J. Parasitol., 94(5), 2008, pp. 1022–1030 ᭧ American Society of Parasitologists 2008

QUANTITATIVE ESTIMATION OF THE COST OF PARASITIC CASTRATION IN A ANCEPS POPULATION USING A MATRIX POPULATION MODEL

N. J. Negovetich* and G. W. Esch Department of Biology, Wake Forest University, Winston-Salem, North Carolina 27109. e-mail: [email protected]

ABSTRACT: Larval trematodes frequently castrate their snail intermediate hosts. When castrated, the snails do not contribute offspring to the population, yet they persist and compete with the uninfected individuals for the available food resources. Parasitic castration should reduce the population growth rate ␭, but the magnitude of this decrease is unknown. The present study attempted to quantify the cost of parasitic castration at the level of the population by mathematically modeling the population of the planorbid snail Helisoma anceps in Charlie’s Pond, North Carolina. Analysis of the model identified the life-history trait that most affects ␭, and the degree to which parasitic castration can lower ␭. A period matrix product model was constructed with estimates of fecundity, survival, growth rates, and infection probabilities calculated in a previous study. Elasticity analysis was performed by increasing the values of the life-history traits by 10% and recording the percentage change in ␭. Parasitic castration resulted in a 40% decrease in ␭ of H. anceps. Analysis of the model suggests that decreasing the size at maturity was more effective at reducing the cost of castration than increasing survival or growth rates of the snails. The current matrix model was the first to mathematically describe a snail population, and the predictions of the model are in agreement with published research.

Life-history theory predicts that an organism will exhibit a mental factors and the presence of certain predators have been strategy that maximizes its reproductive value (Roff, 1992). implicated in causing these changes (Brown et al., 1985; Byrne Predation, for example, is a strong selective force, and the prey et al., 1989; Lam and Calow, 1989; Crowl and Covich, 1990); population must adapt to maintain existence in the habitat. In however, reciprocal transfer studies with C. californica have populations where large individuals are more likely to be cap- also demonstrated a genetic component on which castration can tured by a predator, the prey generally mature at a smaller size, act (Lafferty, 1993a). reproducing before being consumed (Reznick et al., 1990; In snail populations, prevalence of parasites increases with Hutchings, 1993). Similarly, if most predation occurs on small shell length, suggesting that large snails are more likely to be- juveniles, then accelerated growth of the prey population would come castrated than the smaller individuals. Fecundity increases reduce the threat of predation (Crowl and Covich, 1990; Rez- with size of the snail, so castration of the largest individuals nick et al., 1990; Hutchings, 1993). results in an immediate loss of more eggs than if small indi- Parasitic castration is similar to predation in that infected viduals are castrated. Unfortunately, few studies have quanti- individuals are reproductively dead. They persist in the habitat, tatively estimated the cost of castration on the growth rate of but do not contribute to the overall reproductive success of the snail populations. A few investigators have observed that snail population. Snails are the requisite first intermediate hosts for populations are smaller in size when castration occurs (Lafferty, most digenetic trematodes, including of medical impor- 1993b; Fredensborg et al., 2005). In addition, field studies that tance, i.e., Schistosoma spp. Infection by trematodes frequently artificially increased the number of trematode eggs in a habitat results in partial or complete castration of the snail host (Sor- have demonstrated that parasitic infections could significantly ensen and Minchella, 2001). In Charlie’s Pond (North Caroli- decrease the number of snails in an aquatic system (Nagano, na), the snail Helisoma anceps is completely castrated by the 1966; Lie, 1973; Pointier and Jourdane, 2000; Suhardono et al., dominant trematode species Halipegus occidualis (Crews and 2006). However, the decrease in population growth rate caused Esch, 1986). In 1984, prevalence of H. occidualis reached 60%, by a fixed probability of castration remains unknown. indicating that 60% of the snails were lost from the population One approach to estimate the population cost of infection is of reproductively active snails. The high prevalence of castra- with mathematical models. Perhaps the most famous host–par- tion is likely a significant selection pressure, when considering asite model is 1 that examined factors that could regulate both that a 20% reduction of a fish population was sufficient to alter the host and parasite populations (Anderson and May, 1978; the observed growth rate of various fish species (Roff, 1992). May and Anderson, 1978). The model was based on the Lotka– Interest in the population cost of parasitic infection has been Volterra predator–prey equations that used a set of differential focused on comparing snail densities from populations with dif- equations to describe the parasite and host population dynamics. ferent intensities of infection. In marine systems, Cerithidea Differential equations, converted to matrix form, can also be californica and Zeacumantus subcarinatus population densities employed to model age- and size-structured populations (Cas- are negatively correlated with prevalence of a castrating trem- well, 2001). These matrix models have been used in several atode (Lafferty, 1993b; Fredensborg et al., 2005). Furthermore, conservation studies, primarily to determine which life-cycle both species mature at a smaller size when the risk of parasitic castration is high (Lafferty, 1993a; Fredensborg and Poulin, stages should be protected to maintain and promote population 2006). Snails often exhibit phenotypic plasticity in their life- growth (Crouse et al., 1987; Crowder et al., 1994; Marschall history traits, primarily growth rates and fecundity. Environ- and Crowder, 1996). The analyses involved altering life-history traits, and then estimating the corresponding change to the pop- ulation growth rate. Similar analyses can be performed with Received 1 May 2007; revised 12 September 2007; accepted 11 snail–trematode systems. Specifically, population growth rate March 2008. can be calculated in the presence and absence of infection, and * Current address: Department of Infectious Diseases, St. Jude Chil- dren’s Research Hospital, 332 North Lauderdale Street, MS No. 330, changes to the growth rate can be observed for various values Memphis, Tennessee 38105. of key life-history traits. The traits that may offset the cost of

1022 NEGOVETICH AND ESCH—MATRIX MODEL OF H. ANCEPS 1023

TABLE I. Monthly mean fecundity (eggs/month) of snails in each size class. Individuals Ͻ6 mm were not reproductively active, so those data were excluded from the table.

Monthly transition March– April– May– June– July– August– September– October– November– Size class April May June July August September October November March

6–7 0 0 0 0 0 12.86 12.86 5.57 0 7–8 66.86 49.29 0 18.43 9 35.14 15.86 5.57 0 8–9 104.57 70.71 24.86 41.14 44.14 60 23.14 3 0 9–10 129.43 111.86 40.71 51.86 41.57 46.29 9.43 0 0 10–11 181.29 114.43 81.43 61.29 50.14 46.29 0.86 5.14 0 11ϩ 203.14 122.14 123.86 82.71 46.29 29.14 7.71 0 0

infection are growth rate, fecundity, and survival (Sorensen and do not grow into a larger size class. For the Ͻ5-mm size class, SPG Minchella, 2001). was calculated with 3.5 mm. SPG was transformed to millimeters/month for each size class (Table II). The goal of the present study is to determine quantitatively Survival, infection, and self-cure probabilities were estimated from the effect of a castrating trematode infection on the population the capture histories with the use of Program MARK (White and Burn- of H. anceps in Charlie’s Pond. Estimates of life-history traits ham, 1999). Analysis of the capture histories was performed with mul- were previously calculated (Negovetich and Esch, 2008), and tistate models. This allowed estimation of infection and self-cure prob- abilities of snails Ͼ6 mm. Infection status does not affect survival (Ne- they will be used to construct a period matrix product popula- govetich and Esch, 2008), so a second analysis ignoring infection status tion model. With the use of the model, the response of popu- was performed with the use of the Cormack–Jolly–Seber (CJS) model. lation growth rate to various levels of parasitic infection will Estimates of survival from the CJS model were obtained for almost all be investigated. Additionally, the effect of changes in life-his- size classes and months. tory strategy on population growth rate will be determined. Model construction METHODS A period matrix product model was constructed with 8 size classes (Fig. 1) and 9 transition matrices. This model was chosen because it Data collection accounts for the seasonal variation in life-history traits (Caswell, 2001). Life-history traits were calculated from field experiments in Charlie’s All but 1 of the transition matrices describe the change in population Pond, North Carolina; a description of the fecundity experiment and structure from 1 month to the next, and each is unique. The overwinter mark–recapture methods can be found in Negovetich and Esch (2008). matrix describes the change from November to March. The top row of Estimates were made for 2 locations within the pond, but because of each transition matrix represents the number of new individuals that the lack of difference between the sites (Negovetich and Esch, 2008), enter the population (Fij). The remaining elements of the matrices in- the snails from each site were pooled and estimates derived for the clude survival and growth probabilities, in addition to the probability entire pond. Mean fecundity (eggs·snailϪ1 ·moϪ1) was calculated for the of acquiring or losing an infection. For example, the diagonal of the various size classes each month (Table I). Only those snails not shed- matrix is the probability that an individual will survive and remain in

ding cercariae (see Negovetich and Esch, 2007) were used to calculate the current size class (Pij). The structure of the matrix is identical to a the mean. This included snails that did not lay eggs during the week of Lefkovitch matrix (Caswell, 2001). Subscripts of the variables in the isolation in the reproduction platform. The specific growth rate (SPG; matrix refer to the size classes that the snail moves to, or stays in, during

mm/day) was calculated from all recaptured individuals (Negovetich a transition. Specifically, Gij is the probability that a snail survives and and Esch, 2008). Because not all size classes were recaptured each grows into size class i if it was initially in size class j. Likewise, Fij is month, SPG was calculated from the regression of SPG on the recip- the number of offspring of size class i produced from a single snail in rocal of initial size. Specifically, estimated SPG was obtained from the size class j. midpoint of the different size classes. For example, the SPG for a shell Survival probabilities for every month could not be calculated for length of 5.5 mm was the SPG estimate for the 5-mm size class. Growth snails Ͻ5 mm because of the limited number of capture histories. Be- rates for the Ͼ11-mm size class were not calculated because these snails cause of this limitation, August and September were the only months

TABLE II. Specific growth rate (SPG; mm/mo) of the various size classes for each transition. SPG was calculated at the midpoint of each size class. The SPG for snails Ͻ5 mm was estimated from snails 3.5 mm in size. Snails cannot grow out of the 11ϩ-mm size class, so SPG was not calculated for that group of individuals.

Monthly transition March– April– May– June– July– August– September– October– November– Size class April May June July August September October November March

Ͻ5 1.94 0.8 1.5 2.95 3.24 4 3.04 1.36 0.14 5–6 0.92 0.42 0.7 1.44 1.58 2.19 1.45 0.67 0.08 6–7 0.65 0.32 0.49 1.03 1.13 1.57 1.02 0.49 0.07 7–8 0.45 0.25 0.33 0.73 0.8 1.11 0.71 0.35 0.06 8–9 0.29 0.19 0.21 0.5 0.55 0.76 0.47 0.25 0.05 9–10 0.17 0.15 0.11 0.32 0.35 0.48 0.28 0.16 0.04 10–11 0.07 0.11 0.04 0.18 0.19 0.25 0.12 0.1 0.04 1024 THE JOURNAL OF PARASITOLOGY, VOL. 94, NO. 5, OCTOBER 2008

FIGURE 1. Life-cycle diagram showing the general transition probabilities. The first number in the subscript refers to the size class that the snail remains in, moves to, or produces, and the second number in the subscript is the original size class. For example, F15 relates the number of snails in the No. 1 size class produced by a single snail in the No. 5 size class. Similarly, G43 relates the proportion of No. 3 size class snails that move into size class No. 4. Numbers and letters with an apostrophe indicate that those snails are infected. F ϭ fecundity, G ϭ growth, P ϭ remain, I ϭ infected, R ϭ self-cure, 1 ϭϽ5 mm, 2 ϭ 5–6 mm, 3 ϭ 6–7 mm, 4 ϭ 7–8 mm, 5 ϭ 8–9 mm, 6 ϭ 9–10 mm, 7 ϭ 10–11 mm, 8 ϭ 11ϩ mm. when survival could be calculated. Freshwater snails exhibit a Type III for the various size classes was used in the model. Self-cure was as- survivorship curve (Dillon, 2000), with high mortality among the small- sumed to occur only in those snails that overwinter. Moreover, individ- est snails. Thus, survival in this size class was assumed to be lower uals that lost their infection were assumed to regenerate their gonadal than the survival probabilities of the 5-mm size class. The only month tissue and resume reproduction (Goater et al., 1989). where survival was assumed to be larger than that of the 5-mm snails The probability to survive and remain in a size class (Pij) was cal- was in July, which coincides with cohort turnover and an increase in culated as follows. Growth rates were examined and compared to the the number of small snails (Crews and Esch, 1986; Fernandez and Esch, size class of interest. If SPG Ͻ 1 mm/mo, then the probability of re- 1991b; Negovetich, 2003). Estimates of survival are listed in Table III. maining in the size class is 1 Ϫ SPG. This assumes a uniform distri- The magnitude of the infection probability is not dependent on size bution of sizes within each size class. The probability of remaining in class (Negovetich and Esch, 2008). Field collections suggest that snails a size class is multiplied by the probability of surviving during the Ͻ 5 mm are too small to exhibit a patent infection. The majority of monthly transition, thus giving the estimate of Pij for each size class. snails in this size class grow into the next size class within a month, The probability of surviving and growing (Gij) into the next size class whereas prepatency of H. occidualis is approximately 42 days (Goater, is obtained in a similar manner. Here, SPG is the probability of growing 1989). Thus, even if the snail is initially infected when Ͻ5 mm, patency into the next size class if SPG Ͻ 1 mm/mo. The model does not con- will not occur until the snail grows into the next size class. The model strain SPG to a maximum value, so that individuals must grow pro- describes the transition to the infected state as instantaneous, so we gressively through each size class. Thus, when SPG Ͼ 1 mm/mo, the restricted infection to snails 5 mm or larger. The mean of the infection fraction above 1 is the proportion that appears to ‘‘skip’’ a size class. probabilities across size classes was used as the probability of infection In reality, individuals progress through each size class, but during the for each month (Fig. 2). The effect of infection was assumed to be months of increased growth, i.e., June–September, the smallest snails complete castration of the host without an impact on survival or growth can skip 1, or sometimes 2, size classes. rates, which is supported by field data from a previous analysis (Ne- When infection (or self-cure) is accounted for, Pij and Gij in the tran- govetich and Esch, 2008). Self-cure is known to occur in overwintered sition matrix are adjusted for the proportion of uninfected individuals H. anceps (Goater et al., 1989), so the probability of losing an infection that become infected (or lose their infection). Specifically, the propor- was obtained from Program MARK (White and Burnham, 1999). Like tion of uninfected snails that survive and remain in the uninfected size ϭ Ϫ infection probabilities, the mean of the self-cure probabilities ( 35%) class is Pij ·(1 Iij), where Iij is the probability of infection of snails in

TABLE III. Probability to survive during the transition period for each size class. A Cormack-Jolly-Seber model was fit to the set of capture histories, and probabilities were calculated for nearly every size class each month.

Monthly transition March– April– May– June– July– August– September– October– November– Size class April May June July August September October November March

Ͻ5 0.200* 0.240* 0.250* 0.250* 0.240* 0.235 0.138 0.250* 0.050* 5–6 0.403 0.484 0.490* 0.510 0.076 0.311 0.404 0.688 0.094 6–7 0.618 0.774 0.075 0.485 0.089 0.265 0.480 0.456 0.268 7–8 0.542 0.541 0.359 0.442 0.055 0.243 0.464 0.585 0.432 8–9 0.367 0.686 0.427 0.393 0.023 0.408 0.658 0.571 0.515 9–10 0.537 0.656 0.431 0.476 0.030 0.403 0.732 0.524 0.633 10–11 0.591 0.698 0.405 0.598 0.025 0.095 0.800* 0.607 0.529 11ϩ 0.542 0.878 0.379 0.704 0.013 0.155 0.976 0.564 0.575

* Estimated probability. NEGOVETICH AND ESCH—MATRIX MODEL OF H. ANCEPS 1025

FIGURE 2. Monthly mean probability of infection for snails Ͼ5mm (solid line). The dashed line is the maximum infection probability used to measure the effect on population growth rate. The maximum prob- abilities are 150% of the base infection probability (solid line).

Ϫ size class j. Similarly, Gij ·(1 Iij) is the probability of surviving and growing into the next size class without becoming infected. Movement

from uninfected to infected status is Pij ·Iij when snails remain in the current size class, and Gij ·Iij when they grow into a larger size class. Movement from infected to uninfected status follows the same logic, FIGURE 3. The affect on population growth rate (␭) of different com- Ϫ where a fraction of the infected individuals remains infected (1 Rij) binations of the probability of self-cure and percent of base infection. and the remaining fraction self-cures (Rij). For Charlie’s Pond, calculated values of P (self-cure) and the probability An annual transition matrix (ATM) was calculated by taking the mul- of infection (base infection ϭ 100%) estimated ␭ at 1.55, which is tiplicative product of all monthly transition matrices (Caswell, 2001). marked with a point on the graph. This matrix describes the change in population structure from 1 yr to the next in a specific month. The dominant eigenvalue of the ATM ␭ corresponds to the annual population growth rate (Caswell, 2001), ␭ which is equal across the various ATMs. Specifically, ␭ for the ATM risk of infection decreased to 1.43 when self-cure does not beginning in March is equal to the ␭ of the ATM that starts in June. occur. The value of ␭ increases to 1.55 when 35% of the in- For convenience, the ATM corresponds to the snail population for fected individuals lose their infection during winter. With the ␭Ͼ March. The population is growing when 1, but it is decreasing basic model, ␭ exhibits a slight curvilinear decrease as the prob- when ␭Ͻ1. ability of infection increases; i.e., a small increase of infection Model analysis risk is more pronounced in populations with limited risk of castration (Fig. 3). Self-cure offsets castration by linearly in- The basic model was constructed with the life-history estimates cal- ␭ culated from previous work in the pond (Negovetich and Esch, 2008), creasing , and the magnitude of this increase is greater when and served as the starting point for the analysis. The effect of infection the probability of infection is high. and self-cure on ␭ was assessed in a number of ways. First, ␭ was Life-history traits have a pronounced affect on ␭. Altering calculated with the use of the estimates of the probability of infection ϭ the probability estimates by size class reveals that survival and as derived from the mark–recapture study ( base infection probability). Ͻ Second, the infection probabilities were increased or decreased by var- growth are most important for 5-mm snails (Table IV). How- ious proportions, from 1 to 150% of the actual risk of infection. Con- ever, of those size classes susceptible to castration, growth rates comitantly, the percentage of infected individuals that self-cure during winter was modified from 0 to 100% for each infection probability examined. Finally, the probability of infection was increased by 0.1 for TABLE IV. Percent change in the population growth rate to a fixed in- each size class (across months) and for each month (across size class) crease in life-history traits of the various size classes. Survival, growth to assess how ␭ changes when infection risk varies by size class and rate, and probability of infection were increased by 10%. Fecundity was month, respectively. increased by 10% (ϭelasticity), and by 10 eggs/mo (ϭsensitivity). Some Estimates of the life-history traits were altered to understand better values could not be calculated (marked with ‘‘—’’) because of biolog- the relative contribution of each trait to ␭. Survival and growth-rate ical (e.g., reproductive maturity in snails Ͻ5 mm) or model constraints estimates were increased by 10% of the original value used in the basic (10% of 0 is 0). model, and the percentage change of ␭ recorded (elasticity analysis of Caswell, 2001). Fecundity was altered for Ͼ5 mm snails in 2 separate analyses. First, estimates of fecundity were increased by 10 eggs/month Fecundity (equivalent to the sensitivity analysis of Caswell, 2001). Second, the Size class Survival Growth rate Elasticity Sensitivity Infection fecundity estimates were increased by 10%; size classes that did not oviposit during a month remained nonreproductively active. The life- Ͻ5 25.70 52.61 — — — history traits were increased collectively for all months within a single 5–6 4.64 7.06 — 343.87 Ϫ4.67 size class, for all size classes within a month, and individually for each 6–7 8.91 12.78 0.75 198.50 Ϫ7.25 size class every month. 7–8 11.65 2.04 10.24 91.47 Ϫ7.30 8–9 13.36 1.48 5.30 12.22 Ϫ7.55 RESULTS 9–10 8.78 0.29 3.01 4.48 Ϫ4.89 10–11 1.66 0.04 1.12 1.37 Ϫ1.19 The life-history traits used to construct the population model 11ϩ 0.12 — 0.15 0.16 Ϫ0.11 resulted in ␭ϭ2.41 in the absence of infection. Including the 1026 THE JOURNAL OF PARASITOLOGY, VOL. 94, NO. 5, OCTOBER 2008

TABLE V. Percent change in the population growth rate to a fixed in- DISCUSSION crease in life-history traits of the monthly transitions. Survival, growth rate, and probability of infection were increased by 10%. Fecundity was Population model increased by 10% (elasticity) and by 10 eggs/mo (sensitivity). Some values were not calculated (marked with ‘‘—’’) because of biological The current study is the first to describe empirically a snail constraints, i.e., cold water from November to March prevents repro- population with a matrix population model. Comparing ␭ be- duction and infection. tween populations with various probabilities of infection dem- onstrates that castration can cause a significant reduction in the Fecundity population growth rate. When the calculated infection proba- ␭ Growth bilities from 2005 to 2006 were included in the model, be- Month Survival rate Elasticity Sensitivity Infection came Ͼ1, which indicates population growth. However, the population of H. anceps in the pond does not appear to be March–April 7.27 2.79 2.73 5.8 Ϫ7.78 growing as quickly as the model suggests. Instead, environ- Ϫ April–May 6.65 1.74 3.35 7.97 4.2 mental stochasticity has likely caused ␭ to fluctuate around 1. May–June 8.59 5.08 1.41 78.14 Ϫ4.47 For example, fecundity in 2002 was significantly lower than in June–July 8.01 41.55 1.99 263.07 Ϫ6.22 July–August 3.18 6.98 6.82 140.11 Ϫ1.46 2006 (Negovetich and Esch, 2007). Reducing fecundity by half August–September 7.53 0.43 2.47 11.11 Ϫ0.46 for all months except September and October (fecundity was September–October 8.57 8.96 1.42 10.65 Ϫ4.05 not significantly different in these months; Negovetich and October–November 9.94 6.14 0.06 1.56 Ϫ5.65 Esch, 2007) decreased ␭ to 0.43, which implies that the popu- November–March 10 0.41 — — — lation size declined during that year. Interestingly, North Car- olina was near the end of a severe drought at this point in time, which lowered water levels and probably decreased fecundity of the population (Negovetich and Esch, 2007). influence ␭ more in the smaller snails, whereas the intermediate Annual variation in fecundity, and possibly survival and size classes benefit more from an increase in survival. Similarly, growth rates, implies that stochastic processes influence ␭ from ␭ increases with decreasing size of the snails when individuals year to year. For example, a population of Hydrobia ventrosa oviposit 10 additional eggs per month. The effect of increasing fluctuated over 4 yr of observation because of environmental the probability of infection is greatest for the 6–9-mm snails. effects on fecundity (Kube et al., 2006). Both biotic and abiotic Seasonally, the effect of survival on ␭ is about the same for factors are likely causing the fluctuations in Charlie’s Pond. each month, with the largest increase observed in June–July Specifically, pond productivity, water levels, and quantity of (Table V). Changes in the individual estimates of survival, trematode eggs entering the pond all contribute to the observed growth rates, and fecundity reveal similar trends. Snails Ͻ5mm value of ␭. Pond productivity is known to affect reproductive benefit most from an increase in survival from April to Septem- output and growth rate of snails (Brown et al., 1985; Byrne et ber, whereas survival of snails 7–10 mm in size strongly influ- al., 1989; Lam and Calow, 1989; Keas and Esch, 1997). In- ences ␭ from October to March (Table VI). Furthermore, ␭ ex- creasing the food supply late in the year will increase the hibits the largest increase with a 10% increase in growth rate growth rate of H. anceps, thus resulting in larger snails in the of snails Ͻ5 mm in May–October (Table VII). An increase in spring when most egg production occurs. Similarly, snails will reproductive output has the greatest effect on ␭ in the small direct excess energy toward reproduction when aufwuchs is snails in May–August. When a decrease in size at maturity is abundant in the spring (Keas and Esch, 1997). Both situations simulated by allowing nonreproductively active snails to lay 10 will result in increased egg output, the first from larger snails eggs/month, the largest increase in ␭ occurred for snails Ͻ7 in the spring, and the second from the allocation of excess en- mm in June (Table VIII). If the analysis is restricted to the ergy toward reproduction. When periphyton production from reproductively active snails, then a 10% increase of fecundity 1989 and 2002 was compared, significant differences were ob- has the greatest effect on ␭ in the 7-mm size class in July– served between the years (Negovetich, 2003), but it is unclear August (Table IX). how much periphyton production varies from 1 yr to the next.

TABLE VI. Percent change in the population growth rate to a 10% increase in the probability to survive during the monthly transitions.

Monthly transition March– April– May– June– July– August– September– October– November– Size class April May June July August September October November March

Ͻ5 0.08 2.73 4.71 2.91 2.41 6.89 2.47 1.44 0.09 5–6 0.15 0.01 1.38 1.75 0.09 0.12 0.58 0.27 0.22 6–7 1.46 0.32 0.05 2.85 0.29 0.14 1.02 0.80 1.63 7–8 1.71 0.83 0.69 0.12 0.39 0.22 1.43 3.11 2.36 8–9 1.70 1.20 0.82 0.17 Ͻ0.005 0.16 2.93 2.50 2.88 9–10 1.79 1.07 0.64 0.13 Ͻ0.005 Ͻ0.005 0.16 1.77 2.44 10–11 0.36 0.44 0.25 0.06 Ͻ0.005 Ͻ0.005 Ͻ0.005 0.05 0.38 11ϩ 0.02 0.04 0.04 0.01 Ͻ0.005 Ͻ0.005 Ͻ0.005 Ͻ0.005 0.01 NEGOVETICH AND ESCH—MATRIX MODEL OF H. ANCEPS 1027

TABLE VII. Percent change in the population growth rate to a 10% increase in growth rate for each monthly transition. Snails cannot grow out of the 11ϩ-mm size class, so growth rates have no impact on population size.

Monthly transition March– April– May– June– July– August– September– October– November– Size class April May June July August September October November March

Ͻ5 0.63 1.04 4.18 26.87 6.50 0* 6.74 4.44 0.01 5–6 0.21 0.00 0.64 4.96 0.10 0.19 0.71 0.25 0.08 6–7 1.14 0.25 0.04 9.64 0.23 0.12 0.55 0.46 0.16 7–8 0.51 0.27 0.16 0.09 0.15 0.08 0.39 0.44 0.03 8–9 0.19 0.08 0.04 0.00 Ͻ0.005 0.03 0.57 0.51 0.09 9–10 0.09 0.08 0.02 0.01 Ͻ0.005 Ͻ0.005 0.01 0.05 0.04 10–11 0.01 0.02 Ͻ0.005 Ͻ0.005 Ͻ0.005 Ͻ0.005 Ͻ0.005 Ͻ0.005 Ͻ0.005

* The original SPG of 4 mm/mo was at the maximum allowed in the model.

The water level in the pond can also influence population the frog; therefore, increasing the number of eggs in the fecal growth rate through a variety of ways. First, a rapid decline in transmission packet does little to increase the infection proba- water level during severe drought (see Negovetich and Esch, bility of the snail host (Zelmer and Esch, 2000). Instead, the 2007) often exposes substratum in the littoral zone. All snail number of egg-laden fecal deposits establishes the risk of in- eggs that are in that strip of shoreline desiccate and die. Second, fection for the snail host. Comparisons between years have emergent vegetation will be lost from the pond when the water demonstrated a decline in H. occidualis prevalence, which is level recedes. When this occurs, surface area for aufwuchs likely due to a combination of fewer R. clamitans in the pond growth decreases; this reduces the available food supply (Jef- and a general decrease in habitat overlap of the hosts (Nego- fries, 1993; Jones et al., 1999), and negatively impacts the vetich and Esch, 2007). In contrast, the number of eggs present growth rate and fecundity of H. anceps. Moreover, loss of emer- in the pond is more important for those species requiring pen- gent vegetation will limit the habitat overlap of the various etration of the snail host by free-swimming miracidia, e.g., Di- hosts in the life cycle of the primary castrator, H. occidualis plostomulum sp. (Negovetich and Esch, 2007). Increasing the (Zelmer et al., 1999). The net effect is a decrease in the prob- prevalence or intensity in the definitive host will result in more ability of transmission to the snail host, and fewer castrated eggs entering the pond, and thus an increase in the probability individuals in the population. of transmission to H. anceps. Seasonal variation in prevalence Prevalence of castrating trematodes in H. anceps is partially for H. occidualis and other trematode species has been de- determined by the number of transmission stages in the pond. scribed previously (Fernandez and Esch, 1991a; Sapp and Esch, In host–parasite models, the infection rate is the product of 1994), and the fluctuations for allogenic trematode species are encounter rate, infection success, and the number of infective closely tied to the visitation of infected definitive hosts. In es- stages in the environment (Anderson and May, 1978; May and sence, peaks in prevalence were observed during those months Anderson, 1978; Rosa and Pugliese, 2002). The castrating trem- where infected individuals, i.e., waterfowl, were regularly ob- atode, H. occidualis, infects Rana clamitans as the definitive served at the pond. host. Transmission from R. clamitans to H. anceps requires con- sumption of the egg by the snail host. For H. occidualis, the Life-history evolution percentage of infected H. anceps appears to be more closely linked to the prevalence, and not the abundance or intensity of Predator–prey interactions are often used to examine the evo- adult worms in the definitive host (Zelmer and Esch, 2000). lution of life-history strategies (Roff, 1992). When confronted Specifically, the eggs of H. occidualis cluster in the feces of with intense selection pressures, such as an increase in preda-

TABLE VIII. Percent change in the population growth rate to an increase of 10 eggs/month. Snails do not reproduce during winter, so the November– March transition was excluded from analysis.

Monthly transition March– April– May– June– July– August– September– October– Size class April May June July August September October November

5–6 1.29 0.64 68.65 148.34 24.89 2.98 2.57 0.26 6–7 1.69 2.53 2.75 108.74 46.65 3.10 2.57 0.34 7–8 1.11 2.40 3.04 1.55 66.57 3.79 2.73 0.55 8–9 1.05 1.29 1.97 1.91 1.23 1.23 2.81 0.27 9–10 0.57 0.87 1.25 1.21 0.39 Ͻ0.005 0.07 0.12 10–11 0.08 0.23 0.43 0.42 0.20 Ͻ0.005 Ͻ0.005 Ͻ0.005 11ϩϽ0.005 0.01 0.05 0.05 0.04 Ͻ0.005 Ͻ0.005 Ͻ0.005 1028 THE JOURNAL OF PARASITOLOGY, VOL. 94, NO. 5, OCTOBER 2008

TABLE IX. Percent change in the population growth rate to a 10% increase in fecundity. Snails do not reproduce during winter, so the November– March transition was excluded from analysis. Values marked with a dash indicate that those snails were not reproductively active (6–7 mm), or that egg production was not observed during the week of isolation in the reproduction chambers.

Monthly transition March– April– May– June– July– August– September– October– Size class April May June July August September October November

6–7—————0.40 0.33 0.02 7–8 0.74 1.18 — 0.28 5.99 1.33 0.43 0.03 8–9 1.10 0.91 0.49 0.78 0.54 0.74 0.65 0.01 9–10 0.73 0.97 0.51 0.63 0.16 Ͻ0.005 0.01 — 10–11 0.15 0.26 0.35 0.26 0.10 Ͻ0.005 Ͻ0.005 Ͻ0.005 11ϩ 0.01 0.01 0.07 0.04 0.02 Ͻ0.005 Ͻ0.005 — tion risk with size of the prey, the prey population should re- (Table VIII). This is in contrast to a maximum increase of ␭ of spond by increasing its reproductive value to offset the risk of nearly 27% for increased growth rates (Table VII), and 7% for predation (Roff, 1992; Agnew et al., 2000). The response could increased survival (Table VI). Thus, the snail population ben- take the form of increased egg/offspring production by repro- efits most by reproducing at a smaller size. ductively mature individuals, or a decrease in the size/age at In 1984, individuals Ͻ8 mm did not reproduce (Crews and maturity (Roff, 1992). Predation results in an immediate de- Esch, 1986). A few years later, it was suggested that H. anceps crease in population size, and gastropods likely detect the rel- reaches reproductive maturity at 7.5–8.0 mm (Goater, 1989). ative risk of mortality by the quantity of available food (Eisen- The smallest snail that oviposited in 2002 was 5.65 mm (Ne- berg, 1966, 1970; Dillon, 2000). For example, predation de- govetich, 2003); individuals 6 mm in size were laying eggs in creases the population size, thus increasing the available food 2005 and 2006. Based on these observations, it appears that the supply. In cage studies, low-density populations generally ex- size at first reproduction has declined since research began in hibited faster growth rates and were more fecund than high- Charlie’s Pond approximately 24 yr ago. We propose that cas- density populations (Eisenberg, 1966; Dillon, 2000). In high- tration by H. occidualis has driven this shift in life-history strat- density populations, increases in growth rate and fecundity were egy. Prevalence in the snail host exceeded 50% in the 1980s, observed with the augmentation of food (Eisenberg, 1970). but has not exceeded 20% in recent years (Negovetich and Therefore, a feedback mechanism promotes population sustain- Esch, 2007). With 50% of the population reproductively dead, ability even with predation. Specifically, predation reduces pop- the individuals that were most fit are those that reproduced prior ulation size, aufwuchs production increases, and individual to castration. Smaller snails are less likely to be infected than snails exhibit higher fecundity, which offsets the losses to pre- larger snails (Negovetich and Esch, 2007), especially in a sys- dation. tem where individuals are continuously exposed, or nearly so, Parasitic castration is similar to predation, in that individuals to the infective stages of castrating trematodes. Over time, the are removed from the reproducing population, i.e., they are re- population of H. anceps should evolve toward reproduction at productively dead. However, castrated individuals remain in the smaller sizes. Similar shifts in size at reproductive maturity population and continue to consume aufwuchs that would oth- have been observed in the marine snails C. californica (Laffer- erwise be used by noncastrated individuals (Lafferty, 1993b). ty, 1993a) and Z. subcarinatus (Fredensborg and Poulin, 2006). Hypothetically, in the absence of a feedback mechanism, the Specifically, both species mature at smaller sizes in habitats snail population should exhibit an immediate decline in the next experiencing high levels of parasitic castration. Given the con- generation because of the decline in reproductive output. More- stant presence and high prevalence of H. occidualis since 1984, over, if the snail does not alter its life-history strategy and feed- parasitic castration is likely shaping the life-history strategy of back mechanisms do not exist, then the population should be- H. anceps in Charlie’s Pond. come extinct. Mathematical models (May and Anderson, 1978; The model demonstrates that parasitic castration in the snail Blower and Roughgarden, 1987) and experimental evidence host significantly decreases ␭. If an uninfected population is from field studies (Nagano, 1966; Lie, 1973; Blower and nearly stable in the absence of infection (␭ ഠ 1), then castration Roughgarden, 1989; Lafferty, 1993b; Pointier and Jourdane, has the potential to drive ␭ below the level required to sustain 2000; Fredensborg et al., 2005; Suhardono et al., 2006) support the population (␭Ͻ1). Some investigators have hypothesized the notion that castration can lead to population regulation or that trematodes could be used as control agents of snails that local extinction. Our model was used to determine which life- are intermediate hosts of human-infecting parasites (Nagano, history traits most offset the population cost of parasitic castra- 1966; Lie et al., 1968; Lim and Heyneman, 1972; Suhardono tion in the snail host. Increasing egg production by 10 eggs/ et al., 2006). Most were proposed after observing significant month produced a larger percentage change in ␭ than a 10% decreases in the size of a snail population following a year of increase in growth rates or survival probabilities (Table IV). high prevalence of castration. Field experiments on Guadeloupe Although a raw increase (10 eggs/month) was compared to a in the West Indian islands demonstrated the near disappearance percentage increase (10% increase in survival and growth rate), of Biomphalaria glabrata following 15 mo of seeding a pond the change in ␭ demonstrates that a decrease in the size at with the eggs of the castrating trematode, Ribeiroia guadelou- maturity more than doubles ␭ during the June–July transition pensis (reviewed in Pointier and Jourdane, 2000). The addition NEGOVETICH AND ESCH—MATRIX MODEL OF H. ANCEPS 1029 of Echinostoma malayanum and Echinostoma audyi eggs no- the pulmonate snail Helisoma anceps. Journal of Parasitology 77: ticeably reduced the population of Indoplanorbis exustus in Ma- 937–944. FREDENSBORG, B. L., AND R. POULIN. 2006. Parasitism shaping host life- laysia (Lie, 1973). Similar declines in snail population size were history evolution: Adaptive responses in a marine gastropod to in- demonstrated by maintaining ducks infected with Nocototylus fection by trematodes. Journal of Ecology 75: 44–53. attenuatus (Nagano, 1966) and Echinostoma revolutum (Suhar- ———, K. N. MOURITSEN, AND R. POULIN. 2005. Impact of trematodes dono et al., 2006) in snail-infested waters. on host survival and population density in the intertidal gastropod Zeacumantus subcarinatus. Marine Ecology Progress Series 290: The current matrix model is the first to describe mathemati- 109–117. cally the population dynamics of a snail population. Further- GOATER, T. M. 1989. The morphology, life history, ecology and genetics more, castration by larval trematodes was found to decrease the of Halipegus occidualis (Trematoda: Hemiuridae) in molluscan and population growth rate of H. anceps by 40%. A probability of amphibian hosts. Ph.D. Dissertation. Wake Forest University, Win- ston-Salem, North Carolina, 155 p. self-cure in overwintering snails of 35% increased ␭ by 8%. ———, A. W. SHOSTAK,J.A.WILLIAMS, AND G. W. ESCH. 1989. A Depending on the population dynamics of the snail host, cas- mark–recapture study of trematode parasitism in overwintered Hel- tration could theoretically drive ␭ below 1, at which point the isoma anceps (), with special reference to Halipegus oc- snail population must adapt to offset the loss of reproductive cidualis (Hemiuridae). Journal of Parasitology 75: 553–560. output or risk local extinction. Elasticity and sensitivity analy- HUTCHINGS, J. A. 1993. Adaptive life histories effected by age-specific survival and growth rate. Ecology 74: 673–684. ses suggest that decreasing the size at reproductive maturity is JEFFRIES, M. 1993. Invertebrate colonization of artificial pondweeds of more effective at increasing ␭ than increasing growth rate or differing fractal dimension. Oikos 67: 142–148. survival of young snails. The decrease in size at maturity be- JONES, J. I., J. O. YOUNG,G.M.HAYNES,B.MOSS,J.W.EATON, AND tween 1984 and 2006 in Charlie’s Pond, and the differences K. J. HARDWICK. 1999. Do submerged aquatic plants influence their periphyton to enhance the growth and reproduction of invertebrate observed in the marine snails C. californica and Z. subcarinatus mutualists? Oecologia 120: 463–474. (Lafferty, 1993a; Fredensborg and Poulin, 2006) agree with the KEAS, B. E., AND G. W. ESCH. 1997. The effect of diet and reproductive model predictions. maturity on the growth and reproduction of Helisoma anceps (Pul- monata) infected by Halipegus occidualis (Trematoda). Journal of Parasitology 83: 96–104. LITERATURE CITED KUBE, S., J. KUBE, AND A. BICK. 2006. A loss of fecundity in a popu- lation of mudsnails Hydrobia ventrosa caused by larval trematodes AGNEW, P., J. C. KOELLA, AND Y. M ICHALAKIS. 2000. Host life history does not measurably affect host population equilibrium level. Par- responses to parasitism. Microbes and Infection 2: 891–896. asitology 132: 725–732. ANDERSON,R.M.,AND R. M. MAY. 1978. Regulation and stability of LAFFERTY, K. D. 1993a. The marine snail, Cerithidea californica,ma- host–parasite population interactions: I. Regulatory processes. The tures at smaller sizes where parasitism is high. Oikos 68: 3–11. Journal of Animal Ecology 47: 219–247. ———. 1993b. Effects of parasitic castration on growth, reproduction BLOWER, S., AND J. ROUGHGARDEN. 1987. Population dynamics and par- and population dynamics of the marine snail Cerithidea californica. asitic castration: A mathematical model. American Naturalist 129: Marine Ecology Progress Series 96: 229–237. 730–754. LAM,P.K.S.,AND P. C ALOW. 1989. Intraspecific life-history variation ———, AND ———. 1989. Parasites detect host spatial pattern and in Lymnaea peregra (: Pulmonata). I. Field study. Jour- density: A field experiment analysis. Oecologia 78: 138–141. nal of Animal Ecology 58: 571–588. BROWN, K., D. R. DEVRIES, AND B. K. LEATHERS. 1985. Causes of life LIE, K. J. 1973. Larval trematode antagonism: Principles and possible history variation in the Lymnaea elodes. Malacol- application as a control method. Experimental Parasitology 33: ogia 26: 191–200. 343–349. BYRNE, R. A., J. D. REYNOLDS, AND R. F. MCMAHON. 1989. Shell ———, P. F. BASCH,D.HEYNEMAN,A.J.BECK, AND J. R. AUDY. 1968. growth, reproduction and life cycles of Lymnaea peregra and L. Implications for trematode control of interspecific larval antago- palustris (Pulmonata: Basommatophora) in oligotrophic turloughs nism within snail hosts. Transactions of the Royal Society of Trop- (temporary lakes) in Ireland. Journal of Zoology 217: 321–339. ical Medicine and Hygiene 62: 299–319. CASWELL, H. 2001. Matrix population models: Construction, analysis, LIM,H.K.,AND D. HEYNEMAN. 1972. Intramolluscan inter-trematode and interpretation. Sinauer Associates, Sunderland, Massachusetts, antagonism: A review of factors influencing the host parasite sys- 722 p. tem and its possible role in biological control. Advances in Para- CREWS,A.E.,AND G. W. ESCH. 1986. Seasonal dynamics of Halipegus sitology 10: 191–268. occidualis (Trematoda: Hemiuridae) in Helisoma anceps and its MARSCHALL,E.A.,AND L. B. CROWDER. 1996. Assessing population impact on fecundity of the snail host. Journal of Parasitology 72: responses to multiple anthropogenic effects: A case study with 646–651. brook trout. Ecological Applications 6: 152–167. CROUSE, D. T., L. B. CROWDER, AND H. CASWELL. 1987. A stage-based MAY,R.M.,AND R. M. ANDERSON. 1978. Regulation and stability of population model for loggerhead sea turtles and implications for host–parasite population interactions: II. Destabilizing processes. conservation. Ecology 68: 1412–1423. The Journal of Animal Ecology 47: 249–267. CROWDER, L. B., D. T. CROUSE,S.S.HEPPELL, AND T. H . M ARTIN. 1994. NAGANO, K. 1966. Eradication of trematodes by means of parasitic cas- Predicting the impact of turtle excluder devices on loggerhead sea tration. Kitasato Archives of Experimental Medicine 39: 1–8. turtle populations. Ecological Applications 4: 437–445. NEGOVETICH, N. J. 2003. Long-term analysis of Charlie’s Pond: Fecun- CROWL,T.A.,AND A. P. COVICH. 1990. Predator-induced life-history dity and trematode communities of Helisoma anceps. M.S. Thesis. shifts in freshwater snails. Science 247: 949–951. Wake Forest University, Winston-Salem, North Carolina, 69 p. DILLON, R. T. 2000. The ecology of freshwater molluscs. Cambridge ———, AND G. W. ESCH. 2007. Long-term analysis of Charlie’s Pond: University Press, Cambridge, U.K., 509 p. Fecundity and trematode communities of Helisoma anceps. Journal EISENBERG, R. M. 1966. The regulation of density in a natural popula- of Parasitology 93: 1311–1318. tion of the pond snail, Lymnaea elodes. Ecology 47: 889–906. ———, AND ———. 2008. Life-history cost of trematode infection in ———. 1970. The role of food in the regulation of the pond snail, Helisoma anceps using mark–recapture in Charlie’s Pond. Journal Lymnaea elodes. Ecology 51: 680–684. of Parasitology 94: 314–325. FERNANDEZ, J., AND G. W. ESCH. 1991a. Guild structure of larval trem- POINTIER,J.P.,AND J. JOURDANE. 2000. Biological control of the snail atodes in the snail Helisoma anceps: Patterns and processes at the hosts of schistosomiasis in areas of low transmission: The example individual host level. Journal of Parasitology 77: 528–539. of the Caribbean area. Acta Tropica 77: 53–60. ———, AND ———. 1991b. Effect of parasitism on the growth rate of REZNICK,D.A.,H.BRYGA, AND J. A. ENDLER. 1990. Experimentally 1030 THE JOURNAL OF PARASITOLOGY, VOL. 94, NO. 5, OCTOBER 2008

induced life-history evolution in a natural population. Nature 346: SUHARDONO,J.A.ROBERTS, AND D. B. COPEMAN. 2006. Biological con- 357–359. trol of Fasciola gigantica with Echinostoma revolutum. Veterinary ROFF, D. A. 1992. The evolution of life histories: Theory and analysis. Parasitology 140: 166–170. Chapman and Hall, New York, New York, 535 p. WHITE,G.C.,AND K. P. BURNHAM. 1999. Program MARK: Survival ROSA, R., AND A. PUGLIESE. 2002. Aggregation, stability, and oscilla- estimation from populations of marked . Bird Study 46: tions in different models for host–macroparasite interactions. The- S120–S138. oretical Population Biology 61: 319–334. ZELMER,D.A.,AND G. W. ESCH. 2000. Relationship between structure SAPP,K.K.,AND G. W. ESCH. 1994. The effects of spatial and temporal and stability of a Halipegus occidualis component population in heterogeneity as structuring forces for parasite communities in Hel- isoma anceps and Physa gyrina. American Midland Naturalist 132: green frogs: A test of selective treatment. Journal of Parasitology 91–103. 86: 233–240. SORENSEN,R.E.,AND D. J. MINCHELLA. 2001. Snail–trematode life his- ———, E. J. WETZEL, AND G. W. ESCH. 1999. The role of habitat in tory interactions: Past trends and future directions. Parasitology structuring Halipegus occidualis metapopulations in the green frog. 123: S3–S18. Journal of Parasitology 85: 19–24.