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Importance of spatial population characteristics on the fertilization rates of sea urchins.

Introduction

Fertilization success of sea urchins and other free-spawning benthic invertebrates has been the focus of many empirical and theoretical studies over the last 25 years. In particular, a seminal paper by Denny and Shibata (1989) provided a mathematical framework for predicting fertilization rates under field conditions by combining the steady-state solution of the turbulent advection-diffusion equations (Csanady, 1973) with the fertilization kinetics model developed by Vogel et al. (1982). That model, and variations of it, enabled researchers to examine various hypotheses about determinants of fertilization success, including the size and reproductive output of spawning individuals (Babcock et al., 1994; Levitan and Young, 1995; Claereboudt, 1999), the distances between them (Denny and Shibata, 1989; Levitan and Young, 1995; Claereboudt, 1999), and the hydrodynamic environment during spawning (Denny and Shibata, 1989; Young et al., 1992; Levitan and Young, 1995). The original formulation of the model has been progressively modified to approach field conditions, most commonly by incorporating terms for reflection of gametes at the seabed (Claereboudt, 1999) and water surface (Babcock et al., 1994), and the blocking effect of polyspermy (Styan, 1998; Millar and Anderson, 2003). Attempts at validating the Denny and Shibata model (and its derivatives) by means of small-scale manipulative field experiments with sea urchins and sea stars have generally met with limited success (Denny and Shibata, 1989; Levitan and Young, 1995; Metaxas et al., 2002).

Early applications of that model indicated that sperm concentration is likely a major limiting factor, resulting in low fertilization rates except at very high population density. Experimental work with sea urchins subsequently confirmed that even at densities of 144 urchins per square meter, we could expect fertilization rates below 70% (Wahle and Peckham, 1999). Up to this point, most theoretical (Denny, 1988; Denny and Shibata, 1989; Young et al., 1992; Metaxas et al., 2002; but see Levitan and Young, 1995) and empirical (Pennington, 1985; Levitan, 1991; Wahle and Peckham, 1999; but see Levitan et al., 1992) studies had been based on a single or a very few spawning males that contributed to sperm plumes likely to fertilize eggs. This reinforced the notion of sperm limitation in natural populations without necessarily being representative of such situations. Ignoring population size in experimental studies also has been somewhat misleading. For example, Wahle and Peckham (1999) did not vary the number of males in their various "density" trials, but changed the distance between the males and egg batches, falsely assuming that only the closest male could contribute to fertilization. Further, because eggs were enclosed in baskets, fertilization was limited to males spawning upstream from the basket, as these eggs could not encounter downstream males. The results of that study were nonetheless assumed to be representative and have been used to predict fertilization success in wild populations (Meidel and Scheibling, 2001).

Modeling work also led to another assumption: that most fertilization occurs very rapidly once eggs have been released (Denny and Shibata, 1989; Levitan and Young, 1995). Again, early modeling work focused on the fertilization of eggs emanating from a female directly downstream of a single spawning male (Denny and Shibata, 1989; Meidel, 1999). Given that the advection-diffusion model predicts a rapid decrease in sperm concentration as sperm are transported downstream, eggs that are not fertilized rapidly are soon within much diluted sperm and the likelihood of fertilization drops. Therefore, because the model assumes that sperm come from upstream males, fertilization rates are obviously greater in the immediate vicinity of the female (as close to the spawning male as possible), before eggs are advected from the source. Given the longevity of eggs (Meidel and Yund, 2001), however, they could eventually encounter a spawning male as they are transported downstream, where high fertilization rates also could occur. Although informative, single-male models do not provide much insight into fertilization as it would proceed in nature. This weakness highlights the importance of including parameters that are relevant to the population as a whole rather than to the individuals.

Population parameters were included in a model by Levitan and Young (1995) to predict fertilization rate in the echinoid Clypeaster rosaceus. They simulated sperm clouds resulting from multiple males randomly distributed on two-dimensional grids of varying size and examined the fertilization rates along the downstream axis from a spawning female. They demonstrated that population density is important, but that a large population size could compensate for a low density. Claereboudt (1999) used a similar model, but in three dimensions, to examine the effect of density and aggregation on fertilization rate of sea urchins. His model accounts for diffusion of eggs, but not for the fact that a certain proportion of these eggs will already have been fertilized before they reach a given point downstream of the female. Instead, he assumed that the proportion of eggs remaining unfertilized in any cell of his array was related only to fertilization in that particular cell. He obtained results similar to those of Levitan and Young (1995): density is important--but for a given population size, fertilization rate will increase with the degree of aggregation.

Early field experiments did not yield high fertilization rates, which reinforced the conclusion of low fertilization success indicated by most models. Meidel and Yund (2001) criticized the short duration of field experiments, indicating that fertilization in the laboratory can occur for several hours after spawning, whereas most field studies are on the order of minutes. Recent studies have shown that time-integrated (2-4 days) fertilization rates in the field can reach as high as 100% (Wahle and Gilbert, 2002; Gaudette, 2004). These findings cast doubt on the reliability of short-term experiments as indicators of fertilization success in natural populations.

Our study attempts to reconcile empirical evidence for high fertilization rates observed in the field with the traditional Denny and Shibata (1989) model of advection-diffusion and fertilization kinetics. To do so, we incorporate parameters related to individuals as well as to the population as a whole by running simulations over a large spatial scale, and by incorporating sperm contribution from multiple males. We compare predictions of our model for three subpopulations of the green sea urchin Strongylocentrotus droebachiensis Muller 1776--namely those occurring in (1) kelp beds (low urchin density and high biomass of erect macroalgae), (2) barrens (intermediate urchin density and absence of erect macroalgae), and (3) grazing fronts (high-density aggregations of urchins at the kelp bed-barrens interface)--to previous observations in small-scale experiments. We then estimate how fertilization rates are affected by population size and current velocity in each subpopulation, and evaluate parameters most likely to influence the fertilization success.

Materials and Methods

Fertilization model

To predict the concentration of sperm from a single male in a three-dimensional simulation array, we start with the steady-state solution to the turbulent diffusion-advection equation (Csanady, 1973; Denny, 1988):

S(x,y,z) = [[Q.sub.s][bar.u]]/[2[pi][[alpha].sub.y][[alpha].sub.z][u.sub.*.sup.2][x.sup.2]] {exp - ([[[y.sup.2][bar.u.sup.2]]/[2[[alpha].sub.y.sup.2][u.sub.*.sup.2][x.sup.2]]] + [[(z - s)[.sup.2][bar.u.sup.2]]/[2[[alpha].sub.z.sup.2][u.sub.*.sup.2][x.sup.2]]]) + exp - ([[[y.sup.2][bar.u.sup.2]]/[2[[alpha].sub.y.sup.2][u.sub.*.sup.2][x.sup.2]]] + [[(z + s)[.sup.2][bar.u.sup.2]]/[2[[alpha].sub.z.sup.2][u.sub.*.sup.2][x.sup.2]]])} (1)

where S is the concentration of sperm ([m.sup.-3]) from one male at distance x, y, and z from the male, [Q.sub.s] is the sperm release rate per urchin ([s.sup.-1]), [bar.u] is the mean current velocity (m x [s.sup.-1]), [u.sub.*] is the frictional velocity (m x [s.sup.-1]), s is the height at which sperm is released (m), and [[alpha].sub.y] and [[alpha].sub.z] are respectively the horizontal and the vertical coefficients relating friction velocity to directional diffusivity. This equation assumes that gametes are reflected at the bottom (z = 0) and lost above some value of z. We expanded this equation to incorporate a reflective boundary at a height D (D = 12 m, the average water depth at our field sites) by adding a mirror source at the air-water boundary to prevent loss of eggs and sperm at this interface (Babcock et al., 1994). To simulate a large population size in the cross-current direction, we incorporated reflective boundaries in the y-dimension by adding mirror sources on both sides of our simulation grid:

S(x,y,z) = [[Q.sub.s][bar.u]]/[2[pi][[alpha].sub.y][[alpha].sub.z][u.sub.*.sup.2][x.sup.2]] {exp - ([[y.sup.2][bar.u.sup.2]]/[2[[alpha].sub.y.sup.2][u.sub.*.sup.2][x.sup.2]]) + exp - ([(2[Y.sub.1] - y)[.sup.2][bar.u.sup.2]]/[2[[alpha].sub.y.sup.2][u.sub.*.sup.2][x.sup.2]]) + exp - ([(2[Y.sub.2] - y)[.sup.2][bar.u.sup.2]]/[2[[alpha].sub.y.sup.2][u.sub.*.sup.2][x.sup.2]])} {exp - ([(z - s)[.sup.2][bar.u.sup.2]]/[2[[alpha].sub.z.sup.2][u.sub.*.sup.2][x.sup.2]]) + exp - ([(z + s)[.sup.2][bar.u.sup.2]]/[2[[alpha].sub.z.sup.2][u.sub.*.sup.2][x.sup.2]]) + exp - ([(2D - z - s)[.sup.2][bar.u.sup.2]]/[2[[alpha].sub.z.sup.2][u.sub.*.sup.2][x.sup.2]])} (2)

where D is the depth of the water column (m), and [Y.sub.1] and [Y.sub.2] represent the distances (m) between an urchin and the edges (in the y-dimension) of our simulation array (see below). Estimates of parameters relating to turbulent diffusion are difficult to evaluate. For simplicity we assumed that [[alpha].sub.y] = 2.25 and [[alpha].sub.z] = 1.25 (Denny and Shibata, 1989), while [u.sub.*] was set at 10% of the mean current velocity ([bar.u]) as is generally assumed (Levitan and Young, 1995). Diffusion in the x-dimension is negligible compared to advection, and therefore excluded from the model.

To account for multiple males spawning synchronously, we randomly assigned x- and y-coordinates to a number of male urchins (n) in our simulation array (the z-coordinate of each male was set to 0 since adult urchins are on the sea bed). Density of males was chosen to be representative of each of our three subpopulations (Table 1). Our simulation array measured 200 m by 20 m by 12 m (x: parallel to flow, y: horizontal and perpendicular to flow, z: vertical) divided into 1,920,000 cells measuring 0.1 m by 0.5 m by 0.5 m (2000 by 40 by 24 cells). These dimensions for the cells of the arrays provided a good compromise between computation time and loss of accuracy; any further decrease in the cell size resulted in less than 1% change in the model output. We can sum the concentration of sperm obtained from Equation 2 over the total number of males (n) to obtain the total sperm concentration ([S.sub.T]):

[S.sub.T](x,y,z) = [n.summation over (i=1)] [S.sub.i](x - [x.sub.i],y - [y.sub.i],z) (3)

where [x.sub.i] and [y.sub.i] are the x- and y-coordinates of male i in the array.

Fertilization kinetics has traditionally been described by the model developed by Vogel et al. (1982):

F(x,y,z) = 1 - exp(- [phi][tau][u.sub.*][S.sub.T](x,y,z)) (4)

where F is the percentage of eggs fertilized (%), [phi] is the fertilizable surface area of an egg ([m.sup.2]), and [tau] is the sperm-egg contact time (s) (i.e., the amount of time spent in a specific cell). The fertilizable area of an egg is 1% of the egg cross-sectional area (Vogel et al., 1982) calculated from the egg diameter (1.45 X [10.sup.-4] m for Strongylocentrotus droebachiensis; Levitan, 1993). The egg-sperm contact time ([tau]) is equal to the time spent by sperm and eggs in each cell, which is a function of the current velocity (m x [s.sup.-1]) and the length (m) of the cell in the flow direction (c):

[tau] = c/[bar.u] (5)

Because the fertilization rate does not depend on egg concentration, we can calculate the potential fertilization rate for each cell of our simulation area. First, we calculated the proportion of eggs that could be fertilized (P) in any vertical plane j along x, if all eggs emanating from a female reached a plane j in an unfertilized state:

[P.sub.j] = [[40.summation over (y=0)] [24.summation over (z=0)] F(j,y,z) E(j,y,z)]/[[40.summation over (y=0)][24.summation over (z=0)] E(j,y,z)] (6)

where E(j,y,z) is the number of eggs (fertilized and virgin) in each cell of the plane j emanating from a single female located at a given position on the x-axis. The number of eggs from a female in each cell is obtained using Equation 2, and by replacing S by E and [Q.sub.s] by 1. From Equation 4, the fertilization success is independent from the concentration of eggs, and therefore we scaled the egg release rate to 1. From Equation 6, we can calculate the proportion of unfertilized eggs (1 - [P.sub.j]) in a plane that can be transported to the next plane (j + 1). By assuming that mixing occurs between each time step, we can account for a reduced number of unfertilized eggs reaching planes farther away from the female, because of fertilization, by calculating the proportion of eggs not already fertilized in a previous plane that reach plane j, and are fertilized there, as:

[Z.sub.j] = [j.[product].k=1] (1 - [P.sub.k]) (7)

Where [PI] is the arithmetic product from k equals 1 to j. To obtain the total proportion of eggs fertilized from release to any distance d from the female, we can sum [Z.sub.j] from 0 to d.

Fertilization success in different subpopulations

We compared fertilization rates of eggs spawned by a single female (located at x = 0, y = 10, and z = 0 in our array) among three subpopulations of urchins (kelp bed, barrens, and grazing front) at low current velocity (0.05 m x [s.sup.-1]). Urchin density is generally the greatest in grazing fronts and lowest in kelp beds, while the barrens have intermediate densities (Meidel and Scheibling 2001). Urchins at a grazing front also are generally larger than those in the barrens or the kelp beds. We used results from a literature survey of urchins in these subpopulations (Meidel and Scheibling, 2001) to parameterize each simulation (Table 1). The only variable not available from this study is the sperm release rate per urchin in each subpopulation. For that variable we used values from Meidel (1999), who calculated sperm release rate ([Q.sub.s]) as:

[Q.sub.s] = G/[epsilon][delta] (8)

where G is the dry weight of sperm release (g), [delta] is the spawning time (s) and [epsilon] is the dry weight per sperm (3.66 X [10.sup.-12] for S. droebachiensis; Thompson, 1979). Mass of sperm released per urchin and sperm release time were calculated from results of field and laboratory experiments (Meidel, 1999). Sperm release rate per urchin is higher in grazing fronts than in barrens or kelp beds (Table 1).

Effect of environmental factors on fertilization success

We used parameters from the three subpopulations (Table 1) to evaluate the effect of current velocity (0.05, 0.10, and 0.20 m x [s.sup.-1]) on the fertilization rate of eggs spawned by a single female located at x = 0 and y = 10 (on the upstream boundary and 10 m from each along-stream boundary). Although we recognize that kelp beds affect current velocity and modify flow characteristics (Wahle and Peckham, 1999), we do not incorporate a decrease in current velocity in the kelp bed. This decrease would change above the frond canopy and Equation 2 would not be valid. Although our model limits a direct comparison of fertilization rate within and outside of kelps beds under a given ambient flow regime, it does provide estimates of fertilization success for a kelp bed subpopulation over the same range of flow rates as that applied to the other subpopulations. For the sake of simplicity, we assume that current is unidirectional (flowing from x = 0 to x = 200 in our array) and horizontally uniform in our simulation for all subpopulations.

We also evaluated the effect of the position of the female within the population (x = 0, 25, and 50 m from the upstream boundary) on the fertilization success in each subpopulation, using a current velocity of 0.05 m x [s.sup.-1]. Clear boundaries in natural populations are not to be expected (except for a sharp discontinuity at the kelp bed-barren interface), but this allows us to estimate the effect of population size on fertilization rates. A female located at x = 0 and y = 10 would have no male upstream and eggs could be fertilized only by downstream males, whereas eggs released at x = 25 and 50 m could be fertilized by males spawning both upstream and downstream from the point of egg release.

Elasticity analysis

Elasticity analysis is a type of sensitivity analysis that uses small proportional perturbations to measure the response of the model to changes in parameter values (Caswell, 2001). The elasticity ([theta]) of the model prediction to small (10%) increases in parameter (p) values was calculated as:

[[theta].sub.p] = 100 [[[X.sub.p] - [X.sub.0]]/[x.sub.0]] (9)

where [X.sub.0] is the model output given the original value of parameter p, while [X.sub.p] is the model output given modified parameter p (Barbeau and Caswell, 1999). We used distance to reach 90% fertilization (the highest rate reached in all subpopulations after 200 m) as the model output for this analysis. Sperm release rate ([Q.sub.s]), current velocity ([bar.u]), frictional velocity ([u.sub.*]), the coefficients [[alpha].sub.y] and [[alpha].sub.z], the fertilizable area of an egg ([phi]), and the number of spawning males (n) were parameters that were used to examine the model's elasticity.

Results

Fertilization success in different subpopulations

Our model predicts that fertilization rates differ markedly among subpopulations of sea urchins characterized by differences in density, individual size, and reproductive output (Fig. 1). The kelp bed subpopulation had by far the lowest fertilization rate. It took more than 200 m (4000 s at 0.05 m [dot] [s.sup.-1]) for 90% of eggs to be fertilized in kelp beds, compared to 20.5 m (410 s) in barrens and 5.8 m (116 s) in a grazing front. After 1 m of downstream transport (20 s), only 0.2% to 6.1% of eggs were fertilized. After 10 m (200 s), 64% and 99% of the fertilization had occurred in barrens and the grazing front, respectively, while only 12% had occurred in the kelp bed. After 50 m (1000 s), once virtually all eggs were fertilized in both the barrens and grazing front, fertilization rate was about 57% in kelp beds.

Effect of environmental factors on fertilization success

For all subpopulations, our model predicts that urchins located on the upstream boundary of the population (x = 0) had lower initial fertilization rates than urchins located within (x = 25 or 50) the population (Fig. 2). In all sub-populations, most fertilization occurred within 30 m downstream of the spawning female (600 s) when females were located at 25 or 50 m from the upstream boundary of the array. In the grazing front, virtually all eggs were fertilized within 1 m when released at 25 m, while only 6% of the eggs released at 0 m were fertilized after traveling 1 m (Fig. 2). Eggs released at the boundary had a time lag before fertilization could occur (the exponential section of the curves) because there were no upstream males and eggs had to be advected downstream to encounter sperm. The initial effect of distance from edge was not as important in the kelp bed, where after 1 m, 0.5% and 6% of eggs were fertilized when released at 0 and 50 m, respectively.

[FIGURE 1 OMITTED]

Our model predicts that for a finite population size, increasing current velocity decreased total fertilization success (Fig. 3). At low current velocity (0.05 m x [s.sup.-1]), eggs remained over spawning males for longer, resulting in high fertilization rates. At high velocity (0.20 m x [s.sup.-1]), eggs were rapidly advected beyond the array, resulting in low overall egg-sperm contact time and consequently low fertilization rate. The effect of current velocity was particularly important for urchins in the kelp bed subpopulation, where fertilization rates within 200 m decreased from about 90 to about 40% between 0.05 and 0.20 m x [s.sup.-1]. In the grazing front, where fertilization rates were much higher, varying current velocity had no effect within 200 m because all eggs were fertilized. The distance within which 90% of eggs spawned in a grazing front were fertilized increased from 5.8 to 15.9 m between 0.05 and 0.20 m x [s.sup.-1].

For all subpopulations, current velocity had little effect on fertilization rate when considered in relation to the elapsed time since egg release (Fig. 4). The low initial fertilization rate observed at low velocity was due in part to the small distance traveled by these eggs per unit time, which limits encounter rate with spawning males compared to the high-velocity situation.

Elasticity analysis

The distance needed to reach 90% fertilization (the model output we used) was most sensitive to changes in parameter values in the kelp beds, intermediate in the barrens, and fairly robust in the grazing front (Fig. 5). Increasing the density of spawning males, the sperm release rate, and the fertilizable area of an egg by 10% all decreased the distance (and thus increase fertilization rate) by 1.8% to 13.5%, whereas increasing hydrodynamic parameters that affect gamete dispersal ([bar.u], [u.sub.*], [[alpha].sub.y] and [[alpha].sub.z]) increased the distance (i.e., decrease fertilization rate) by as much as 16.2%. The model output was most sensitive in all subpopulations to an increase in current velocity, which increased the distance needed to reach 90% fertilization success by 10.7% to 16.2%.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Discussion

Fertilization success in different subpopulations

Our model predicts higher fertilization rates than those reported by Meidel and Scheibling (2001) using the same population parameters. On the basis of results of a field experiment by Wahle and Peckham (1999), Meidel and Scheibling (2001) predicted fertilization rates of 14.6%, 43.3%, and 62.0% in kelp bed, barrens, and grazing front subpopulations, respectively. Our model predicts these values after eggs are advected for only 3.6, 6.7, and 11.5 m respectively. These distances are negligible given the spatial extent of populations of Strongylocentrotus droebachiensis, which can span hundreds of kilometers of coastline and extend hundreds of meters to kilometers offshore (e.g., Wharton and Mann; 1981, Moore and Miller, 1983; Brady and Scheibling, 2005). These results suggest that the Meidel and Scheibling model underestimates fertilization success by using Wahle and Peckham's (1999) data from small-scale field experiments. However, our model assumes that mixing occurs between each time step (equation 7), inflating the probability of an egg being fertilized. Nonetheless, given the rapid and high level of fertilization obtained in our simulations, this assumption would likely not affect the general behavior of our results.

Our model uses a specific spatial area occupied by a subpopulation (200 m by 20 m). The actual area occupied by each subpopulation in nature likely varies between sub-populations and would be difficult to estimate. The size of the grazing front (2 m wide; Lauzon-Guay and Scheibling, 2007) would always be smaller than our simulation array; nevertheless, because our model predicts greater than 90% fertilization success in the first few meters after spawning, the spatial extent of our array will have little effect on the model prediction for the grazing front. The spatial extent of a barren or kelp bed subpopulation will have a greater effect on the fertilization success of eggs spawned. In these cases, population size may be more important than population density or individual size in controlling fertilization success. Small, dense aggregations of urchins have exhibited lower fertilization success than larger aggregations of lower urchin density in field trials (Gaudette, 2004). The smaller the population, the faster eggs are transported away from a source of sperm and the less likely they are to be fertilized. This rapid decrease in fertilization rate as eggs are transported away from a source of sperm led Denny and Shibata (1989) to suggest that more than 90% of the fertilization will take place during the first 20 s after eggs are released. In comparison, our model suggests that it would take between 116 s (grazing front) and 4000 s (kelp bed) for 90% of the fertilization to occur. The cause of the difference between the two predictions is that their model allowed fertilization to occur only from sperm originating from a male located upstream from the female, but our model includes multiple males both upstream and downstream of the spawning female. Therefore, the general notion that fertilization occurs rapidly may not be warranted except in the artificial case of a single spawning male upstream of the female.

[FIGURE 5 OMITTED]

Position within the population

The position within the population also affects the rate at which eggs are fertilized: as a female is moved away from the edge, sperm from a greater number of males can fertilize her eggs as soon as they are released. This observation supports the notion that it is not only the closest upstream male that contributes to the fertilization, but that males located many meters upstream of the spawning female also contribute. For example, in field experiments with the sea star Oreaster reticulatus, Metaxas et al. (2002) reported 20% fertilization success as far as 32 m downstream from the closest spawning male. Including upstream males in our simulations does increase the initial fertilization, but the time needed to obtain 90% of the fertilization still remains far longer than the previously suggested 20 s. Even for females located 20 m inside the population, 90% of the fertilization occurred in the first 16 to 20 m (320 to 400 s), which is 16 to 20 times longer than that reported by Denny and Shibata (1989). This result highlights the importance of the time scale of fertilization assays (Meidel and Yund, 2001; Wahle and Gilbert, 2002; Gaudette, 2004). Also, because our understanding of factors affecting the survival of gametes in the wild is inadequate, it is difficult to accurately predict fertilization success. As the time needed for gametes to become fertilized increases, the more likely it is that gametes will be lost (e.g., though predation, other types of mortality, or passive transport), resulting in lower fertilization ratios than those predicted by our model.

Current velocity

Previous models (Denny and Shibata, 1989; Levitan and Young, 1995) predicted that current velocity has a negative effect on fertilization rate, especially when that rate is low for other reasons. Similarly, although he did not test this in his simulations, Claereboudt (1999) suggested that current velocity would have an effect opposite to that of sperm release rate in his model because sperm concentration is proportional to sperm release rate but inversely proportional to current velocity (Equation 1). However, according to our fertilization kinetic model (Equation 4), fertilization rate should be independent of current velocity for an infinitely large population. Any decrease in sperm concentration due to increased current velocity is compensated by a proportional increase in frictional velocity, such that the exponential term of the fertilization equation remains unchanged (i.e., [proportional] [bar.u.sup.-1], [u.sub.*] [proportional] [bar.u], and F [proportional] 1 - exp ([u.sub.*]S)). While the concentration of sperm decreases with increasing current velocity, the proportion of eggs fertilized for a constant sperm concentration also decreases. However, because egg-sperm contact time ([tau]), which is inversely proportional to the current velocity (Equation 4), decreases as fast as frictional velocity ([u.sub.*]) increases with current velocity, the fertilization rate in each cell will decrease with current velocity. This results in a lower fertilization rate after a given distance, because eggs in high current velocity will travel that distance more rapidly. However, if we look at the proportion of eggs fertilized as a function of time since they were spawned, we obtain very similar results across a range of current velocities (Fig. 4). This differs from the model of Levitan and Young (1995) because our fertilization rate is a function of frictional velocity whereas theirs is a function of sperm swimming speed. Sperm swimming speed (0.088 mm x [s.sup.-1]; Levitan 1993) is orders of magnitude lower than frictional velocity (0.1 [bar.u]), which is more generally used to calculate the probability of a sperm-egg encounter.

As previously mentioned, fertilization success is largely a consequence of whether or not eggs are carried away from the spawning population before being fertilized, which in turn is determined by the size of the population and the current velocity. Even when individuals are small and at low urchin density, and the current is strong, eggs will have a high probability of eventually being fertilized if the population is large. This conclusion is supported by recent evidence extending the period of egg viability in S. droebachiensis to days after spawning (Meidel and Yund, 2001), compared to previous estimates that ranged in hours (Epel et al., 1998; Wahle and Peckham, 1999). The possibility that eggs are initially retained on the aboral surface of the female urchin because of their high viscosity (Thomas, 1994: Yund and Meidel, 2003) also may increase fertilization success (Bishop, 1998) by subjecting eggs to high sperm concentrations for a longer period before they are advected from the population. Information on retention time of eggs under varying flow conditions, and the extent of sperm penetration of the egg matrix, is needed to evaluate how retention affects fertilization success, but it would most likely decrease the importance of population size.

We intentionally modeled gamete dispersion using a simplified version of the flow dynamics observed in nature. The change in flow characteristics within a kelp bed was not included in our model, although it likely has an important effect on fertilization success (Wahle and Peckham, 1999). A reduction in flow velocity below the canopy could result in fertilization occurring over a shorter distance; however, the time required for fertilization to occur may not be affected. We focused our model on the fertilization in unidirectional flow although many shallow populations in exposed areas are subject to oscillatory flow conditions. Such a flow regime would have dramatically different effects on the dispersion and mixing of gametes. Also, we ignored the effect of a boundary layer that could retain gametes near bottom and reduce their rate of advection, especially eggs that are negatively buoyant (Thomas, 1994). An increased concentration of gametes in the boundary layer could greatly increase fertilization rates. These are important aspects of flow that warrant further investigation.

Elasticity analysis

Changing the number of spawning males (n) or the sperm release rate ([Q.sub.s]) has the same effect on the total number of sperm present in the system (Equations 2 and 3). However, these factors will have different effects on the fertilization rates observed. Increasing the number of males increases the probability of having a male very close to the spawning female and increases the probability of having eggs fertilized rapidly (i.e., before eggs are transported to an area of lower sperm concentration). This effect, combined with the increased number of sperm present in the system, makes the model output slightly more sensitive to the number of males than to the sperm release rate. The number of males spawning can also be seen as spawning synchrony. Our results suggest that spawning synchrony would have a large effect on fertilization success, especially in barrens and kelp bed subpopulations. Mass spawning of S. droebachiensis has been observed during a multi-species mass spawning (C.P. Dumont, Universite Laval, pers. comm.). Like an increase in the number of spawning males, aggregative behavior associated with spawning (Young et al., 1992) could have a positive effect on fertilization success, except for urchins in a grazing front that are continuously at high density. The elasticity of the model output to changes in current velocity suggests that a low-density subpopulation (e.g., in a kelp bed) would be particularly influenced by strong current velocity. Population size would also be a critical factor. Eggs spawned by a female in a small population would be rapidly transported outside of the population, which would decrease their fertilization success. As discussed above, this effect would become less pronounced as the size of the population increased.

Our current understanding of the long-term behavior and survival of gametes renders any prediction of zygote production difficult and site-dependent. Even if we assume a high survival rate for gametes, uncertainty about population size within a given area compromises the accuracy of our prediction of fertilization success. However, given that the areal extent of urchin populations is typically on the order of 1000 m2, we could expect fertilization success on the order of 90% to be common for S. droebachiensis in the field. This success rate is much higher than that previously proposed for this species on the basis of theoretical approaches. This result emphasizes the need to incorporate ecologically relevant parameters other than the "single spawning male, upstream from the female" model often used in the past. It also suggests that more focus should be put on factors likely to control survival, development, and settlement of larvae, because these might be more important than fertilization success in controlling recruitment.

We have demonstrated that by including parameters at the population level, traditional models of advection-diffusion and fertilization kinetics can predict fertilization rates in accordance with those suggested by recent field observations. These findings underscore the importance of using the appropriate scale in modeling ecological phenomena in order to obtain relevant theoretical predictions.

Acknowledgments

We thank James Watmough and Myriam Barbeau for helpful discussions, and two anonymous reviewers for comments on a previous version of this manuscript. This research was funded by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) to RES. JSLG was supported by a Canadian Graduate Scholarship from NSERC, a Vaughan Graduate Fellowship, and a Magee Postgraduate Merit Award.

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JEAN-SEBASTIEN LAUZON-GUAY (1,*) AND ROBERT E. SCHEIBLING (2)

(1) Biology Department, University of New Brunswick, Bag Service 45111, Fredericton, New Brunswick E3B 6E1, Canada; and (2) Department of Biology, Dalhousie University, Halifax, Nova Scotia B3H 4J1, Canada

Received 27 October 2006; accepted 13 February 2007.

* To whom correspondence should be addressed, at Department of Biology, Dalhousie University, Halifax, Nova Scotia B3H 4J1, Canada. E-mail: js.lauzon@unb.ca
Table 1 Description and values of variables used for our three
subpopulations (KB: Kelp bed, PB: Permanent barrens, GF: Grazing front)
of urchins (Strongylocentrotus droebachiensis)

Variable name           Symbol           Unit             All

Horizontal diffusion    [[alpha].sub.y]  --               2.25
  coefficient (1)
Vertical diffusion      [[alpha].sub.z]  --               1.25
  coefficient (1)
Sperm release rate (2)  [Q.sub.s]        [s.sup.-1]       --
Current velocity        [bar.u]          m x [s.sup.-1]   0.05, 0.10,
                                                            or 0.20
Frictional              [u.sub.*]        m x [s.sup.-1]   0.1 [bar.u]
  velocity (1)
Population density (4)  N                [m.sup.-2]       --
Density of spawning     --               [m.sup.-2]       --
  male (4)
Number of spawning      n                4000 [m.sup.-2]  --
  males in array
Height of gamete        s                m                0.05
  release
Fertilizable surface    [phi]            [m.sup.2]        1.65 x
  area of an egg (3)                                        [10.sup.-10]
Depth of water column   D                m                12
Length of cell in flow  c                m                0.5
  direction
Egg concentration       E                [m.sup.-3]       --
  emanating from a
  female
Sperm concentration     S                [m.sup.-3]       --
  emanating from one
  male
Total sperm             [S.sub.T]        [m.sup.-3]       --
  concentration from
  all males in array
Percentage of eggs      F                %                --
  fertilized
Percentage of eggs      [Z.sub.j]        %                --
  fertilized in plane
  j
Sperm-egg contact time  --               s                --

Variable name           Value KB      PB            GF

Horizontal diffusion    --            --            --
  coefficient (1)
Vertical diffusion      --            --            --
  coefficient (1)
Sperm release rate (2)  1.9 x         2.7 x         2.9 x
                          [10.sup.7]    [10.sup.7]    [10.sup.7]
Current velocity        --            --            --
Frictional              --            --            --
  velocity (1)
Population density (4)  14            71            136
Density of spawning     0.67          3.87          18.40
  male (4)
Number of spawning      2680          15480         73600
  males in array
Height of gamete        --            --            --
  release
Fertilizable surface    --            --            --
  area of an egg (3)
Depth of water column   --            --            --
Length of cell in flow  --            --            --
  direction
Egg concentration       --            --            --
  emanating from a
  female
Sperm concentration     --            --            --
  emanating from one
  male
Total sperm             --            --            --
  concentration from
  all males in array
Percentage of eggs      --            --            --
  fertilized
Percentage of eggs      --            --            --
  fertilized in plane
  j
Sperm-egg contact time  --            --            --

(1) Denny and Shibata, 1989; (2) Meidel, 1999; (3) Levitan, 1993;
(4) Meidel and Scheibling, 2001.
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Author:Lauzon-Guay, Jean-Sebastien; Scheibling, Robert E.
Publication:The Biological Bulletin
Geographic Code:1CANA
Date:Jun 1, 2007
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