Behavioral Ecology Vol. 10 No. 5: 572-577
© 1999 International Society for Behavioral Ecology
A dynamic model of group-size choice in the coral reef fish Dascyllus albisella
Department of Evolution, Ecology, and Organismal Biology, Aquatic Ecology Laboratory, The Ohio State University, 1314 Kinnear Road, Columbus, OH 43212-1156, USA
Address correspondence to F. A. Martinez. E-mail: martinez.38{at}osu.edu .
Received 4 April 1998; revised 29 September 1998; accepted 17 February 1999.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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We developed a dynamic programming model of group size choice for settling coral reef fish to help understand variability in observed group sizes. Rather than calculating optimal group size, we modeled optimal choice and calculated the acceptable group sizes that arose from this choice. In the model, settling individuals weigh the fitness value of settling in a group against the expected fitness of searching another day and encountering other groups, choosing the option with the higher value. Model results showed that individuals settling on any given day in the settling season have several acceptable group sizes in which they can settle. The range of acceptable group sizes also changes across the season. Early in the season, when there is still adequate time to grow, large groups (with higher survival) have the highest fitness. Late in the season, when the ability to grow fast becomes more important, small groups, which convey fast growth rates (although riskier), have higher fitness. Thus, according to our model, even when fish all make the same, simple decisions, a variety of outcomes are possible, depending on the specific options encountered and temporally changing ecological pressures. Even when all fish behave optimally, initial variability in group sizes will persist.
Key words: coral reef fish, coral reef fish settling, dynamic programming, group size variability, growth-survival trade-off, optimal group size, recruitment.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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The adaptiveness of grouping behavior and group size can be investigated using an optimality approach (Pulliam and Caraco, 1984
). By contrasting the costs and benefits associated
with grouping, individuals can choose whether to enter a group
(Pulliam and Caraco, 1984) and
what size group to enter (e.g., Baird and
Dill, 1996). But the definition of an "optimal" group
size (Caraco and Wolf, 1975;
Giraldeau, 1988;
Pulliam and Caraco, 1984;
Ranta, 1993) can be
problematical. Expectations of optimality are often based on unrealistic
assumptions, such as perfect knowledge, no cost to sampling, or availability
of all possible options. Because individuals will differ in their experiences
(e.g., knowledge of environment or local availability of different options),
it is important to model the decision process rather than the decision
outcome. Thus, we believe it is important to separate the concept of an
optimal "group size" from that of an optimal "choice"
given constraints of individual experience
(Sibly, 1983).
Coral reef fish provide an excellent opportunity to investigate optimality
in choice of group size. Many species have a dispersive larval stage in the
pelagic ocean waters and then return to the reefs where they settle into
groups with conspecifics (Sale,
1980). For these fish, group size can affect an individual's
growth rate and probability of survival
(Booth, 1995;
Forrester, 1990;
Jones, 1987a,
b). As group size increases,
growth decreases (Booth, 1995;
Forrester, 1990;
Jones, 1987a); however, as
group size increases, the probability of survival increases
(Booth, 1995;
Forrester, 1990;
Jones, 1987b). Therefore, some
coral reef fish are faced with a trade-off between growth and survival as a
function of group size at the time of settlement (e.g.,
Booth, 1995;
Forrester, 1990). Given a
description of how these two rates change with group size, we can describe an
"optimal" group size for settling larvae. However, in nature,
group-living coral reef fish of any given species occur over a range of group
sizes under relatively similar environmental conditions
(Shapiro, 1991). In this
study, we sought to understand how differences among individuals and
differences in their experiences might drive some of this variability in group
sizes.
Here we look at how the trade-off between growth and survival can influence group size choice by settling larvae of the coral reef fish Dascyllus albisella. We specifically address three main aspects: (1) can variability in group-size choice arise under a specific set of unchanging conditions in a simple optimal decision model? (2) Is the range of acceptable group sizes influenced by the date of settlement? (3) Does body size at settlement affect the range of acceptable group sizes?
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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To study group size selection by settling D. albisella, we developed a dynamic programming model. In this model, fish arrive at the reef and sequentially encounter groups in which they might settle. At each encounter, the newly arrived fish must decide whether to settle with that group or continue searching for a different group. A larva's decision is made by comparing the values of fitness that would accrue by settling in the currently encountered group to the potential fitness achieved by searching further. We defined fitness in this case as the product of size-specific fecundity of a female and the probability that it survives and grows to reach maturity by a specified time. Because larvae can settle over a protracted period of time during the year, we also asked how group size choice should vary throughout a single settlement season. We used data from the literature for D. albisella to set the timing of events and the values of parameters in the model.
Subject species
The larval stage for Dascyllus albisella lasts approximately 4
weeks (Wellington and Victor,
1989). After larval development in the pelagia, D.
albisella return to the reef where they settle into branching corals
(e.g., Porites compressa and Montipora verrucosa) as
juvenile groups (Booth, 1992).
When juveniles mature, they leave their groups and join the adult population
in different areas of the reef (Booth,
1991). This allows for processes affecting juveniles and adults to
be considered separately, without the effects due to interactions between life
stages (Booth, 1995).
Dynamic programming model
We used dynamic programming techniques (after
Mangel and Clark, 1988) to
develop a model of group size choice. The model is a discrete time, sequential
decision model, in which we represent the decision-making process of settling
larvae of D. albisella. Upon arriving to the reef, a modeled larva
encounters a potential settlement site containing i = 0-9 fish. A
site with 0 fish is merely an unoccupied but otherwise acceptable coral head.
Upon encountering a potential site, the larva can either settle or continue
searching for another site. If the larva settles, the group becomes size
i + 1.
An individual can arrive on any given day of the settling season. Arrival
at the reef starts near the end of April and continues until mid-October
(Booth, 1991). We assumed that
spawning starts on April 1 and the planktonic phase lasts 26 days
(Booth, 1991), such that
arrival begins on April 26 and continues until October 15. We denote an
individual's arrival date as t0 (t0 =
1, 2,..., 187), where 187 is the last possible day of arrival. We assumed in
the model that fish arrive at the reef with a limited energy reserve and feed
only at low levels until they settled into a group. Considering expected
starvation times for larvae of this size
(Miller et al., 1988) and
allowing for some low level of feeding, we assumed that larvae had six days to
find and settle into a group. If, after six days of searching, the larva had
not settled into a group, we assumed it had died. Note that this assumption
does not imply that larvae actually search for six days (predation risk is
high while searching), just that their energy reserves allow them to search no
longer than six days. Thus, an individual's settling date was between
t0 and t0 + 5. In the model, terminal
time (T) was set as the number of days (T = 432) from the
beginning of the settling season to the mid-point of the spawning season of
the year after arrival.
The expected fitness of an individual that has arrived at the reef on day
t0 (t0 = 1, 2,..., 187) and is
searching over the reef at body size x on day t (t
= t0, t0+1,...,
t0+5) and encounters a group of size i
(i = 0, 1,..., 9) is
Fx(t0,i,t), the
maximum of the expected fitness from settling with that group,
Vsettle(x,i,t), and the expected fitness from continuing
to search another day,
Vsearch(x,t0i,t):
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Expected fitness of settling
Expected fitness from settling on day t, given encounter with a
group of size i, is equal to the probability that the larva survives
from day t to T multiplied by its expected fecundity at
terminal time:
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i+1 is the daily probability of surviving in
a group of size i+1 (the encountered group size plus the newly
settled individual), and R[X(x,i+1,t)] is
the fecundity of a fish of size X(x,i+1,t), which
is the size at time T of a fish that settled at size x into
a group of size i on day t,
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Expected fitness of searching
The expected fitness of continuing searching on day t given an
encounter with a group of size i and arrival at the reef on day
t0 is the probability of surviving 1 more day of search
multiplied by the probability of encountering a group of any given size on the
next day, which is then multiplied by the expected fitness given that it makes
the optimal decision (i.e., search or settle) upon encountering each potential
group size:
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j
is the probability of encountering a potential settlement site having a group
of size j on the following day. If t =
t0+5, then
Vsearch(x,t0,i,t) = 0, so
that Fx(t0,i,t0+5)
= Vsettle(x,i,t).
Model parameters
Arrival size
Dascyllus albisella settles at body lengths of 10 to 16 mm TL
(total length) (Booth, 1992).
In our model, we varied larval size at arrival from 10 to 16 mm to assess
body-size-dependent effects in group size choice. Because most larvae settle
within a short time after arriving at the reef
(Leis, 1991), we considered
growth while searching as negligible; thus size at arrival was the same as
size at settlement.
Growth rate and probability of survival after settling
Booth (1995) estimated mean
juvenile growth rates of 0.16-0.30 mm/day and mean survival time (as
persistence after settling) of 32-44 days. Daily growth and survival after
settling are both functions of group size
(Booth, 1995). We derived these
parameters for our model indirectly from data reported in the literature.
Using Booth's estimates for number of days (di+1)
and proportion survival (si+1) to maturity of
D. albisella as a function of group size, i+1
(Booth, 1995), we calculated
daily survival rates (i+1) using the equation:
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Daily growth rates as a function of group size were determined by taking the difference between size at maturity (70 mm) and the median size at settling (13 mm) and dividing it by the number of days to reach maturity for each group size. The equations for daily growth and survival for the model were obtained by running least-squares regressions on our estimates (Table 1).
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Probability of survival while searching
No field estimates of larval mortality are available
(Leis, 1991). Mortality from
predation is considered to be an important factor while larvae are in the
pelagic zone (Leis, 1991).
However, Johannes (1978)
suggested that the pelagic stage of coral reef fish developed as a mechanism
to escape high predation pressure in the reef environment. We have thus
assumed that larvae returning to the reef will be subject to high predation
risk before settling. Coral heads in the reef are used as refuges
(Hixon and Beets, 1993), and
so settling into the reef reduces predation risk.
Because we found no estimates of larval mortality and had made the assumption that mortality while searching over the reef is higher than that for larvae that have settled, we also assumed that the daily survival rates for larvae when they reach the reef must be lower than their daily survival rates after they have settled as juveniles. Therefore, we set the probability of survival of the largest possible recruit (i.e., 16 mm) while searching equal to 95% of the y-intercept value of the survival function for settled individuals. We then assumed the daily probability of survival while searching to be a declining function of size at the time of arrival (see Table 1).
Group-size distribution
Group sizes in Dascyllus albisella can range from 1 to 15
individuals, and mean group size ranges from 3.3 to 7.5 individuals
(Booth, 1995). However, no
estimates of group-size frequency distributions for D. albisella are
available. Booth (1992)
reported only 30% occupancy of coral heads within one reef. Thus, we assumed a
negative binomial probability distribution of encountered group sizes to
reflect the high probability of encountering an empty coral head (i =
0). We used a mean encountered group size of five. Probabilities were
normalized such that
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Maturation size and fecundity
Dascyllus albisella are considered juveniles when 15-70 mm TL.
Sexual maturity is attained at around 65-70 mm TL
(Booth, 1991), and maximum
adult size is approximately 140 mm
(Stevenson, 1963). We modeled
size at maturity as 70 mm.
Size-specific fecundity was estimated from data on length and number of
eggs reported by Stevenson
(1963). Stevenson
(1963) measured fecundity
values ranging from 12,700 to 43,700 eggs/female; however, sample size was low
(n = 9 females) and the size range of the females limited (104-125
mm). We used an exponential function that approximated the fecundity values
reported by Stevenson (Table
1).
Sensitivity analysis
The trade-off between growth and survival likely is driving many of the
model results. Therefore, we need to know if inaccuracies in our estimates of
the growth and survival parameters could lead to different model responses. We
also recognize the possibility that interactions among parameters might be
important. For instance, if we change growth rates, it not only might affect
the growth—survival trade-off, but also might interact with the effect
of size-specific fecundity. In addition, our estimate of the length-fecundity
relationship for D. albisella is limited by a low sample size and
narrow range of lengths; thus we assessed the influence of the shape of the
length—fecundity relationship on the model's performance.
We looked at model sensitivity using Yates' algorithm, a
2k factorial design of changes in parameters
(Box et al., 1978). For this
analysis, we changed five parameters: the slope and y-intercept
values of both the group-size—dependent growth and survival functions,
and the shape of the length-fecundity relationship (see
Table 1). New estimates of the
parameters for growth and survival functions were obtained by decreasing and
increasing the original values by 10%. A new length—fecundity
relationship function was established by fitting Stevenson's
(1963) fecundity data to a
linear function. In the analysis, a factor or a combination of factors was
considered to have a significant effect on model response if the proportional
change between a baseline observation and the change in the value of the
response attributed to the factor(s) was greater than the proportional change
of the original parameter(s) value(s) for growth and survival (i.e., 20%). For
the length—fecundity relationship, we considered a 10% minimum change to
be a significant effect.
Because of the uncertainty of our original parameter values, we were interested in the qualitative patterns of group size choice as a test of the conceptual framework of our model more than in the precise fitness values. We looked at two responses: the "last day" for which only one group was acceptable for settling and the range of acceptable group sizes on day 75 of the model run. Day 75 was arbitrarily chosen because it is close to the middle of the range of days for which a larva can settle and still reach maturity by terminal time.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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On any given day during the settling season, for an individual of a given length (13 mm) just arriving to the reef, a range of group sizes is acceptable (Figure 1). This means that when an individual encounters a group in this range of sizes, it should choose to remain there rather than continue searching, even if a better group might be encountered the following day. The sudden discontinuity in value of settling at, for example, group size = 7 on day 119, is an artifact of using a single terminal time in the model. A more probabilistic approach to assigning terminal times would result in a less discrete step in fitness values (see Discussion).
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Effect of time of settlement on group size choice
Early in the settling period, D. albisella individuals just
arriving to the reef would do better settling into the first group of
conspecifics encountered (except a group of 0) rather than searching for a
group with a higher fitness value (Figure
1a). Later during the settling period, the fitness of settling
into large groups drops below the value of continuing searching. When
individuals encounter these groups, they maximize fitness by searching another
day for a smaller group rather than settling
(Figure 1b). As the settling
season progresses, the range of acceptable groups continues to narrow by
shifting further toward small groups
(Figure 1c).
Effect of size at settlement on group size choice
Fitness from settling into a given group size on a given day is greater for
large than for small individuals (compare bars between Figure
2a,
b). Fitness for continuing to
search on a given day is also greater for larger than small individuals
(compare lines between Figure
2a,
b), such that qualitative
patterns in ranges of acceptable group sizes are similar for large and small
individuals (Figure 2).
However, later in the settling season, the range of acceptable group sizes
narrowed earlier for small than for large individuals (Figure
2c,
d).
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Effect of reef residence time on settlement choice
The longer an individual has been searching on the reef, the less value
there is in continuing searching. Because there is a 6-day energetic limit to
the number of days a fish can search, by the fifth day on the reef, searching
becomes a risky choice. The number of days an individual has been searching on
the reef does not affect the expected fitness from settling into a group of a
particular size, so more group sizes may become acceptable as the larva spends
more days searching.
Parameter sensitivity
Overall, qualitative patterns in time dependence and body size dependence
of ranges of acceptable group sizes were robust to changes in parameter values
and functions. Output from runs of the model using modified parameter values
for group-specific growth rate, group-specific probability of survival, and
fecundity showed the same qualitative patterns of group size choice by D.
albisella as runs with the original parameters. An increase in growth
rate extended the number of days in the settling season during which a fish
could settle and still reach maturity by terminal time in the model
(Table 2). Decreasing the
growth rate reduced the number of days a fish could settle and still reach
maturity. Changes in growth rate and fecundity had effects on the range of
acceptable group sizes for day 75 of the settling season. A decrease in growth
rate, as well as switching the length-fecundity relationship from an
exponential to a linear relationship, had the effect of narrowing the range of
acceptable group sizes by making the larger groups unacceptable
(Table 2). An increase in
growth rate had no effect on the range of group sizes. Changes in the
probability of survival had no effect in either of the target responses.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Organisms that live in groups often face a trade-off between the associated costs and benefits (Booth, 1995
; Pulliam and Caraco,
1984). Living in a large group can increase survival through
better predator detection (Caraco,
1979; Lima, 1995;
Rasa, 1989) or a dilution
effect (Dehn, 1990;
Foster and Treherne, 1981;
Roberts, 1995;
Wrona and Dixon, 1991).
However, it can also lower feeding rate due to competition for food between
members of the group (Clifton,
1990; Coates,
1980; Forrester,
1991; Jones,
1987a; Rasa,
1997). Although we could calculate an "optimal group
size" from measured relationships between growth and survival (e.g.,
Booth, 1995), we chose to model
optimal "choice" instead. All individuals modeled used the same
optimal choice rules, yet faced differences in individual experience (i.e.,
differences between individuals in the actual group size encountered on a
given day). Model results indicate how variation in the optimal group size
chosen could arise on a single day from otherwise identical individuals and
thus how initial variation in group sizes can persist. Our model of group size choice for Dascyllus albisella showed that early in the settling season, settling into large groups provided a higher fitness than settling into small groups. Despite slow growth in the large groups, the high probability of survival coupled with adequate time to grow before their first spawning season made large groups more valuable. Early in the season, a group size of 10 individuals is the optimal group size (e.g., Figure 1a). However, early in the settlement season, when faced with the decision of continuing searching or settling into the most recently encountered group, settling is always a better choice (regardless of the group size and its availability) than the survival uncertainties and growth delay from searching for another group. As the settling season progresses, the value of large groups declines because fast growth becomes more important than an increase in survival probability. So later in the season, when faced with a decision of settling into a large group or continuing searching, individuals should continue searching to find smaller groups (Figure 1b, c).
The results discussed above are in part driven by specific quantitative assumptions in the model, and thus it is important to discuss their qualitative generality in the face of changed assumptions. If a fish cannot grow to maturity in a particular group by the middle of the next spawning season, we set its fitness to zero in that group. In reality, a fish will have future spawning times available, so that fitness should not actually drop to zero. But expected fitness would indeed be greatly reduced, and the emphasis late in the season should still be on maximizing growth. Thus, our qualitative results of bias toward small groups late in the season should still stand under relaxation of the "zero fitness" assumption.
Although our results are specific to Dascyllus albisella because
we used parameter values derived from studies of this species, general
patterns in results will be similar for the many coral reef fish species
having planktonic larvae that settle into groups as juveniles on the reef. The
most useful generality from this study, however, is in the approach of
modeling the decision rather than the outcome. Previous investigators have
demonstrated this, primarily in investigations into why "stable"
group sizes tend to be larger than "optimal" group sizes
(Clark and Mangel, 1984;
Giraldeau, 1988;
Sibly, 1983). In those
studies, it was noted that group size is determined by decisions of
individuals external to the group. As long as joining the group is better than
being solitary, the group will increase, regardless of the optimal group size.
Whereas those studies were focused on explaining how differences between a
single stable and an optimal group size could arise, ours was focused on
explaining how multiple acceptable group sizes rather than a single optimal
size could arise, though both problems emphasized similar mechanisms. In our
study, we were interested specifically in how individual experience (e.g.,
random encounters with groups) and conditions (e.g., body size and time of
arrival to reef) produce variability in group sizes chosen, again, through
decisions made by individuals not yet in a group.
Summary
In this study, we developed a dynamic programming model of an optimal
decision process of group size choice that considers the trade-off between
growth and survival as a function of group size in the coral reef fish
Dascyllus albisella. Despite only moderate confidence in quantitative
estimates of some parameters, the qualitative patterns that arose from the
model proved to be robust in a sensitivity analysis of the model. According to
our model, when fish make optimal decisions, the result is a variety of
possible outcomes that depend on the specific choices encountered and
temporally changing ecological pressures; even when all fish behave optimally,
we should see variance in group sizes chosen.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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We thank D. J. Booth for his comments and suggestions regarding our interpretation of his data during the development of our model. Suggestions from R. Ydenberg and two anonymous reviewers greatly improved the manuscript. This study represents a portion of the dissertation work of F.A.M. The work was funded in part by a Graduate Teaching Associateship from The Ohio State University to F.A.M., a National Science Foundation grant (DEB 9410327) to E.A.M., and a National Oceanic and Atmospheric Administration grant (NA86RG0053) to E.A.M.
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