Indicators from LEK Interviews

Researchers interviewed 81 fishers in the towns of Puerto Peñasco, Golfo de Santa Clara, Rodolfo Campodónico, Bahia Kino, Desemboque, and Puerto Libertad (Sonora) between April and June 2008 and September 2008 and February 2009. This represents around 2% of the estimated 3,800 artisanal fishers in the northern Gulf of California (P. Turk-Boyer, Centra Intercultural de Estudios de Desiertos y Océanos [CEDO], unpublished data). Interviewees ranged in age from 20 to 89 years and had expertise in the following gear types: gill net, cast net, shrimp and finfish trawls, longline, hand line, harpoon, compressor diving, and traps. On average, respondents had 28.5 years of fishing experience. The interview forms used are available from C. H. Ainsworth.

Interviewees were asked to characterize the abundance of fish, invertebrates, mammals, birds, and reptiles into one of three categories (low, medium, or high) for each of the six decades since 1950. Species were aggregated into groups; the structure was chosen to be compatible with species-aggregated, trophicmodeling approaches for fisheries research. The format of the groups generally aggregates invertebrate species, and it represents species of commercial interest and ecological importance, such as keystone species, in more detail. A statistical test is used here to show whether fishers' perceptions of species abundance varies depending on whether or not they rely on the stock for their livelihoods. For this, respondents were divided into two pools, those that targeted a given species group and those that did not. Relative abundance trends from LEK interviews were developed for 24 species groups (Table A.1 in the appendix) for each decade and used as input in the fuzzy logic procedure.

In addition to the abundance trends, simple indicators were collected in the form of yes/no questions. The questions for each species group were whether the respondents had noticed localized depletions or extirpations of the species and whether they had noticed a reduction in average body size. The yes/no responses were obtained for 36 species groups (Table A. 1). They were not recorded by decade; a single response was assumed to represent the net change over a respondent's career.

Catch per Unit Effort

Catch-per-unit-effort trends were determined based on CPUE or catch-and-effort data from Mexican government statistics (Diario Oficial de la Federación 2004, 2006 [see Table A.2]; CONAPESCA 2009). Additional information was obtained from unpublished statistics (V. M. Valdéz-Ornelas and E. Torreblanca, Pronatura Noroeste). Data from a concurrent study to estimate unreported catch provided port-level information (S. Perez-Valencia, CEDO, unpublished data), while a study of fishery logbooks also provided high-quality information for a small number of vessels (J. Torre and M. Rojo, Comunidad y Biodiversidad, unpublished data). Finally, the remaining data gaps were filled by other literature sources listed in Table A.2.

Catch data for major commercial species were taken from Diario Oficial de la Federación (2004, 2006) and CONAPESCA (2009), while the unpublished information, surveys, logbooks, and literature sources provided estimates of catch for minor targets and bycatch species. Where statistics were available by state, total catch in the northern gulf was assumed equal to that of Sonora and Baja California combined. The catch values also include estimates of unreported catch and discards, which are largely based on expert opinion (L. E. Calderón-Aguilera, Centra de Investigatión Científica y de Educación Superior de Ensenada, personal communication). Effort trends in artisanal fisheries were based on the number of vessels operating (Galindo-Bect 2003); this approach has the advantage that it can capture trends for unregistered vessels, which may constitute as much as one-third of the fleet, judging by recent aerial surveys (Rodríguez-Valencia et al. 2008). Other effort series were located for lobsters (Vega-Velásquez 2006), small pelagic organisms (Arreguín-Sánchez et al. 2006), groupers (Arreguín-Sánchez et al. 2006), totoaba Totoaba macdonaldi (Lercari and Chávez 2007), and shrimp (Galindo-Bect 2003). As a last resort, effort was based on human population growth rate in two Sonoran cities, Puerto Peñasco and San Felipe (INEGI and Government of Sonora 2008a, 2008b). Catch and effort references are listed in Table A.2. Full documentation of development of catch, effort, and CPUE series is available from the author (cameron.ainsworth@noaa.gov). The CPUE series were developed for 34 species groups (Table A.1) as annual trends (units vary) and averaged to the decade level for input into the fuzzy logic procedure.

Harvest Model

A logistic growth model with harvests provides biomass (B) estimates at each year t, as in equation (1) (Schaefer 1954; Hilborn and Walters 1992),

The model was initialized at 1950 using B₀ and run to 2008, calculating annual biomass estimates B_t in t/km². These were then averaged to produce decadal values (B₁₉₅₀, B₁₉₆₀, …, B₂₀₀₀) and used as inputs to the fuzzy logic algorithm. Series were developed for 26 species groups (Table A.1). Between averaging these biomass values over decades, using fuzzy sets to describe them, and combining these estimates with other abundance series, the method presented here should be insensitive to the large uncertainties involved in these calculations.

Vulnerability Index

As a final indicator of species abundance, the relative vulnerability to fishing of each exploited species group is estimated based on the method of Cheung et al. (2005). Those authors combined life history characteristics, including maximum length, the Von Bertalanffy growth constant, maximum age, fecundity, age at first maturity, and natural mortality, in a fuzzy logic framework to produce a composite vulnerability-to-fishing score. The index was shown empirically to predict species status better than common alternative proxies. Their method was recreated here. Life history information for Gulf of California species was taken from FishBase and other sources (see Tables A.3, A.4); then, using the published membership functions and expert rule set, the data were collated to produce a final vulnerability score for each species in a procedure analogous to the one described in this article. The only notable difference between this treatment and the one by Cheung et al. (2005) is that the geographic range of a species was not considered as an indicator of vulnerability because it is not relevant when considering a localized area like the Gulf of California. The vulnerability scores were calculated for individual species and averaged to the level of species groups. The available life history data allowed the calculation for 21 species groups (Table A.1). The scores are time independent, and so the same score is used for each decade in the analysis.

Fuzzy Logic Overview

Fuzzy set theory, also called fuzzy logic (Zadeh 1965; Bellman and Zadeh 1970), emulates an expert's judgment by combining inputs through a heuristic IF-THEN rule matrix to reach a conclusion regarding the data. It is a means of computing with words (Zadeh 1999), where “linguistic variables,” representing a wide range of possible data types, combine according to relationships similar to “rules of thumb” contained in a rule matrix. However, where classical logic requires that a variable be categorizable into a single class (e.g., a Boolean variable belonging exclusively to a yes or no category), the linguistic variables in a fuzzy set can hold varying degrees of membership in multiple classes. For example, if “purple” is a fuzzy set that describes all colours composed of red and blue light, then indigo, violet, and fuchsia could be said to hold increasing membership in the “red” category relative to the “blue.”

A Worked Example

A simple example (Figure 1), in which two data streams (abundance from interviews and CPUE) are combined to produce relative abundance, demonstrates the fuzzy logic procedure. The procedure begins by assigning an analog abundance indicator (x), such as the abundance score derived from interviews (Figure 1A). The analog indicator is then translated into fuzzy sets containing several linguistic abundance categories (Figure 1B). In the case of abundance from interviews, there are five categories ranging from low to high. The partial membership (µ) in each abundance category (n) is determined by consulting membership functions. The membership function µ_n(x) € [0, 1] describes the degrees of membership of x in linguistic categories 1 though N, where Σ_n µ_n(x) = 1. Piecewise linear membership functions are used for simplicity throughout this study.

Membership in the linguistic variable categories determines what rules operate (or “fire”) in the rule matrix. In this paper, the strength at which the rules fire is determined by the algebraic sum of the intersecting memberships (Figure 1C). Many conclusions may be reached simultaneously (Figure 1D) with varying degrees of belief (firing strengths). Each time a cell in the rule matrix fires, it strengthens our belief in the corresponding conclusion. There are 50 different possible conclusions. Numbered 1 to 50, each conclusion represents a linear interpretation of abundance, so that a conclusion of 40 indicates twice the abundance of one numbered 20. Whenever a cell is fired, the partial membership that elicited that action is added to a running total for the corresponding conclusion category (Figure 1E). In this way, conclusions reached repeatedly (or fired with large partial memberships) will accrue a high score.

After all the information is passed through the matrix for a certain group—period combination, we are left with an array of 50 elements; the partial memberships in each category represent our relative confidence, or degree of belief, in that abundance conclusion. The partial memberships are normalized as in Figure 1F, after which they are passed as inputs to the defuzzification membership function (Figure 1G). Defuzzification is the process by which partial memberships are converted to a single number representing relative abundance (i.e., a “crisp” number that is no longer part of a fuzzy set). It is therefore the reverse of the procedure described earlier to calculate memberships (i.e., Figure 1B). The defuzzification function in Figure 1G is a simplification; the actual one employed has 50 categories with triangular functions. The centroid-weighted average method of Cox (1999) is used, in which we multiply the centroid of each category (0.02, 0.04, 0.06, …, 1.00) by the relative weighting (or confidence) in that conclusion to obtain a weighted average between 0 and 1 that represents the relative abundance for that species group and period.

FIGURE 1.

A worked example of the fuzzy logic method. A two-dimensional analysis is presented for clarity; it processes two data streams, abundance from interviews and CPUE (values are hypothetical). The algorithm presented here is repeated for each species group and time period.

FIGURE 2.

Fuzzy set membership functions. These relationships convert an analog indicator of abundance (x-axis) to partial memberships in linguistic abundance categories (y-axis). The partial memberships sum to 1 and represent the degree of belief in the abundance categories. Panels (A) and (B) show the membership in abundance categories under minimum and maximum variance between interview scores (low [L], medium—low [ML], medium [M], medium—high [MH], and high [H]); panel (C) shows the membership in CPUE categories (very low [vL], low [L], medium [M], and high [H]); panels (D) and (E) show the membership in yes/no (Y/N) categories under minimum and maximum variance between interview scores; and panel (F) shows the membership in biomass categories predicted by the harvest model (underexploited [U], moderately exploited [M], fully exploited [F], overexploited [O], and critically overfished [C]).

Fuzzy logic implementation.—In the worked example, the rule matrix uses two dimensions (i.e., abundance and CPUE in Figure 1D). Scaling up, the implementation for both abundance and exploitation indicators uses three dimensional matrices. The next section describes the membership functions used to characterize the six data series involved. Later, combinatorial rules, the rule matrices, and defuzzification are discussed.

Membership functions.—The abundance information obtained in interviews was evaluated according to the membership functions in Figure 2A, B. The x-axis in Figure 2A, B represents the analog abundance (x) score obtained from the interviews. It is averaged across respondents, “low” responses being assigned a value of 0, “medium” ones being assigned 0.5, and “high” ones being assigned 1.0. An average abundance score of 0.5 could be achieved through (at least) two scenarios: one in which every fisher reported medium abundance and one in which half of the fishers reported high abundance and half reported low abundance. To account for agreement between respondents, a dynamic membership function was used in which the angle subtended by the triangular functions increases from a minimum (Figure 2A) to a maximum (Figure 2B) if there was high or low agreement between respondents, respectively.

As a measure of agreement, the standard error of the mean was determined for each group-period combination as SE = σ²/n, where n is the number of respondents. Variance (σ²) was calculated assuming a binomial distribution in which the majority response category (low, medium, or high) was considered “correct” and all other responses were considered “incorrect”; thus σ² = np(1 - p), where p is the fraction of correct responses. Using the SE in this way corrects for the varying number of respondents per decade (e.g., fewer people contributed to the 1950s estimates than to the 2000–2009 estimates). The SE was next standardized so that the group—decade combination that had the best agreement (lowest error) adopted the membership function in Figure 2A, that with the poorest agreement adopted the function in Figure 2B, and other responses adopted intermediate functions, where slopes and intercepts were scaled linearly between the extremes.

Membership was evaluated in each of five fuzzy set abundance categories: low (L), medium—low (ML), medium (M), medium—high (MH), and high (H). Categories L and H use trapezoidal forms: beyond a certain threshold, full membership was assigned to these extreme categories. This allowed us to ignore the influence of a small number of responses that contradict the majority belief. Although the level of fishing effort, fishing skill, gear efficiency, catchability, and other factors will affect the amount of catch a fisher obtains for any given level of fish abundance, this methodology trusts that fishers can integrate over a wide range of externalities and thus have valid cognitive models of resource supply. Averaging their responses to obtain x should also eliminate many possible biases. For this reason, we restricted the analysis to consider only group-period combinations that had at least eight respondents.

The membership function used to categorize (normalized) CPUE per species group and decade is provided in Figure 2C. It categorizes the analog CPUE score into four linguistic categories: very low (vL), low (L), medium (M), and high (H). The membership functions used to categorize “yes/no” depletion and body size indicators into yes (Y) and no (N) categories again use a dynamic membership function to reflect the level of agreement between respondents (Figure 2D, E). As with the abundance indicators from interviews, the form of the membership function varies according to SE, which was calculated assuming a binomial distribution of yes and no answers. The membership function used to evaluate biomass predictions from the logistic harvest model is provided in Figure 2F. Here, membership is evaluated in five linguistic categories: overexploited (O) or critically overfished (C) if the decadal biomass value (e.g., B₁₉₅₀, B₁₉₆₀, …, B₂₀₀₀) was below B_MSY, or underexploited (U), moderately exploited (M), or fully exploited (F) if biomass was above B_MSY. In the case of the Cheung et al. (2005) vulnerability index, our input membership function is identical to their output (defuzzification) membership function; it is equivalent to Figure 2C with four linguistic categories: low (L), medium (M), high (H), and very high (vH).

Combining the data series.—Having determined the partial memberships in linguistic categories through the use of membership functions, we next consult the rule matrices to combine the information; Tables 1A and 1B show the abundance and exploitation matrices, respectively.

Membership in the three indicators (on X, Y, and Z axes of each matrix) leads us to a conclusion regarding the abundance for each species in each time period. The conclusions are found inside the matrix cells (color coded in Table 1). Each cell fires with a strength proportional to the algebraic sum of the intersecting memberships. All axes are assumed to be independent, and the strength of memberships is combined using the probability-OR operator for three variables, namely,

Various parts of the rule matrix were accessed for each time period (1950, 1960, 1970, 1980, 1990, and 2000). If the value of the depletion indicators (i.e., stock depletion or body size reduction) is “yes,” this has the effect of lowering the relative abundance score of the conclusion for recent periods (1980 to 2000) but increasing the abundance score of the conclusion for older periods (1950 to 1970) (i.e., the slope between 1950 and 2000 becomes more negative; Table 1). Matrices for the intermediate periods (1960, 1970, 1980, and 1990) are not shown, but they apply a smooth linear transition between the extreme values in 2000 (top row of Table 1) and 1950 (middle row of Table 1). If the value of the depletion indicator is “no,” then the bottom row of Table 1 is accessed regardless of decade.

After all data series pass through the matrix, we are left with 50 different abundance conclusions with varying degrees of belief (partial memberships), as described in the worked example. The partial memberships are combined through defuzzification and are converted to a single crisp number representing the relative abundance. This process is repeated for each species group and period to produce time series of relative abundance that can be compared with observational series (Table 2).

Summary.—The data series used here for abundance indicators consisted of decadal abundance trends from LEK interviews, annual CPUE data averaged to decades, and a yes/no population-level depletion indicator from LEK interviews that referred to all decades. The data series used for the exploitation indicators consisted of annual stock biomass from a logistic growth harvest model averaged to decades, a vulnerability-to-fishing index that referred to all decades, and a yes/no body size change indicator from LEK interviews that referred to all decades. These numerical indicators were translated into memberships in ordinal linguistic categories (e.g., low, medium, or high) using membership functions, where membership in each category represents our relative belief in that category's being true. The memberships in each category determined the firing strength of cells in a rule matrix, where each cell leads to a particular conclusion regarding species abundance. Finally, a crisp abundance score was determined through defuzzification, the reverse of the process that created the membership scores from the numerical indicators. Abundance per decade is plotted for each species group, representing its stock status since 1950.

TABLE 1.

Heuristic rule matrices used by the fuzzy logic algorithm to combine data sources. Two parallel matrices deal with abundance and exploitation indicators, respectively. Abundance indicators include abundance estimated by interviews, catch per unit effort (CPUE) estimated from literature values, and stock depletion suggested by interviews. Exploitation indicators include exploitation status suggested by the Schaefer harvest model, vulnerability to fishing based on life history characteristics, and body size reduction suggested by interviews. The conclusions resulting from these X—Y—Z linguistic variables are identified in the cells. They represent a linear abundance index that ranges from 1 to 50. The presence of depletion or size-reduction indicators upgrades the abundance conclusion for past periods (e.g., 1950) and downgrades it for recent periods (e.g., 2000). Matrices for the intermediate periods (1990, 1980, 1970, and 1960; not shown) apply a smooth transition between the extremes 2000 and 1950. Abbreviations are as follows: L = low, ML = medium—low, M = medium, MH = medium-high, H = high, vL = very low, vH = very high, C = critically overfished, O = overexploited, F = fully exploited, M = moderately exploited, u = underexploited, Y = yes, and N = no.