Mathematical model describing the population dynamics of Ciona intestinalis, a biofouling tunicate on mussel farms in Prince Edward Island, Canada

A mathematical model was used to describe the population of the aquatic invasive species, Ciona intestinalis in the presence of cultured mussel production. A differential equation model was developed to represent the key life stages: egg, larva, recruit, juvenile and adult. Stage transition rates were calculated from time spent in a stage and transition probabilities. Because surface availability for the settlement phase is a key determinant of population growth, dead juvenile and dead adult stages were also modelled, together with their drop-off rates. This model incorporated temperature dependencies and an environmental carrying capacity. Model validation was carried out against field data collected from Georgetown Harbour, in 2008. Relative sensitivity indices were calculated to determine the most influential factors in the model. The results indicated that the modelled outputs were broadly in agreement with the observed data. Under baseline conditions the number of C. intestinalis increased from early September to late October, after which they reached a plateau at an abundance of approximately five individuals per cm. Sensitivity analyses revealed that a reduction in spawning interval or the development time of larva accelerated C. intestinalis population growth. In contrast, decreasing either carrying capacity or the percentage drop-off of live juvenile and adult stages resulted in a decline in the population. This research provides the first detailed model describing population dynamics of C. intestinalis in mussel farms and will be valuable in exploring effective treatment strategies for this invasive species.


Introduction
Mussels accounted for 66% of total Canadian shellfish production (38,646 tonnes), with an estimated market value of CAD$39 million, in 2011 (Statistics Canada 2012). The Prince Edward Island (PEI) blue mussel (Mytilus edulis Linnaeus, 1758) industry produces approximately 80% of all mussels cultured in Canada (Statistics Canada 2012). Over the past 15 years the industry has encountered increasing challenges related to aquatic invasive species, especially tunicates. These biofouling species compete for food and space, reducing water flow rates from the species overgrowth, jeopardizing mussel health and yield, which can cause significant economic losses to mussel farmers and processors as a result of the costs associated with controlling their population growth as well as the additional labour costs during the mussel cleaning process at processing plants (Carver et al. 2006;Locke et al. 2009). Four species of invasive tunicates are found in PEI (Fisheries and Oceans Canada 2006;Macnair 2005): clubbed tunicate (Styela clava Herdman, 1881), vase tunicate (Ciona intestinalis Linnaeus, 1767), golden star tunicate (Botryllus schlosseri Pallas, 1766), and violet tunicate (Botrylloides violaceus Oka, 1927). Of these, the vase tunicate is considered to be the greatest threat for PEI aquaculture. Two years after the first identification of C. intestinalis in Montague River, PEI in the autumn of 2004, it became a dominant fouling species, causing severe problems for the PEI mussel industry (Carver et al. 2006;.
C. intestinalis is a solitary tunicate, with a short-lived planktonic stage before settling on a suitable substrate during metamorphosis and becoming a sessile filter feeder (Carver et al. 2006). The growth and reproductive rates are strongly temperature dependent (Dybern 1965;Carver et al. 2006); exhibiting rapid growth in the summer, before decreasing with declining temperature (Carver et al. 2006). A study of C. intestinalis populations in Atlantic coast of Nova Scotia, Canada estimated 12,000 eggs were produced per a 100-mm long individual over a season (Carver et al. 2003). Another study in Japan gave an estimate of 100,000 eggs per individual (Yamaguchi 1975). With its high fecundity and ability to reproduce rapidly (Carver et al. 2006), a mussel sock can be covered with C. intestinalis individuals in a short time; increasing the biomass on the mussel socks and resulting in mussel mortality through fall-off. To mitigate this, farmers remove tunicates from mussel socks by chemical and mechanical methods including 4% acetic acid treatment and high-pressure washing with water for C. intestinalis (Carver et al. 2003;Carver et al. 2006;. There is a need to compare the efficacy of treatments to find the best mitigation strategies in terms of time and frequency of treatment. A conventional approach involving field trials has been conducted for C. intestinalis (unpublished) and B. violaceus (Arens et al. 2011) to carry out such comparisons. However, these trials require considerable time to execute and are both cost and labour intensive. As an alternative, computerbased modelling can be used to mimic the population dynamics of a particular species (e.g. C. intestinalis) and subsequently to explore the likely effect of different control measures. While field-based experiments continue to provide a vital role, both in establishing the value of key parameters as specified within any model as well as in validating modelled outputs, a key advantage of such models is that they provide a mechanism to explore a range of possible intervention strategies in an inexpensive and timely manner.
Mathematical models are based on a set of equations with fixed parameters to describe a system. They have been applied to a wide range of problems associated with parasites and their control (Ebert et al. 2000;Jerwood and Saporu 1988;Luis et al. 2010;White et al. 2011). They can be used to represent complex phenomena and interactions, including those found in aquatic contexts (Fenton et al. 2006;Ford et al. 1999;Kanary et al. 2011;Murray 2011;Revie et al. 2005;Robbins et al. 2010;Thebault et al. 2007). These models typically predict the number of parasites/ species of interest or the rates of change in the numbers of a species at a given time. Applying such approaches to model the population dynamics of C. intestinalis should provide the basis for a better understanding of population growth over time, as well as an ability to compare modelled results among various scenarios. The objective of this study was to develop a mathematical model that could describe the population dynamics of C. intestinalis in areas with mussel production, to better understand the growth of these populations under different temperature conditions.

C. intestinalis population dynamics
The life cycle of C. intestinalis consists of egg, larva, recruit, juvenile, and adult stages. To capture the seasonal variation, six compartments representing the life stages of C. intestinalis were identified: egg (E), larva (L), recruit (R), juvenile (J), spring adult (A sp ), and autumn adult (A au ). Because surface area on which C. intestinalis can settle is a key determinant of population growth, two additional compartments to model dead stages were set up: dead juvenile (DJ) and dead adult (DA) (Figure 1). Egg refers to the C. intestinalis egg which has already been fertilized. Larva is the stage after the eggs hatch and become free-swimming larvae. Recruit refers to the tadpole that settles on a surface and develops through a process of metamorphosis. Those recruits that fail to metamorphose are assumed to detach from the surface. Juvenile is the stage at which the animal is completely metamorphosed but before it reaches sexual maturity. Spring adult refers to an animal that reaches its sexually mature size between May and September, and has the ability to reproduce. Autumn adults are animals that reach their adult stage between October and April. Two aggregate stages were also estimated: a surface-occupying stage (N SO ) and a visible surface-occupying stage (N VO ). N SO comprises any individual that is attaching to the available surface, including R, J, A sp , A au , and the two dead stages, DJ and DA, which continue to occupy space regardless of their mortality status until they drop-off of the surface, releasing more space for recruits; while N VO , which was only used for model validation purposes, represents the total number of individuals in a visible stage; which are those included in N SO but excluding recruits. The dead others stage (DO) in Figure 1 was not captured in the model since individuals transferred to this stage do not result in any reduction in settling surface.
A sexually mature adult C. intestinalis spawns eggs repeatedly throughout its lifespan (L Asp days for spring adult and L Aau days for autumn adult). On average, α eggs are released per individual every G SI days. The eggs are then fertilized with sperm externally at the rate of F f (T˚). These fertilized eggs hatch at the rate of F h (T˚) and develop to larvae in G E (T˚) days. The egg is viable for up to L E days before it is removed. After becoming a free-swimming larva, the tadpole turns into a recruit by finding a substrate to settle on at a percentage of F s (T˚) which is capacity dependent and is thus adjusted by a capacity adjusting factor (γ(a,t); details in a subsequent section). The larval phase can last for up to G L (T˚) days before settling and larva that do not settle in L L days will die. The recruit stage takes G R days to metamorphose to become a juvenile. The percentage of recruits that undergo the metamorphosis process is F m (T˚), whereas 1-F m (T˚) of recruits die in G R days. Each individual in the juvenile stage takes G J (T˚) days to become a sexually mature adult and has a daily mortality rate of m J . Space occupancy is released through a drop-off process, which occurs at the daily rates of µ DJ and µ DA for dead juveniles and dead adults, while juveniles and adults drop-off of mussel sock at the daily rates of µ J and µ A . A dichotomous variable, x, was used to control whether adult can produce eggs with the cut-off temperature at 4˚C (Eq. 1; Table 1). A similar approach was applied for spring and autumn adult Table 1. List of mathematical equations for Ciona intestinalis population dynamic model including differential equations, temperature model, capacity adjusting factor, and average relative sensitivity function. compartments (Eq. 5 and 6; Table 1). A dichotomous variable, y, was created to define whether the model was in spring (y=1) or autumn season (y=0). This allows the model to assign animals from juvenile stage to spring or autumn adult compartments depending on time of the model. The C. intestinalis life cycle can be described by a set of differential equations (Eq. 1-8; Table 1). The model was set to run for one calendar year, with Day 1 being the 1 st of May until the termination of the model at Day 365 and was initialized with an initial autumn-adult presence of 0.002 individual·cm -2 (or approximately 12 adults per mussel sock) based on field observations made by the Atlantic Veterinary College (AVC) shellfish research group; all other life stages were initially set to zero. Spring was set to begin on the 1 st of May and last for 120 days before switching to the autumn season. The model time step was set to 0.001 of a day.

Parameter estimation
A total of 20 parameters were identified, with seven of these being temperature dependent (see Table 2 and Figure 1 for details). Parameters were estimated based on values reported in the scientific literature. In cases where a range was reported (e.g. number of eggs laid per spawning (α) and larval lifespan (L L )) values were randomly selected from a uniform (for α) or triangular (for L L ) distribution, while for parameters derived from more than one source, the average value based on these sources was estimated. Similarly, the average values of estimates at different temperatures were determined for the temperaturedependent parameters, i.e. development times and percentage of individual successfully making the transition to a new phase for each stage (full details of the sources for these parameters are provided in Tables S1 and S2). All temperaturedependent parameters were linearly interpolated between reported values. Where no estimates could be found (e.g. temperatures less than 6 ˚C or greater than 24 ˚C in Figure 3) parameter values were assigned to the nearest reported value for %fertilization (F f (T˚)), %hatchability (F h (T˚)), %settlement (F s (T˚)) and %metamorphosis (F m (T˚)) ( Figure 3), while values for development time of each stage were linearly extrapolated ( Figure 2).

Temperatures
Average daily sea water temperatures were assumed to broadly follow trends that could be modelled by a sine wave and were estimated using trigonometric regression (Beer 2001;Cox 1987). The dependent variable was daily average temperature, whereas the independent variable was day of the year expressed in terms of sine and cosine functions, sin(2j t/p) and cos(2j t/p), where j is an integer, representing the number of sine and cosine terms, t represents day of the year ranging from 1 to 365, and p is the period, which is assumed to be 365 days in this model (Eq. 9; Table 1). The best fit temperature model was one comprising three sine/cosine terms (Rsquared = 0.97). However for the sake of parsimony, a model with one sine and one cosine term was used (R-squared = 0.95). After acquiring the regression coefficients, the model was simplified to the simple sine function in Eq. 10 (Table 1)

Environmental carrying capacity
The model assumes that the settling rate of larvae is density dependent, varying with the proportion of N SO (t) to environmental carrying capacity (K) of a surface area a, which is the maximum number of C. intestinalis that the system can accommodate per cm 2 multiplied by the total surface area (a cm 2 ) of mussel socks in a bay. The model used an estimate of 40 individuals per cm 2 (Ramsay et al. 2009) for K. Mussel sock density in PEI was estimated to be 630 mussel socks per acre in the 500-acre bay of Georgetown Harbour (AVC shellfish research group pers. comm.). Each mussel sock was assumed to have a cylindrical shape with a length of 180 cm and a diameter of 10 cm. The capacity adjusting factor γ(a,t) is the proportion of available surface area to the total surface area at time t. It is used to adjust the settling rate and is defined in Eq. 11 (Table 1).

Model validation
Model fit was assessed by comparing the modelled number of each C. intestinalis life stage to observed field data collected by the AVC shellfish research group during May to November in 2008 at Georgetown Harbour, PEI. Two datasets relating to larval concentration, and population development were used in the assessment. A third dataset, population recruitment, is provided in the supplementary materials. For larval concentration, data collection was carried out using the larval sampling method described in . In brief, a 150-litre water sample in the water column between water surface and the first 2 meter depth was collected using bilge pump every two weeks from the study area and concentrated to 10 millilitres. This was then evaluated under a stereo microscope to identify and estimate the number of larvae. The data thus provided a representation of larval concentrations across the season. Due to these differences in measurement scale, direct comparison between the modelled output and observed data was not carried out and validation was based on a comparison of temporal patterns.
For population development, the number of C. intestinalis accumulating during the reproductive season was investigated by deploying fifteen 100 cm 2 PVC plates at 2-3 meter depth below the water surface on May 1 st , 2008. One plate was then randomly retrieved every two weeks to evaluate the numbers of C. intestinalis present at that time point. The samples were visually evaluated, so individuals smaller than 5 mm were not included in the count (for details see Ramsay et al. 2009). The data thus represented the total number of settled C. intestinalis per cm 2 over the season which was compared to the modelled N VO . Based on the extensive field experience of one of the co-authors (Davidson) it was estimated that only around half of the juvenile population would be visually detectable in a field setting and that the dead recruit stages do not take up space since they detach from the surface after they die.
Population recruitment data were collected according to the method used by Ramsay et al. (2009). PVC plates measuring 100 cm 2 were left at 2-3 meter depth below the water and retrieved after a two-week period. The numbers of recruiting C. intestinalis were identified under a dissecting microscope. This procedure was repeated every two weeks over the study period (May to November, 2008). These data provide estimates of the numbers of early recruiting C. intestinalis over time, but can only sensibly be compared to the modelled numbers in the recruit stage for the initial two weeks when the recruitment occurred.

Sensitivity analysis
Sensitivity analysis was carried out to analyse the influence of each parameter on the N SO using a relative sensitivity function ( ) which is the percentage of change in modelled output relative to a certain percent change in input. Each parameter (p) was increased or decreased by 20% of its default value one at a time, except for L Asp and L Aau that were changed simultaneously. was calculated and averaged over time to obtain an average relative sensitivity index ( ̅ ) for each parameter (Eq.12; Table 1). F(t) denotes modelled N SO at any time t when a parameter value is varied, while F b (t) refers to the modelled N SO when all parameters were set to their default values.

What-if scenarios
Four temperature conditions were modelled for what-if scenarios: baseline (replicating the temperature from Georgetown Harbour in 2008), cold year, long summer, and warm summer. Three parameters with high ̅ values, together with environmental carrying capacity and the drop-off rates of live juvenile and adult stages were further evaluated for their influences on N SO . These parameters were varied by a 20% increase or decrease on their default values and the outcomes assessed under the four temperature conditions.

Results
The observed and modelled average daily temperatures of Georgetown Harbour from May 2008 to May 2009 are illustrated in Figure 4. The modelled temperature was 3.3 ˚C at the start of the model with a mean of 7.1 ˚C, had a minimum of -2.7 ˚C and reached a maximum of 16.9 ˚C in late August.
The modelled total number of egg (eggs×10 9 ), larva (larvae×10 9 ), as well as the abundances per cm 2 of recruit, juvenile, adult, and visible occupying stages (N VO ) are illustrated in Figures 5-9. Each stage started to become active in early June when the temperature reached 8 ˚C, but the numbers were so low that they can hardly be detected in the summary plots. The modelled numbers of eggs rose from mid-August and reached a peak of 47×10 9 eggs in mid-October ( Figure 5). The number of larvae began to increase just after the rise in egg abundance, as would be expected, reaching a peak of 4.8×10 9 larvae around mid-September at around the same time as the observed larval counts reached their maximum ( Figure 6). The shapes of the observed and modelled larvae abundance over time are broadly similar though there is a limitation in comparing their magnitudes, as the two quantities are represented on quite different scales. Recruits followed a similar pattern to larvae, once again reaching a peak (4.5 recruit·cm -2 ) in mid-September ( Figure  7). The abundance of juveniles began to rise in late August and reached its peak (4.3 juvenile·cm -2 ) in early October, while the abundance of adults increased from early September and reached the highest levels (1.9 adult·cm -2 ) in mid-December (Figure 8). A comparison between N VO for the  observed and modelled data is shown in Figure  9. The observed N VO gradually increased from late July until mid-October with a rapid increase in late October. The modelled N VO broadly followed the shape of the observed curve, reaching a plateau at an abundance of around 5 individ.·cm -2 . However, the modelled N VO started to increase a month later than the observed data and showed a much more rapid rise after this initial increase than was the case for the observed data.
The modelled outputs using parameters fitted to the observed temperatures are presented in Figures S1-S5. The outputs of each stage appear to show an initial peak or early rise around late July (recruit and juvenile stages) to early August (egg, larval, adult, and N VO stages). Additionally, the modelled outputs (bases observed temperature) result in values around 5-6 times higher than those seen in the outputs using a simple sine curvebased temperature model and as high as two orders of magnitude for egg and larval stages. Figure 10 demonstrates the average relative sensitivity indices ( ̅ ) calculated from Eq. 12 (Table 1) for the 20% increase/decrease models at the baseline temperature. It shows the impact of changes in parameter values on the modelled output. The further the value of ̅ is from zero, the more influential a parameter is. The sign of ̅ explains the direction of the modelled output with respect to changes in an input parameter. For instance, the ̅ of 2.88 associated with a 20% decrease in spawning interval, the most influential parameter to the model (Figure 10), will generate on average a modelled output (N SO ) that is 57.6% higher than when the default parameters are used. In contrast, the effect of increasing the spawning interval by 20%, will generate a 28.8% decrease in the output. It can also be seen that the sensitivity in modelled output to changes of up to 20% in the mortality rate of juveniles and natural drop-off of dead     Table 2).  Table  1).  Warm summer juveniles and adults was very low, with ̅ values close to zero ( Figure 10). The effects of any two parameters can be compared by dividing ̅ of one parameter by the other. For example, the effect of a 20% reduction in G L (T˚) is 1.15 times greater than the effect of a 20% reduction in G E (T˚). To show the temporal variation of relative sensitivity index ( ), which cannot be seen in Figure 10, the plot of for each parameter over time when parameters were decreased by 20% is also shown (Figure 11). It is not until Day 40 that the values start to show an increase or decrease, and this change continued up to around Day 160, after which point they remained constant. Parameters with positive such as those related to the development time appeared to have higher magnitudes when compared to those with negative values ( Figure  11). Mapping values over time for a 20% decrease in parameter values, indicates a similar and opposite trend to those found for the 20% increase model (data not shown).
The four temperature scenarios explored are shown in Figure 12. Figure 13 illustrates the N SO stages for a baseline temperature scenario based on 20% variation in three parameters with high ̅ values: spawning interval (G SI ), development time of larva (G L (T˚)), and number of laid eggs per spawning (α), as well as an increase to 1% and 2% of the drop-off of live juvenile and adult (µ J , µ A ). As expected, increases in G SI or G L (T˚) caused a reduction in N SO , while an increase in α increased the N SO compared to default value (see Figure 13). However, when these variations were explored in a warm summer year (data not shown) the overall change in N SO was marginal by comparison. Similarly changes in environmental carrying capacity (K) resulted in little or no change in N SO stages for baseline and cold year scenarios. On the other hand there were more pronounced changes in the output for both long and particularly warm summer scenarios ( Figure  14) when K was altered. The results also indicate that the model is highly sensitive to changes in temperature condition. Looking at the default values (dash lines in Figure 14), the modelled N SO varied from only 0.004 individuals·cm -2 in a cold year ( Figure 14B) to the maximum capacity of 40 individuals·cm -2 in a warm summer ( Figure  14D). Figure 15 applied variation to the %dropoff of live juvenile and adult (μ J and μ A ) to demonstrate its effect under different temperature conditions on the N SO stages. When comparing %drop-off between the two temperature conditions, warm summer showed higher N SO than long summer for every level of %drop-off. Additionally, the decrease in N SO for a warm summer occurred later and with a larger relative change than was observed for the long summer scenario.

Discussion
The C. intestinalis populations model has demonstrated a capacity to address a number of the objectives of this study. It is flexible and can be adapted to a range of different temperature conditions. The model, in general, provided similar outputs to the observed data based on a comparison of temporal patterns. Although differences in scale between the observed and modelled larval counts prevented any direct comparison, the model provided an accurate prediction as to the timing of both the growth and the decline of the larval stage.
For the recruit stage focussing on the first two weeks of recruitment period, the model using a simple sine curve to represent sea water temperature was unable to capture an initial moderate rise in mid-July. The likely explanation for this is that the temperature model did not capture the high temperatures seen during late June and July which would have affected the settling rate. When fitted to the observed temperature profile from Georgetown Harbour in 2008, as opposed to the simulated baseline (sine-wave model), the model was able to capture this initial rise in the recruit population ( Figure S3).
A separate study of recruitment patterns of C. intestinalis took place in the Montague River, PEI in 2006 (Ramsay et al. 2009). The observed water temperature from the end of May to December in 2006 ranged from 6 ˚C to 18 ˚C which was quite similar to the modelled baseline temperature used in our study, though the model estimates were consistently around 1 ˚C lower. This field study reported the first recruitment of recruits in the second week of June when the temperature was nearly 9 ˚C and found one recruitment peak when the temperature was at its maximum (17.7 ˚C) in late August. Our model indicated a similar single recruitment peak pattern, though the peak was reached around one month later as the temperatures were not so high in the current study as compared to temperatures in the Montague River in 2006. This supports the argument that the model can adequately predict the recruitment timing of C. intestinalis given suitable temperature profiles.   As noted in Materials and Methods, comparing the observed and modelled recruitment data cannot strictly be justified, except during the first two weeks when recruitment occurs. This is due to the fact that in the observed data the surface on which recruits were counted was always based on fresh plate, which would result in higher estimations of settling rate when compared to the modelled data. Furthermore the study design used to collect the observed recruitment data performed sampling every two weeks, which may not be frequent enough as the larval stage will last for only around 10 days at the temperatures involved. In addition our model assumes that there was no external pressure affecting the modelled population, i.e. that all recruits derive from eggs and larvae produced by modelled adults in the population. However, within the observed data such external pressures are unknown and may therefore result in significant differences between the observed and modelled data.
The population development data (detailed in materials and methods) represented C. intestinalis stages that were attached on a surface and were of detectable size (larger than 5 mm) over the season. Due to the limitation of information on the life stages in the observed data (i.e. proportions of each stage were not identified) it was not easy to compare the modelled visible surface occupying stages to this dataset by simply adding up the numbers of juvenile, adult, dead juvenile and adult stages, since the modelled juveniles might have included individuals that were less than 5 mm in size. To validate the model against this population development data, the model was built under the assumption that only half of the juvenile stage individuals would be of detectable size. The model did not reflect the modest rise in N VO seen early in the season in the observed data for the same reason that it could not simulate the initial rise of recruits, which are the source of juveniles, as discussed previously. The model also illustrated that the population remained relatively static after its peak in the middle of October. The abundance stopped increasing because the individual growth rate of C. intestinalis in cold temperatures is very low (Dybern 1965;Yamaguchi 1975), yet it did not significantly decrease because the dropoff of dead individuals is also low (AVC shellfish research group pers. comm.).
Temperature was modelled using a single sine term, though an equation utilising three sine terms provided a better fit. This was because the study aimed to create a model that explains how C. intestinalis populations behave under a range of temperature conditions, which requires a model that can be easily modified to different contexts and is not over-fitted to one specific set of temperatures. On the other hand, if the objective is to make specific predictions, real water temperature data from a given year or season may provide better results. The single sine term model provides the flexibility of changing parameter values in a sensible way, but in this case study failed to capture the higher temperatures that occurred during July and early August which will influence the modelled temperature-dependent flow rates. However, fitting the parameters to the observed temperatures resulted in an earlier increase in the population of each stage (when compared to those seen using the simple temperature 1% (warm summer) 2% (long summer) 2% (warm summer) model); indicating that the model can produce adequate outputs for different temperature scenarios. The estimation of temperature-dependent parameters assumed linear interpolations when no values were reported between 6 ˚C and 26 ˚C. Where temperature was beyond this range, imputation, using either the nearest value or linear extrapolation, was carried out. As can be seen from Figure 2, a reasonable number of estimates exist at a range of temperatures for development times of egg, larval and juvenile stages which allowed for a sensible degree of interpolation, except at lower temperatures. However, there appeared to be more discontinuity when considering estimates of the percentage of individual successfully making the transition to a new phase for various life stages ( Figure 3) and therefore the interpolations adopted are inevitably more open to debate and refinement. In practice this was only a concern at lower temperature as sea water temperatures in the PEI coastal area rarely rise above 23 ˚C. Nonetheless, a reasonable amount of evidence that C. intestinalis do not develop at temperatures below 6 ˚C mitigates this as a serious concern.
The plateau patterns observed in the modelled output (Figure 9 and 13-15) were a consequence of two factors: temperature and space availability. At high temperatures, the life stages grow quickly and rapidly reach maximum capacity. Although the population can still grow, there is no space available to accommodate the new recruit stages. In low to moderate temperatures (e.g. the modelled baseline and cold year scenarios), the population tends to grow more slowly. In these scenarios the maximum capacity is not reached prior to a time at which the temperature begins to decrease and thus limits the growth in the population. This strong relationship between growth rate and temperature is a characteristic of ectothermic organisms (Guarini et al. 2011), such as C. intestinalis. Under warm weather conditions such as the warm summer scenario presented in this study, the populations will grow faster and reach the maximum capacity very quickly (Gillooly et al. 2001). In contrast, when modelling the population under a cold weather scenario very few individuals successfully develop, indicating that the temperatures observed in this study represent values close to the lower thermal range limit for C. intestinalis. This is in agreement with Dybern (1965) who found no Ciona species in the sub-Arctic and Arctic regions, where temperature records are seldom higher than 3-4 ˚C This study used a relative sensitivity function to evaluate the influence of changes in parameter values on the overall outcomes of the model. Although this method is known to have limitation, it is a relatively simple way to compare the effects of different parameters (Smith et al. 2008). Under the baseline temperature scenario, the model is particularly sensitive to development time and percentage of individual successfully making the transition to a new phase early life stages (i.e. egg and larva), as well as to spawning interval and number of eggs laid per spawning. As expected increasing spawning interval or development time slows down the growth of the C. intestinalis population, while increasing the number of eggs laid per spawning positively affects population growth. The sensitivity of the model to changes in the percentage drop-off of dead juvenile and dead adult stages was low; however, these drop-off rates only relate to the dead stages. A range of mitigation strategies to control C. intestinalis focusing on the removal of the occupying stages have been suggested elsewhere (Carver et al. 2003;Carver et al. 2006;Edwards and Leung 2009;Gill et al. 2007). The predictions from this model suggest that changes in environmental carrying capacity have a larger impact on population growth under warm summer or long summer-like conditions as compared to what would be the case in cold years. Although drop-off of live C. intestinalis rarely occurs naturally (AVC shellfish research group pers. comm.), the model indicates that changes in this parameter can have major impacts on modelled outputs. The results suggest that increased dropoff of live C. intestinalis, as would be the case under certain mechanical treatments, can act to limit population growth and is worthy of further investigation in combination with space and time control.
The use of this model is currently limited to a one-year scenario. To use this model for multiyear scenarios, factors related to mortality during the winter would be required to adequately model the correct number of initial adults at the start of a new yearly cycle. The population may increase exponentially in a subsequent year if there is very low mortality during winter. On the other hand if high mortality of adults occurs there would be few initial adults to initiate the reproduction cycle, resulting in an outcome not dissimilar to the single year scenario modelling in this paper. Therefore, the effects of temperature on physiological rates and development stages of this species, particularly in the colder winter months, require further study.
Overall, this mathematical model provides reasonable predictions around the dynamics of C. intestinalis populations on mussel farms in PEI. This approach should prove useful for farm management and can be adapted to model populations in different region or of other invasive species. Future studies will explore its application to an evaluation of the effectiveness of combining treatment and space management at different temperature profiles to develop mitigation strategies for the control of C. intestinalis populations and to improve bay management plans that might be implemented by mussel producers.