Model framework

A general CJS (Cormack 1964; Jolly 1965; Seber 1965) type model framework for CMR data was first defined, in which a cohort of marked lobsters are released, and

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subsequent sampling used to recapture the ‘survivors’ (those that remain within the study area) (See; Appendix III, for RCode provided by M. Bell). Recaptures and newly marked individuals were re-released at approximately the same place and time of capture, and this process was repeated over eight separate fishing occasions.

Consequently each marked individual may be captured several times over the course of the study, generating a capture history (CH) where one of three observed states was recorded for each day after first release: 0 not observed; 1 captured and released; -1 captured and removed from the study. A value of 0 was recorded if no traps were hauled that day or if the tagged lobster was not observed on a haul occasion. A value of -1 was recorded if the lobster was so damaged that it would impact the survival or catchabilityof the individual, and was then removed from the study site.

The probability of a particular CH occurring was the product of a series of probabilities of the possible fate of the individual over each day following marking and first release (Lebreton, Burnham et al. 1992). Given an individual’s availability within the capture area during the study (‘Area over which traps exert an influence and the area around

traps from which a lobster could potentially enter the area of influence’ (Bell, Eaton et al. 2003)), three possible fates could be defined: (1) the lobster does not enter a trap,

but remains in the capture area; (2) the lobster enters a trap and is observed; (3) the lobster does not enter a trap and permanently emigrates from the capture area. The probabilities of one of these fates occurring can be described by three parameters describing fishing and population processes between release occasions; probability of capture ( ), probability of survival ( ) and fishing effort ( ). Given that the study period was short it was assumed that movement processes would dominate over survival processes. Therefore is hereafter referred to as site fidelity (not emigrating) (Lebreton, Burnham et al. 1992).

For short-term trap fishing, where traps are hauled and immediately reset, the capture process is complex and considered continuous (i.e. capture could occur at any point between one haul and the next). Effectively the model treated an occasion, the hauling and setting of traps, as a single point in time. Calculations within the model outlined here, were therefore conducted on a continuous scale, with capture process

parameters cast in continuous terms and population processes described as instantaneous rates operating simultaneously (Dunnington, Wahle et al. 2005). To

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generate continuous terms the initial probabilistic parameters and (constrained between 0 and 1 to give meaningful probabilities (Lebreton, Burnham et al. 1992)) were transformed logistically to continuous parameters of catchability ( ) and

mortality ( ) respectively. However, due to the assumption that movement processes dominated survival processes, is hereafter referred to as rate of loss. Continuous parameters were used for model calculations and the construction of the reduced m- array tables (Table 2.2). Re-casting the parameters from probabilistic to continuous forms, also allowed for easier incorporation of the unequal sampling intervals (Bell, Eaton et al. 2003; Dunnington, Wahle et al. 2005).

The CMR model required the following key assumptions to be made (Lebreton, Burnham et al. 1992): (1) tagged individuals mix freely with the untagged population; (2) tags remain present and are always detected in the catch or the rate of tag-loss is known; (3) capture and tagging does not alter the probability of survival or behaviours that would change the probability of capture, relative to untagged or non-captured individuals; (4) individuals that leave the study area do not return to the study area; and (5) interspecific interactions within and around the trap do not affect capture probability, i.e. effort exerted and probability of capture is equal across all traps and all animals.

The model (See; Appendix III) was formulated in terms of CMR data summarised in tabular reduced m-array format (Table 2.2). Each row represents recaptures for a

Table 2.2 The reduced m-array format of CMR data. R, is the number of lobsters released at occasion i, and m

the number of lobsters recaptured on occasion j.

Occasion Releases Recaptures Not

recaptured j = 2 j = 3 … J = 1 … ∑ = 2 … ∑ … … … … … ∑

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particular release cohort. The release totals (Ri) comprised both newly tagged lobsters

and recaptures, and multiple recaptures were pooled with first recaptures from a new release cohort (Burnham, Anderson et al. 1987). CHs recorded in a reduced m-array allowed for expected (𝐸) values of each recapture-cell ( ; Table 2.2) to be

calculated. For example, expected value for CH [101] (i.e. released, not observed, and then observed again), for occasions -1, and respectively, may be calculated as:

[Eq. 2.1]

where is the number of lobsters that were released on occasion and recaptured on occasion ; , is the total number of marked lobsters released on occasion , and the final two terms are probabilities ( ) of the two fates leading up to being

recaptured on occasion . is the probability of remaining available for capture within the capture area (i.e. not dying or emigrating) from the occasion of release , up to and including occasion -1, without being captured. can be expressed in terms of the parameters of catchability ( ) and rate of loss ( ) as:

[Eq. 2.2]

where is the effective fishing effort over the time between occasion and occasion -1, and the catchability on occasion -1. As fishing effort, , effectively scales by the time over which traps are set, soak time is not included in this

expression ( ). The second expression ( ), includes , the soak time ( ) in days between occasions and -1, and , the rate of loss on occasion -1. (Eq. 2.1) is the probability of being caught, given the lobster’s

availability in the capture area, between occasion and . This can also be termed the rate of harvest, expressed in terms of continuous parameters as:

[Eq. 2.3]

Equation 2.3 assumes that , , and occur simultaneously and compete with each other. The last term, expresses the proportion of losses due to fishing, and the first term expresses the total number of losses. This expression is derived from the

[ ] = = ( ∑( ) ) = ( ( ( )))

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Baranov catch equation, in which the term is equivalent to fishing mortality and is equivalent to natural mortality during the time between occasions (Baranov 1918). Given the assumption that the fate of each individual lobster is independent, but the identity of parameters between individuals within the same release cohort are the same, the appropriate model for the data is a multinomial one (Lebreton, Burnham et

al. 1992); this gives the probability of any particular combination of a number of fates

for the various cohorts. The kernel of the log-likelihood of parameter for the model, , can be calculated as:

[Eq. 2.4]

where the probability of the ‘recaptured’ cell , in the reduced m-array table (Table 2.2), [ ], can be summarised using expectations from equation 2.1:

[Eq. 2.5]

The probability of the ‘not recaptured’ cell for row , [ ∑ ], the number of lobsters released in cohort that are never seen again, can be calculated as:

[Eq. 2.6]

Estimated parameters from the model were scaled by effective fishing effort, creating meaningful constraints between soak times of different length (See; section 2.2.5). To find the values of and that maximise the log-likelihood value, a quasi-Newton algorithm was used (Press, Flannery et al. 1989). For the purposes of interpretation, the and parameters were transformed back to scales of probability; leading to parameters of probability of capture per effective effort exerted by traps on occasion , (not the same as ) and probabilities of site fidelity on each day of the interval leading up to occasion , :

[Eq. 2.7] [Eq. 2.8] = = ∑ ( ∑ [ ] ( ∑ ) [ ∑ ] ) [ ] = [ ] [ ∑ ] = ∑ [ ] =

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The variance-covariance matrixes for the logistically transformed parameters were calculated numerically and for derived parameters the delta method was used to obtain approximate standard errors (s.e.) (Press, Flannery et al. 1989; Burnham and Anderson 2002; Dunnington, Wahle et al. 2005). The goodness-of-fit (GoF) tests, generated in programme MARK use the second part derivative method to generate derivatives numerically (Cooch and White 2011).

Once estimates of parameters , , and were obtained, the population size (𝑁) could be estimated through the following calculation, allowing for appropriate scaling of catch data per occasion, :

[Eq. 2.9]

where 𝑁is the population of lobsters over the entire soak time from which the observed catch at occasion , 𝐶, is drawn. The variance covariance matrix was then used to obtain s.e. and confidence intervals (CI) for the population size estimates for each sex over each occasion after the first occasion.

Sexes were treated as two separate groups during this study, males (Group 1) and females (Group 2), this allowed for differences in catchability and rate of loss to be modelled and population estimates between sexes compared. Due to the short time period of the study, each population estimate was essentially a separate estimate of the same population; making it possible to derive a single, mean population estimate for each group. As s.e. could not be aggregated into the mean, the standard deviation of all estimates was used to gain s.e. and 95% CI’s of the range of values.

The population estimates was given in terms of abundances within the capture area. To transform these estimates into densities requires some information about the size of the capture area from which the catches were drawn..

To estimate capture area, the trapping area must first be estimated; defined as the area within which the probability of capture of a lobster, during the deployment time of the trap, was greater than 0 (Bell, Addison et al. 2001). Trapping area of a single trap was estimated as the area of a circle with radius equal to that of both the area of bait influence ( ) and average home-range size of a lobster ( ) (Fig. 2.3). Theoretically this represents the maximum distance a lobster could travel to enter a trap (Watson, Golet

=

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et al. 2009). However, in reality this area is influenced by many factors, such as soak-

time, movement and foraging behaviour, habitat type, water movement, temperature and other functions that may vary spatially, temporally and among individual lobsters. It was beyond the scope of this study to determine trapping area from fieldwork. Therefore, an estimate for , from Watson et al. (2009), of 11m radius was used. However, from Watson et al. (2009) of 30m radius was considered small. H.

americanus may move about 100-300 m d-1 (Krouse 1980; Watson, Vetrovs et al. 1999), and further albeit during seasonal migrations, reportedly walking 1-4km-day and covering 30-100km in one season (Dow 1974; Fogarty, Borden et al. 1980a; Campbell and Stasko 1985; Campbell and Stasko 1986; Estrella and Morrissey 1997). For European lobster Moland et al. (2012) found home-ranges to have 10-250m radius (n=10). Therefore with some reservations related to the country and species

differences within the limited evidence available, trapping area radius of a single trap was set at a nominal 100m; this takes into account the uncertain size of the lobster’s home-range, but does mean that there is considerable overlap between the trapping areas of individual traps. A minimum convex polygon was drawn around the

experimental traps on this basis, covering an area of ca. 0.42km2 (Fig. 2.4).

Figure 2.3 Theoretical trapping area (A) of a single trap (black square ), the home range

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