A simulation study of abalone fertilization.

ABSTRACT A computer simulation was conducted to study the effects of densities, abundances, and aggregations of adult abalone populations on the fertilization success (amount of fertilization per spawning female). Impacts of densities were examined by having different numbers of adult abalone in spawning grounds of the same size. Impacts of abundances were studied by having different numbers of adult abalone in spawning grounds of different sizes with constant densities. Impacts of aggregations were investigated by having different spatial distributions for adult abalone with constant densities and sizes of spawning grounds. The fertilization success increased with the adult density, and the rate of increase in the fertilization success decreased with increasing densities, exhibiting a typical Allee effect. If the density remained the same for a single population, changes in the adult abundance had minimal impact on the fertilization success unless the abundance was really low. Adult aggregation would only considerably increase the fertilization success at relatively low adult densities. The fertilization success was greater for a large population than the combined fertilization success for two smaller and isolated populations with the summed abundance of the two small populations equivalent to the abundance of the large population.

KEY WORDS: fertilization, free-spawner, abalone, density, aggregation, simulation

INTRODUCTION

Abalone species (Haliotis spp.), like many marine sedentary and sessile invertebrates, use free-spawning (broadcast-spawning) as their fertilization strategy. Free-spawners usually have either limited or no mobility at the adult stage, and they release their gametes (sperm and eggs) into the water column where fertilization occurs through transportation of gametes by the water movement. The adopted strategy may suggest that these species were designed for fertilization to occur among individuals living at some distance apart without having to aggregate. However, laboratory experiments have demonstrated that successful fertilization requires high sperm concentrations (Levitan et al. 1991, Babcock & Keesing 1999). When sperm concentration falls below a threshold level, fertilization rate will be minimal (or close to zero). Studies on the fertilization ecology of marine free-spawners indicated that some population parameters, such as density, abundance, and degree of aggregations, are important factors for the fertilization success. These studies can generally be grouped into three categories: experimental research (Pennington 1985, Levitan et al. 1992), surveys of natural spawning events (Grant et al. 1998, Gaudette et al. 2006), and theoretical modeling (Levitan & Young 1995, Claereboudt 1999).

Most of the studies on the egg fertilization of marine invertebrate free-spawners were conducted by field experiments, in which only a small number of individuals were induced to spawn for fertilization assays during a short period. These experiments generally showed that fertilization rates dropped quickly when the distance between sperm and egg sources increased to only a few meters apart because of rapid sperm dilution, indicating the importance of aggregation in fertilization. However, these small and manipulative experiments may not adequately reflect gamete interactions during the natural spawning (Yund 2000). In these small and manipulative experiments, eggs were placed in fixed spots and there was usually only one sperm source. Effects of egg advection and supplies of sperm by many individual males were ignored.

Eggs of marine free-spawners often have a long viability. Eggs of abalone species are viable for a few hours, whereas eggs of the green sea urchin, Strongylocentrotus droebachiensis, were reported to be viable for at least 24 h (Pennington 1985) or even for 2-3 days (Meidel & Yund 2001). Eggs could be entrained in flow structures and be fertilized throughout their period of viability if sperm subsequently become available (Yund & Meidel 2003). When multiple sperm sources were used in the experiment, the fertilization rate was found to be higher (Levitan et al. 1992). Meidel and Yund (2001) also found that fertilization rates increased significantly between 15-min and 1-h samples when eggs were placed where multiple males naturally spawn. Supplies of sperm by multiple males appear to have an accumulative effect on fertilization. The study by Meidel and Yund (2001) actually used an interesting hybrid design between an experiment and a natural survey with fixed egg positions but natural sperm release.

Surveying natural spawning events provides a complementary study on fertilization ecology of marine free-spawners. These spawning surveys generally reported higher fertilization rates than field experiments, and likely result in more reliable estimates of fertilization rates, because the natural spawning strategies were automatically incorporated (Yund 2000). However, natural surveys are difficult because of limited knowledge of the timing and spatial extent of spawning (Gaudette et al. 2006). Impacts of different population densities, abundances, and degrees of aggregations on fertilization successes are difficult to assess by natural surveys.

A third alternative is to conduct theoretical modeling to simulate large spawning events. The modeling overcomes the logistical problems of conducting natural spawning surveys or large-scale experiments involving many spawning individuals in a natural setting, and can easily incorporate egg advection and multiple sperm supplies into the analyses. Mathematical model frameworks have been developed to predict gamete spatial distributions in a 3- dimension world (Denny 1988, Denny & Shibata 1989, Babcock et al. 1994), and to estimate fertilization rates at various sperm concentrations (Vogel et al. 1982). Few theoretical modeling studies on the fertilization of marine free-spawners have been reported in the literature. Levitan and Young (1995) conducted 2-dimension simulations on the sea biscuit, Clypeaster rosaceus, and Claereboudt (1999) conducted 3-dimension simulations on benthic flee-spawners in general.

In this paper, theoretical modeling fertilization was applied, as a case study, to an abalone population. Many abalone stocks have either collapsed or are at greatly reduced sizes (Campbell 2000, Karpov et al. 2000, Woodby et al. 2000). Shepherd and Partington (1995) reported that a South Australian abalone population may experience a depensatory (Allee) effect at low stock sizes, because recruitment may decrease to a greater degree than the amount of decrease in spawning stock biomass caused by reduced fertilization rates at low densities. Although reproductive failure is inevitable below certain low densities, the critical range of threshold densities remain to be resolved.

Abalone species, in general, share many reproductive features among themselves. They release, on average, from 2.5 x [10.sup.11] to 1.0 x [10.sup.12] sperm, and from 9.0 x [10.sup.5] to 5.9 x [10.sup.6] eggs (Clavier 1992, Babcock & Keesing 1999, Litaay & De Silva 2001, Grubert et al. 2005). The eggs were viable for at least 2 3 h (Clavier 1992, Encena et al. 1998, Babcock & Keesing 1999). Sperm viability is short, particularly at low concentrations in which sperm use up their available energy reserves and age more rapidly (Chia & Bickell 1983). Sperm were viable for about two hours (Encena et al. 1998, Babcock & Keesing 1999) at a concentration of 1.0 x [10.sup.5] per mL but started to show aging after 45 min at a lower concentration of 1.0 x [10.sup.3] per mL (Babcock & Keesing 1999). However, Baker and Tyler (2001) found that fertilization success decreased rapidly within 30 min of gamete release. Effective egg fertilization generally occurs with a sperm concentration between 1.0 x p[10.sup.3] and 1.0 x [10.sup.6] per mL, and fertilization rates increase with sperm concentration within this range (Kikuchi & Uki 1974, Leighton & Lewis 1982, Clavier 1992, Babcock & Keesing 1999). Little fertilization occurs when sperm concentration falls below this range. Fertilization rates also drop when sperm concentration is too high because of a function of polyspermy (Stephano 1992). The most comprehensive fertilization kinetic model for an abalone species was presented by Babcock and Keesing (1999). The estimates of the model parameters were used for the current simulation study.

The current simulation study was used to investigate the fertilization success, which was defined as the amount of fertilization per spawning female, at different densities, abundances, and degrees of aggregation of an abalone population with four major objectives. The first objective was to determine if there exist some critical densities below which the fertilization success would drop greatly because of the depensatory mechanism (Allee effect). The second objective was to examine the amount of increase in the fertilization success for aggregated populations compared with randomly distributed populations at various densities. The third objective was to examine how the fertilization success changed with the population abundance, whereas the density remained the same. The fourth objective was to determine which population structure was more beneficial for the fertilization success; a large population or two smaller separate populations with the summed abundance equal to the abundance of the large population. Understanding of the impacts of the studied population parameters on the fertilization success is important for effective management of the resources.

MATERIALS AND METHODS

This study was designed to calculate fertilization success by computer simulation of abalone spawning events at various densities and abundances, and two levels of aggregations. Adult abalone were distributed in the simulated spawning ground either with an intermediate degree of aggregation or a random fashion. Abalone spawned synchronously. Gamete diffusion followed the turbulent diffusion model and the amount of fertilization was calculated using the fertilization kinetic model. Density, abundance, and aggregation were all referred to the spawning adult population. Density was expressed in number of adult abalone per [m.sup.2], abundance indicated the number of adult abalone in the simulated spawning ground, and the degree of aggregation was measured by the method of nearest neighbor distance (Clark & Evans 1954).

Spatial Distribution of Simulated A balone

The dimensions of the simulated spawning ground were set to 1,000 m long, 5 m wide, and 5 m deep for the fertilization study at different densities and aggregations. The length of the spawning ground was varied between 25 m and 1,000 m for the fertilization study at different population abundances with the same densities. Abalone adult densities were varied from 0.02-1.6 per [m.sup.2]. The computer software for individual-based modeling, Netlogo (Wilensky 1999), was used to generate abalone distributions on the spawning ground. All abalone were assumed to spawn at the same height of 0.1 m (1 dm) above the substrate. Therefore, abalone were distributed in a 2-dimensional area (1,000 x 5 [m.sup.2]), known as the spawning floor. To generate random distributions, each abalone was simply placed on the spawning floor in a random manner. To generate aggregated distributions, abalone were first randomly placed in the spawning floor, which was partitioned into patches of 1 x 1 [dm.sup.2] each. Spatial aggregations resulted from movement of abalone in searching for mates. Each abalone was made to release some amount of male or female chemical "cues" on the patches within a surrounding circle to attract the opposite sex. The mean radius of the circle was set between 5 and 10 dm for the simulation program, and the radius (r) for each abalone was randomly generated from a normal distribution with a coefficient of variation of 20%. Within the circle, the scent-strength decreased with the distance between the center of a patch and the individual abalone. The scent-strength was set to be 1 - (d/r) for patches with a distance between d and d +1 dm away from the abalone, where d is a positive integer smaller than r. The overall male and female scent-strength for each patch is the sum of scent-strengths from individual males and females.

The abalone were made to search for mates before spawning. Individuals located on patches containing the opposite sex cue were considered to have found mates. The other individuals were considered to be searching for mates. These two kinds of individuals moved in two different ways. The former set their headings randomly towards one of the eight neighboring patches with equal or stronger scent-strength of opposite sex cue than the patches they occupied, and then moved 1 dm in the heading direction. The latter randomly set their new heading between -60[degrees] and 60[degrees] relative to their current headings, and then either moved by 1 dm at population densities [greater than or equal to] 0.4 per [m.sup.2] or jumped by 5-20 dm at population densities <0.4 per [m.sup.2] in the new heading directions. Jumping at low densities greatly increased the chance to find mates, so that the spatial distributions would become as patchy as desired more easily. The jumping distance was set between 5 and 20 dm for the simulation program. If an individual had to go beyond the edge of the spawning floor by the amount of move or jump, it simply turned 180[degrees] and then moved or jumped in this opposite direction. The number of moves or jumps was set to be between 30 and 50 for the simulation program.

The degree of aggregation was measured by calculating [rho] = [bar.m] x 2 x [square root of d], where [bar.m] is the mean of distances of all individuals to their nearest neighbors, and d is the population density (Clark & Evans 1954). When the population is randomly distributed, P would be around 1. If all animals are located in the same spot, [rho] would be 0. The patchier the spatial distribution, the closer [rho] to 0. As the spatial distribution becomes more uniform, the value of P approaches 2.1491. All aggregated distributions were generated to have [rho] around 0.5, and the random distributions had P around 1 for densities [greater than or equal to] 0.1 per [m.sup.2] and 1.2 -1.5 for densities <0.1 per [m.sup.2]. Aggregated distributions were set at this intermediate level, as Claereboudt (1999) showed that high levels of aggregation had little additional contribution towards fertilization success compared with intermediate levels. The generated aggregated distribution had a larger number of occurrences of either no abalone or multiple abalone per [m.sup.2] than the random distribution, whereas the generated random distribution had a larger number of occurrences of one or two abalone per [m.sup.2] than the aggregated distribution (Table 1).

Turbulent Diffusion Model and Fertilization Kinetic Model

Abalone were assumed to spawn synchronously at 0.1 m above the substrate, and gametes were diffused according to the turbulent diffusion model presented by Denny and Shibata (1989) and modified by Babcock et al. (1994) and Babcock and Keesing (1999). The model allows predictions of gamete concentrations at any spots downstream from the point of release, and the modified model allows gametes to diffuse back to the water column when reaching the water surface. To facilitate calculations, a point source shedding gametes was located 1 dm above the origin of x and y axes with x-axis parallel to the flow of water current, which was set to be parallel to the longitudinal axis of the spawning ground. A third axis, z-axis, was set perpendicular to both x-axis and y-axis. The gamete density (number/[m.sup.3]) at a downstream location from the point of release was estimated by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where Q is the gamete release rate (number/s), U is the current velocity (m/s), h is the height of gamete release (0.1 m), [sigma] and [[sigma].sub.z] are the standard dispersion coefficients, and d is the water depth in m at which gamete would diffuse back. For the calculation of sperm concentration, d was set to be 5 m, the depth of water. For the calculation of egg concentration, d was set to be 1 m, because eggs are negatively buoyant and are restricted below 1 m of water depth at the substrate bottom (Claereboudt 1999). The standard dispersion coefficients were calculated as [[sigma].sub.y] = [a.sub.y]([U.sup.*]/U)[x.sup.B] and [[sigma].sub.z] = [a.sub.z] ([U.sup.*]/U)[x.sup.B] where [a.sub.y] and [a.sub.z] are the dispersion coefficients describing the shape of the plume, [U.sup.*] is the friction velocity as a measure of the turbulence intensity (Levitan & Young 1995), B is the rate constant describing the rate of growth of the eddies acting to disperse the plume as it spreads out.

Fertilization rates (r) were calculated using the fertilization kinetic model proposed by Vogel et al. (1982):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [beta] and [[beta].sub.0] are two fertilization rate constants, [S.sub.0] and [E.sub.0] are sperm and egg concentrations (number/mL) respectively, and is the sperm half-life in seconds (s), namely the age when 50% of eggs are fertilized. When sperm-egg contact time is less than [tau], the former can be substituted for the latter (Vogel et al. 1982, Levitan et al. 1991).

Values for the parameters of the turbulent diffusion model and fertilization kinetic model were set according to the estimates by Babcock and Keesing (1999) (Table 2). In the current study, sperm viability was set at 0.25 h (15 min), and egg viability (longevity) varied from 0.25-2 h.

Gamete Distribution

The 3-dimensional spawning ground with axes x, y, and z was partitioned into small blocks of 1 x 1 x 1 [dm.sup.3] each. The gamete concentration for a block was represented by the calculated concentration at the central point of the block using Eq. (1). Preliminary tests showed that the concentration at the center could not accurately represent the concentration for an entire block when the distance between the center of the block and the point of gamete release was smaller than 2 m on the x-axis. In this case, the block was subdivided into 64 subblocks of 2.5 x 2.5 x 2.5 [cm.sup.3] each. Gamete concentrations were calculated for each subblock, and the concentration for the block was the mean of the calculated concentrations for the 64 subblocks. The overall sperm or egg concentration for a block was the sum of concentrations of sperm or eggs from all upstream males or females whose gametes would still be viable on reaching the block.

Fertilization Calculation

Fertilization rates were calculated in individual blocks as eggs traveled through each block. Sperm-egg contact time was equivalent to the time eggs would travel through a block. Eggs were assumed to travel at the same speed as the current velocity (0.05 m/s) following the x-axis, and the sperm-egg contact time was, therefore, 2 sec in each block. Eggs in each block were either drifted down from upstream blocks or newly-released from females. A proportion of the drifted eggs may already have been fertilized, and the newly-released ones were all unfertilized eggs. The proportion of fertilized eggs among the drifted eggs (p) in a block were assumed to be same as in the immediate upstream block. Thus, the numbers of fertilized and unfertilized eggs in this block were, respectively, [E.sub.x] = p x [D.sub.o] and [E.sup.0] = (1 - p) x [D.sub.o] + [D.sub.n], where [D.sub.o] and [D.sub.n] are the numbers of drifted and newly-released eggs respectively. The fertilization rate, r, was calculated using Eq. (2). The number of eggs which were fertilized within the block was N = r x [E.sub.0], and the new proportion of fertilized eggs was p = (r x [E.sub.0] + [E.sub.x]/([E.sub.0] + [E.sub.x]). The fertilization success was expressed as the sum of fertilized eggs over all blocks divided by the total number of the spawning females. The fertilization success is proportional to the percentage of fertilized eggs.

In this paper, the term, the amount of fertilization, was used to indicate the relative number of fertilized eggs. Calculation of absolute numbers relies on the time length of gamete release. Assuming there is no variation in this time length among individual males or females, the calculated relative amounts would be proportional to the absolute values. Therefore, the relative amounts are sufficient for comparing the fertilization successes among different population settings.

RESULTS

The amount of fertilization increased with the adult population density and egg viability for aggregated and random distributions (Fig. 1). Given an adult population density, the amount of fertilization was higher with longer egg viability. Given a fixed egg viability, the amount of fertilization increased with the density in a concave-up fashion, and the rate of the increase was positively correlated with the density, exhibiting a depensatory function.

[FIGURE 1 OMITTED]

The fertilization success, as measured by the amount of fertilization per spawning female, increased with the adult population density in a concave-down fashion, and the rate of increase in the fertilization success, therefore, declined with the density (Fig. 2). The fertilization success also increased with the egg viability. For instance, at an adult density of 0.2/[m.sup.2], the fertilization success was doubled when the egg viability increased from 0.25-2 h.

The fertilization success was, in general, higher for aggregated distributions than for random distributions, and the differences between the two spatial distributions decreased with increasing adult density (Fig. 3). The differences were relatively high (> 10%) at low densities ([less than or equal to] 0.2/[m.sup.2]), and minimum (<5 %) at densities of 0.4/[m.sup.2] or higher. At the low densities, the differences in the fertilization success between the aggregated and random distributions were, in general, higher with shorter egg viabilities. For instance, at a density of 0.02/[m.sup.2], the fertilization success for the aggregated distribution increased by 75% with an egg viability of 0.25 h, and by 50% with an egg viability of 2 h (Fig. 3). In short, the amount of increase in the fertilization success caused by aggregation was negatively correlated to the fertilization success at the random distribution. When the fertilization success was higher at the random distribution, aggregation would make a smaller amount of contribution towards the fertilization success.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

With the same adult population density, the fertilization success remained almost unchanged when the adult population abundance, or the length of the spawning ground, was relatively large (Fig. 4). The length of the spawning ground at which the fertilization success started to decline was positively correlated with the egg viability. At a density of 0.2/[m.sup.2], the decline started at a spawning ground length of 200 m with the egg viability of 1 h, and at the length of 100 m when the egg viability was 0.25 h.

More eggs were fertilized for a large population in one spawning ground, compared with the summed number of eggs fertilized for two smaller populations in isolated spawning grounds with the same sizes as the one for the large population, even though the summed abundance of the two smaller populations was equal to the abundance of the large adult population (Table 3). The differences increased with the decreasing abundance or density of the large population. Also, fewer eggs were fertilized when the abundances or densities of the two smaller populations were more similar. For instance, when the two adult densities were 0.6/[m.sup.2] and 0.2/[m.sup.2], the number of fertilized eggs was reduced by 22% relative to the number of fertilized eggs for the large population with an adult density of 0.8/[m.sup.2]. When the two densities were both 0.4/[m.sup.2], the reduction was 28% (Table 3).

[FIGURE 4 OMITTED]

DISCUSSION

The current study simulated the natural abalone spawning event with the incorporation of some important factors, namely the diffusion and advection of sperm and eggs and sperm supplied by many male individuals, that are often logistically difficult to be included in the experimental studies. This study resulted in a number of new findings, which have some important applications to management practices.

The study found that a higher population density would provide a larger benefit for fertilization success. There was a positive relationship between the population density and the fertilization success throughout the practical density ranges. Several earlier studies have also shown that the fertilization successes were positively correlated with adult population densities. In small scale experiments, Levitan (1991) and Levitan et al. (1992) demonstrated that urchin fertilization rates were higher at higher densities, even though no egg diffusion or advection was considered. A theoretical simulation also showed such a relationship between fertilization rates and adult densities (Levitan & Young 1995).

A shortage of sperm, known as "sperm limitation," was often blamed for low fertilization (Levitan & Petersen 1995). In fact, sperm is rarely in a short supply in the natural environment, as sperm released by a male usually far out-number eggs released by a female. The number of sperm released by a male abalone is about 5 orders of magnitude higher than the number of eggs released by a female abalone (Clavier 1992, Babcock & Keesing 1999). However, effective fertilization requires a high sperm concentration. If sperm-concentrated spots are limited within the range, which eggs may travel through in the time length of viability, fertilization would be insufficient. The number of such concentrations within the range is directly related to the population density, as high sperm concentrations mostly occur in close vicinities of the individual spawning males. Spawning of male abalone lasted for about one hour (Clavier 1992) or even a few hours (Babcock & Keesing 1999). Therefore, viable sperm continue to be supplied to the sperm-concentrated spots for up to several hours, although sperm diffuse away and shortly lose viability in the diluted states. A higher population density would produce more sperm-concentrated spots within the range, and an egg would have a higher probability to pass through one or more such spots and get fertilized. This probability shall be practically independent of the number of eggs. Fertilization of eggs would not affect sperm concentrations, because only a small proportion of sperm is consumed in the egg fertilization process and new sperm are continually being supplied. Consequently, the percentage of fertilized eggs is positively correlated with the adult population density.

Although the fertilization success increased with the adult population density, the rate of increase in the fertilization success decreased with increasing densities, exhibiting a typical Allee effect (Allee 1931). This finding is different from the finding from a simulation study conducted by Claereboudt (1999). Claereboudt (1999) showed that the proportion of fertilized eggs increased linearly with adult densities, implying that the expected decreasing rate in the fertilization success is the same at low adult population densities as at high adult population densities. The length of the simulated spawning area appeared to be only 20 m long in the Claereboudt's study, although gametes could travel hundreds of meters alive. This conclusion from the Claereboudt's study may just be an artefact from the setting of such a small simulated spawning ground. The finding from the current study indicated that the strength of the Allee effect on the fertilization process is stronger at lower adult densities. Therefore, more cautious measures need to be taken to prevent failure of fertilization process when the adult population density is low.

The current simulation study demonstrated that the Allee effect on the fertilization success was in effect over all practical densities, and there did not appear to be a presence of some critical but relatively small population size only below which Allee effect would be effective for the fertilization process. However, it must be noted or stressed that existence of the Allee effect on the fertilization process does not necessarily mean the effect will be manifested in the amount of recruitment (number of animals at certain age or size). In fisheries science, we may observe or estimate the amount of recruitment, but we are rarely able to measure or estimate the amount of fertilized eggs, because larval mortalities are largely variable and commonly several orders of magnitude higher than fertilization rates (Yund 2000).

The recruitment process involves two major processes, the fertilization process and subsequent survival process. If the survival rates for fertilized eggs, larvae, and juveniles are largely density independent, we may expect to see reduction in recruitment rate (amount of recruitment per spawner) when population densities are decreasing over all practical densities. Mortality rate is, however, usually density-dependent for marine species. The two most widely used stock-recruitment models, the Beverton-Holt and Ricker models, were derived from the assumption that mortality rate was density dependent (Quinn & Deriso 1999). The effect of a lower fertilization success at a lower adult population density on recruitment process would be buffered or even over-compensated, if fertilized eggs, larvae, and juveniles have to go through a density-dependent mortality process. This would result, consequently, in a critical range of relatively small population size only below which the Allee effect would be effective for the recruitment process. For instance, hypothetically assuming a density-dependent mortality model for abalone: M = 0.3 + 0.005*(Number of fertilized eggs), we'll get the critical adult population density to be 0.4 per [m.sup.2], and only below this point we may detect the Allee effect on the recruitment process caused by declining recruitment rates. The location of this critical population density is a function of the strength of the density-dependent effect and the extent of densities over which the densities have an effect on the mortality rate. Shepherd and Partington (1995) reported that Allee effects played a role in the collapse of the abalone populations in Waterloo Bay and West Island, Australia. On the other hand, Atkins et al. (2004) reported that densities of immature Northern abalone, Haliotis kamtschatkana, have been fluctuating despite significant decreases in large abalone densities, since the British Columbia abalone fishery closure because of low abundances in 1990. The survival rate for young Northern abalone appeared to be density-dependent even at low population densities.

The effect of population size on recruitment for a free-spawning population is likely to be the combination of the Allee effect on the fertilization process and compensatory effect for the survival of young animals. A population subject to an Allee effect on recruitment is unlikely to persist in the range of population sizes where that effect is manifested. A large number of species using the free-spawning strategy suggests that this reproductive strategy must have its underlying mechanism to ensure a long-term survival of the species. The influences of the Allee effect and intraspecific competition on population dynamics are often considered not to overlap (Levitan & Young 1995). The former is believed to act only at small population sizes, when mates are limiting, and the latter in much larger populations, when resources are limiting (Levitan & Young 1995). The interaction of Allee (depensatory) effects and compensatory effects may be much more common than originally believed.

The current study found that a large population would result in more fertilized eggs than two smaller isolated populations with the summed abundance equal to the abundance of the large population. The study also discovered that, overall, higher fertilization would occur in two isolated populations that had larger differences in their densities, whereas the summed abundance remained the same. This clearly indicated the benefits of a large number of spawners in one spawning ground to the fertilization process.

Population segments of marine free-spawners are often interconnected through the flow of planktonic larvae to form a metapopulation. There are two harvesting strategies that would produce the same amount of yield. In the first strategy, all population segments are harvested more or less evenly every year. In the second strategy, an entire segment is divided into a number of sections, and these sections are harvested in turn one at a time (year) with higher fishing intensities. This fishing strategy is known as the rotational fishing scheme, in which the fishing process rotates back onto the section of each segment that was harvested in the 1st year after all sections have been harvested in one rotation. A rotational fishing strategy would result in more eggs to be fertilized for the metapopulation than the first fishing strategy, which tends to generate more even densities for the population segments. When densities of two segments are low, transplanting of mature animals from one segment to another produces a higher amount of fertilization, if the transplant-related mortality is minimal.

When the population density remains the same, a decrease in the population abundance or the size of the spawning ground does not affect the fertilization success unless the abundance or the size of the spawning ground is unrealistically small. In the simulation study, only when the length of the spawning ground is shorter than 100-200 m, do we see reductions in the fertilization successes. Levitan and Young (1995) reported that the population abundance had an important effect on the fertilization of the sea biscuit in their simulations. However, their simulated spawning ground was only up to 128 m long. Eggs released close to the extreme end of the spawning ground are less likely to be sufficiently fertilized, because the range where the eggs may become fertilized is shorter and many eggs may advect out of the spawning ground unfertilized. The proportion of eggs released near the extreme end of the spawning ground is usually small, and the impact on the fertilization success for the entire population is minimal. Only when the population abundance is really small, (i.e., the spawning ground length was really small) would this proportion have a considerable weight, resulting in noticeable reductions in the fertilization successes. Also, the weight of this proportion is positively correlated with the egg viability. When the egg viability is longer, we would expect to see the starting of the reduction in the fertilization success at a larger abundance.

The current study found that there was little impact of aggregation at this intermediate level ([rho] [approximately equal to] 0.5) on the fertilization success at relatively high densities ([greater than or equal to] 0.4/[m.sup.2]). The aggregation would result in higher fertilization successes than the random distribution only at low densities, especially when egg viabilities are low. Aggregation would overall increase population densities in locations where spawning individuals are present. Even though localized densities are low in a randomly distributed population, eggs would still have higher chances to pass over one or more spawning males in some distance downstream and get fertilized at higher densities. The benefit of being aggregated would, therefore, diminish. Only at low densities where eggs have small chances of being fertilized at distances, increases of the localized densities would considerably boost the amount of fertilization. Consequently, a lower fertilization would provide a larger scope for improvement, likely making the aggregation more beneficial. Small-scale experiments showed that aggregation increased the fertilization success (Levitan et al. 1992). The theoretical simulation also indicated that the benefit of aggregation for fertilization, but the benefit would decrease with the increasing population density (Levitan & Young 1995).

The aggregation was generated in different ways between these two studies and the current study. In the cited studies, higher aggregations were produced by increasing the density, namely by reducing the size of the area and maintaining the same number of spawning individuals. In the current study, higher aggregations were produced by making the animal's spatial distribution clustered and maintaining the size of the spawning ground and the number of abalone. Aggregation is not a universal behavior for mobile free-spawning invertebrates during spawning period. Although H. kamtschatkana or H. Laevigata were reported to exhibit some contagious distributions (Breen & Adkins 1980, Shepherd 1986), no such behavior was observed for H. tuberculata during the spawning season (Clavier 1992).

In this simulation study, impacts of population densities, sizes, and aggregations on the fertilization were studied under some "ideal" conditions, which may not occur in the real world. Gamete diffusion and advection may be reduced by a kelp forest, and gametes may be entrained in current eddies (Dayton & Tegner 1984). The current flow direction may not be aligned with the longitudinal axis of the spawning ground. In these cases, we would expect to see a higher positive impact of aggregation on fertilization, because eggs may have a shorter distance to travel to meet with concentrated sperm. All adults were assumed to spawn synchronously in the current study. A reduction from the full synchrony would be equivalent to a decrease in the density of spawning individuals. In addition, male abalone spawn in a series of 30-70 contractions over the spawning period, unlike some other marine free-spawning species, such as sea urchin, which spawn in a continuous stream (Clavier 1992). The impact of this spawning pattern on sperm distribution was not accounted for by the steady-state model used in the study. However, these variations would not invalidate the general conclusions about the relative importance of population parameters in the fertilization success, as over- or under-estimation were applicable to all simulations. The current study provides a general mechanical framework about the fertilization process at different adult population densities, abundances, and aggregations for a marine free-spawning population in general and an abalone population in particular.

ACKNOWLEDGMENTS

The author thanks Alan Campbell and Claudia Hand for the constructive comments that helped to improve the quality of this paper.

LITERATURE CITED

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ZANE ZHANG

Shellfish Stock Assessment Section, Pacific Biological Station, Fisheries and Oceans Canada, 3190 Hammond Bay Road, Nanaimo, British Columbia, V9T 6N7, Canada

E-mail. [e-mail]

TABLE 1.
Density profiles for an aggregated and a random distributions
with a density of 0.8/[m.sup.2].

                               Number of Occurrences

      Number of           At Aggregated           At Random
Abalone per [m.sup.2]     Distribution          Distribution

0                             1,498                 1,129
1                              469                   870
2                              252                   363
3                              145                   109
4                              81                    28
5                              37                     1
6                               7                     0
7                               8                     0
8                               2                     0
9                               1                     0

TABLE 2.
Model parameter settings.

Symbol                      Description                  Value

Turbulent Diffusion Model:
U                  Current velocity (m/s)          0.05
U *                Friction velocity               5% U
D                  Water depth (m)                 5
h                  Gamete release height (m)       0.1
[Q.sub.S]          Sperm release rate (per s)      5 x [10.sup.7]
[Q.sub.E]          Egg release rate (per s)        1 x [10.sup.2]
B                  Rate constant                   0.5
[a.sub.y]          Diffusion coefficient           7.79
[a.sub.z]          Diffusion coefficient           0.77
d                  Water depth (m) at which        5 for sperm
                     gamete would diffuse back.    1 for eggs
Fertilization      kinetic model:
[beta]             Fertilization rate constant     3.35 x [10.sup.-5]
                     ([mm.sup.3]/s)
[[beta].sub.0]     Fertilization rate constant     2.94 x [10.sup.-3]
                     ([mm.sup.3]/s)

TABLE 3.
Percentage of the sum of fertilized eggs for two small isolated
populations over the number of fertilized eggs for one large
population with abundance equal to the summed abundance for
the two small populations. The spawning ground for each
population was 1,000 m long, 5 m wide and 5 m deep. The sperm
and egg viabilities were 15 min and 1 h respectively,
and animals were distributed in aggregated fashions.

                                           Percentage

                                   At Densities of Two Small
 At Density of a                Populations (number/[m.sup.2])
 Large Population
(number/[m.sup.2])   0.1 + 0.1  X + 0.2  Y + 0.4  Z + 0.6  0.8 + 0.8

       0.2             0.63
       0.4                       0.67
       0.6                       0.73
       0.8                       0.78     0.72
       1.0                       0.83     0.76
       1.2                       0.85     0.79     0.77
       1.4                       0.87     0.80     0.78
       1.6                       0.89     0.83     0.80      0.80