Método propuesto

2.3

Literature review on selectivity estimation in fisheries

In the context of trawl surveys, the capture probability of the fishing net is usually referred to as thegear selectivityorcatchability. It has been found in a large number of studies that the gear selectivity is typically a function of fish size, which in most cases is measured by the length of fish (see Munro & Somerton, 2001; Madsenet al., 1999, for example). This section briefly reviews a general statistical methodology for the estimation of size-selection (see Millar & Fryer, 1999, for details).

This section starts with a list of explicit definitions of selection processes, specified with underlying assumptions and limitations. Then a family of logistic regression models is introduced to estimate the gear selectivity as a function of fish length. Note that in this section, only length of fish is considered as a predictor for selectivity. However in practical applications, there might be other predictors depending on the data from the experimental survey.

2.3.1 Definition of selectivity curves

Selectivity curves are used to quantify the probability of a fish being captured given its size. Intuitively, the selectivity curve can be expressed as a function of fish size for the fishing net. The size of a fish is usually measured by its length, because fish length is easy to measure in the field. As a result, the gear selectivity is often modelled as a function of fish length.

The size selectivity can be partitioned into three parts, and each part is defined by a selectivity curve. The three parts of the whole selection process are defined as

• thepopulation-selectioncurve,s(l), is the probability that a fish with lengthl

from the population is captured, which quantifies the differences between the catch and the entire population;

• theavailable-selectioncurve,a(l), is the probability that a fish with lengthlis captured given that it was available to the gear, which quantifies the differences between the catch and the fish available to the gear; and

18 Chapter 2. Capture probability in anglerfish survey

• the contact-selection curve, r(l), is the probability that a fish with length l

is captured given that it contacted the gear, which quantifies the differences between the catch and the fish coming into contact with the gear.

The above three selection curves differ from each other in terms of the population from which the fish are selected, and the above list of definitions is presented in order so that the population or sub-population which the catch is relative to is decreasing in terms of its range, i.e., the whole population, the sub-population available to the gear and the sub-population contacting the gear. The three selection curves defined above are related by

a(l) = r(l)×P{fish of lengthlcontacts the gear given

that it is available to the gear}, (2.2)

s(l) = a(l)×P{fish of lengthlis available to the gear}. (2.3) Note that the probability components P{}on the right-hand sides of (2.2) and (2.3) depend on the fish behaviour, and these probabilities vary for different species (see Millar & Fryer, 1999, p. 92).

The above relations are then illustrated in consideration of the anglerfish survey data. Given the net retention probability defined in Section 2.1, the contact-selection curve for anglerfish survey is the retention probability r(l). Figure 1.2 shows that, in addition to the fish from the haul path between the wings, the fish that contact the gear also include the fish herded by the doors into the path between the wings, i.e., some fish were herded from v2i intov1i in Figure 2.1. Therefore, the available- selection curve a(l) for anglefish survey is r(l)×(v1i +hv2i)/(v1i +v2i), where (v1i +hv2i)/(v1i +v2i) is the probability that a fish of length l contacts the gear given that it is available to the gear. Therefore, the population-selection curve s(l) for anglerfish survey is

s(l) = r(l) | {z } contact × v1i+hv2i v1i+v2i | {z } available × v1i+v2i As | {z } population selectivity , (2.4)

whereAsis the surface area of the stratums.

The population-selection curve for anglerfish survey given in (2.4) is the probability that an anglerfish is captured from the population within stratums. Note that (2.4) is

2.3 Literature review on selectivity estimation in fisheries 19 also the probability for haulithat a fish located somewhere in stratumsis included in the sample. This probability is the inclusion probability in the Horvitz-Thompson estimator of abundance (Horvitz & Thompson, 1952), which will be used in Part III for the anglerfish abundance estimation. A more general study of the Horvitz- Thompson method for abundance estimation will be presented in Part IV.

2.3.2 Length-based retention curves

As the primary tool used for anglerfish application, a family of logistic regression models have been introduced by Millar & Fryer (1999) for the estimation of the contact-selection curve,r(l). In the context of anglerfish survey data, anglerfish may go beneath the footrope of the towed gear, and the fish finally being captured are those fish that end up with staying in the cod-end of the gear.

In most applications of fisheries research, the contact selectivity is affected by the size and/or the shape of the mesh openings in the cod-end. The larger the fish are, the more easily they are retained in the cod-end. Therefore, it is usually assumed that the contact-selection curve is a monotonically nondecreasing function of fish size, which is usually measured by length of fish. This leads to the usage of logistic curves for a mathematical description of contact-selection curve of a fishing net. The following lists three types of logistic curves for a fish of lengthl:

• linear logistic is expressed as

r(l) = exp(β0+β1l) 1 + exp(β0+β1l)

, (2.5)

equivalently

logit(r(l)) = β0+β1l, (2.6)

which is symmetric about the median oflwith an upper asymptote of unity; • asymptote-logistic is expressed as r(l) = γ exp(β0+β1l) 1 + exp(β0+β1l) , (2.7)

20 Chapter 2. Capture probability in anglerfish survey

which can be viewed as an extended form of linear logistic curve given by (2.5), with an extra parameterγallowing the asymptote to be less than1; and • asymmetric logistic is expressed as

r(l) = exp(β0+β1l) 1 + exp(β0 +β1l) 1 κ , (2.8)

which is another extended form of the above linear logistic curve withκmod- elling the asymmetry of the curve about the median of lengthl.

The three logistic curves, (2.5), (2.7) and (2.8), are solutions to simpler cases of the Richards curve given by (2.13) in Appendix 2.A. The Richards curve is also known as the generalized logistic curve, and is one of the most flexible functions to model the growth rate for population dynamics. An alternative way to construct flexible logistic regression models is to use a wide parametric class of link functions introduced by Aranda-Ordaz (1981).

When using logistic curves to model contact selectivity as a function of fish size, the situation is usually not as complicated as when modelling the growth rate using (2.13). Therefore, simpler cases of the Richards curve are considered in Appendix 2.A to obtain (2.5), (2.7) and (2.8), from the simplest case to more complicated ones. In this way, we do model assessment step by step to avoid fitting too complex a model to the data we have.