturation delay interacts with density dependence via adult
density in Eq. 4, such that the effects of differences in the delay are nonlinear. An alternative
assumption would be that density dependent mortality occurs in the larval dispersal stage. In this
alternative, the recruitment rates (Rdi* and Rni*) would be nonlinear functions of the density of
adults via their contribution to the larval population. We do not consider this possibility here.
The fraction pi of larvae produced by adults in patch i recruit locally, and the remainder
disperse to the other patch. The equilibrium recruitment rates are
∗
∗
𝑅𝑛1
= 𝑝1 𝑓𝑁𝑎1
(5)
∗
∗
𝑅𝑑1
= (1 − 𝑝2 )𝑓𝑠𝑁𝑎2
(6)
∗
∗
𝑅𝑛2
= 𝑝2 𝑓𝑁𝑎2
(7)
∗
∗
𝑅𝑑2
= (1 − 𝑝1 )𝑓𝑠𝑁𝑎1
(8)
where f is adult production of offspring that survive to settlement, and s is the relative probability
of survival to settlement for larvae that disperse between patches. For example, if s=0.1, the
probability of survival to settlement for larvae that disperse is 10% the probability for larvae that
settle in their natal patch.
We find the equilibrium by inserting Eqs 3-4 into Eq. 2. With the assumption that the two
patches are identical and the fraction of larvae that recruit locally is the same in both patches
(p1=p2=p), we have a single equation for the adult equilibrium (Na1*=Na2*=Na*),
∗
∗
𝑝𝑓𝑒 −(𝛿𝑗+𝛼𝑁𝑎)𝜏𝑛 + (1 − 𝑝)𝑓𝑠𝑒 −(𝛿𝑗+𝛼𝑁𝑎 )𝜏𝑑 − 𝛿𝑎 = 0
(9)
which we solve numerically. For the case in which all recruits have a maturation delay equal to
the mean of non-dispersers and dispersers (mean=(n+d)/2), the equation for the equilibrium
adult density is
∗
∗
𝑝𝑓𝑒 −(𝛿𝑗+𝛼𝑁𝑎)𝜏𝑚𝑒𝑎𝑛 + (1 − 𝑝)𝑓𝑠𝑒 −(𝛿𝑗+𝛼𝑁𝑎)𝜏𝑚𝑒𝑎𝑛 − 𝛿𝑎 = 0
which can be solved analytically:
1
𝑁𝑎∗ = 𝛼 (𝜏
1
𝑚𝑒𝑎𝑛
𝑙𝑛 (
(𝑝+(1−𝑝)𝑠)𝑓
𝛿𝑎
) − 𝛿𝑗 )
(10)
We present the ratio of solutions to Eq. 9 and Eq. 10 in Fig. 2, using parameter values in Table
A1. These values are intended to represent a small reef fish such as Forsterygion lapillum, but
the qualitative results depend only on the conditions that survival to maturity is greater for nondispersers than for dispersers (n<d) and long-distance dispersers have lower survival than
larvae that recruit to their natal patch (0<s<1).
As described in the Results, the proportion of dispersers vs. non-dispersers among settlers
is key to the predictions in Fig. 2. The proportion of non-dispersers entering subpopulation i at
equilibrium is (from Eqs 5-8):
∗
𝑅𝑛𝑖
∗ +𝑅∗
𝑅𝑛𝑖
𝑑𝑖
𝑝
= 𝑝+(1−𝑝)𝑠
(11)
which increases with p and decreases with increasing s (Fig. A1). In general, the heterogeneous
adult equilibrium is greater than the homogeneous equilibrium when the proportion of nondispersers is large. However, the heterogeneous equilibrium adult density does not switch from
less than to greater than the homogeneous equilibrium at a fixed threshold value of the
proportion of non-dispersers among settlers (i.e., in Fig. A1 the transition between white and
shaded areas does not coincide with a contour for Rni*/(Rni*+Rdi*). Instead, the threshold
proportion of non-dispersers decreases (from approximately 0.4 to 0.3) as p and s increase.
Further details of our model may be found in Noonburg et al. (in press).
References
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Table A1. Model parameters and values for numerical results in Fig. 2.
Parameter

a
j
d
n
p
s
f
Value
10
1
0.01
2
1
variable
variable
10
Description (units)
Strength of adult-induced mortality on of dispersers in the juvenile stage (year-1
m2)
Adult mortality rate (year-1)
Density independent mortality rate of dispersers in the juvenile stage (year-1)
Maturation delay for dispersers in the juvenile stage (year)
Maturation delay for non-dispersers in the juvenile stage (year)
Fraction of non-dispersing larvae
Relative probability of survival for larvae that disperse between patches
Adult production of offspring that survive to settlement
Figure A1. Proportion of non-dispersers among settlers in each subpopulation at equilibrium
(contours). White background indicates the region of parameter space where the heterogeneous
equilibrium is less than the homogeneous equilibrium. Shaded background indicates the region
where the heterogeneous equilibrium is greater than the homogeneous equilibrium. These
regions are identical to Fig. 2 in the main text.