kinas mote final

THE IMPACT OF INCIDENTAL KILLS BY GILL NETS ON THE

FRANCISCANA DOLPHIN (PONTOPORIA BLAINVILLEI) IN

SOUTHERN BRAZIL

Paul G. Kinas

ABSTRACT

The franciscana dolphin, Pontoporia blainvillei, is endemic to the western South Atlantic Ocean. Its distribution is restricted to waters up to 30 m depth, making it vulnerable to anthropogenous impact. In southern Brazil the artisanal gillnet fisheries have increased since the early 1980s, and entanglement mortality of franciscanas has become a source for concern. Strandings have been documented for about 20 years, but the impact of incidental captures has remained unknown. In the 1990s numbers of franciscanas incidentally killed by the artisanal gillnet fleet, population size, and the intrinsic growth rate were estimated. I integrated all available information into a population model in order to quantify the impact of incidental kills. The statistical analysis was conducted within a Bayesian framework to maintain coherence while current biological knowledge was weighted with observational data. My analysis indicated a 99% probability that the population is decreasing. For analysis of future impact of the incidental kill the population was defined as being at 'quasi collapse' when it reached 10% of its current size, and the probability of quasi collapse within 30 years was calculated under alternative scenarios. Results indicate that current levels of entanglement mortality cannot be sustained and that protective measures are needed.

The franciscana dolphin, Pontoporia blainvillei, is a small cetacean endemic to the western South Atlantic Ocean, ranging from Espírito Santo, Brazil (18°25'S) (Moreira and Siciliano, 1991) to Golfo Nuevo, Península Valdés, Argentina (42°35'S) (Crespo et al., 1998). Its distribution is restricted to waters up to 30 m depth (Pinedo et al., 1989), making it vulnerable to anthropogenous impact. Because of continued incidental mortality in fishing nets throughout most of its range (see, e.g., Praderi et al., 1989), the franciscana seems to be the most threatened small cetacean species in the western South Atlantic Ocean.

In southern Brazil the coastal gillnet fisheries have increased substantially since the early 1980s (Haimovici, 1997), and the entanglement mortality of franciscanas caused by these fisheries has become a source of concern. The estimation of incidental mortality and population abundance as well as morphometric and genetic studies to define stock discreteness have been established as research priorities for this species by several meetings and workshops carried out over the last decade (see, e.g., Perrin et al., 1989; Crespo, 1992; Pinedo, 1994). The effect of gill nets and high-seas drift nets on marine mammals has been a source of concern elsewhere as well (Mangel, 1993; Lennert et al., 1994; Caswell et al., 1998).

Morphometric (Pinedo, 1991) as well as genetic studies (Secchi et al., 1998) suggest two populations, one to the north of Santa Catarina state (I) and the other to the south (II). On the basis of a phylogeographic approach (Dizon et al., 1992), Secchi (1999) suggested a subdivision of the southern population (II) into two stocks for management purposes, the northern stock inhabiting the coast of Rio Grande do Sul state and Uruguay (IIa), and the southern stock on the coast of Argentina (IIb). My study will refer only to the stock designated IIa.

An aerial survey conducted in southern Brazil in 1996 (Secchi et al., 2000) provided first estimates for the stock (IIa). The estimated number of franciscanas per square kilometer was obtained by distance sampling (Buckland et al., 1993) in a small

area close to Rio Grande and than multiplied by the area inhabited by the stock to provide abundance estimates. These numbers must be used with caution because the area covered by the transects might not be representative for the range of the stock's distribution.

Kinas and Secchi (1998) estimated the numbers of franciscanas killed by entanglement in bottom-set gill nets from data obtained with volunteer fishermen from the gillnet fishing fleet operating out of Rio Grande. Depending on model choice, bootstrap 95% confidence intervals indicated total catches of 188,318 or 114,220 individualsper year. Data were inconclusive about best choice of model. The potential dangers of relying on volunteers is acknowledged and discussed by the authors. On the basis of a series of informal observations about the fleet, they argue that there are no apparent differences between volunteers and other fishermen.

Age-specific fecundity rates mx and longevity w estimates were used by Secchi

(1999) to estimate the populations’ intrinsic growth rate r through Lotka’s equation

∑

To examine uncertainties in this estimate of r, he used a Monte Carlo approach. In each replicate of the simulation, the (unknown) age-specific survival rates lx were randomly

selected from a range of plausible survival curves. Also, observed values of mx were

combined with associated sample sizes to define parameters of appropriate Beta distributions from which sample fecundity rates were taken. Values of w were also chosen randomly from a prespecified interval ranging from 17 to 21 years. The resulting Monte Carlo sample of intrinsic growth rates r reflected the uncertainties induced by the lack of knowledge about all these parameters. In most of the scenarios analyzed by Secchi (1999) the values of r suggested a declining stock.

Strandings of franciscanas have been documented by Pinedo and Polacheck (1999) for the period from 1979 to 1998, with a four year gap from 1988 to 1991. The authors confront these data (Table 1) with an indicative fishing effort calculated for the fleet to observe that, although documentation effort has increased, stranding rates have fallen, suggesting a decrease in population size (Fig. 1).

Finally, the real impact of the coastal gillnet fishery operating off Rio Grande remains unknown. The goal of my study is to integrate published information about the population biology of franciscanas in an effort to determine whether current levels of entanglement mortality can be sustained by this dolphin stock. Analysis was conducted within a Bayesian framework (Berger, 1985; Gelman et al., 1998) intended to preserve coherence as biological knowledge about this dolphin population was weighted with observational data. Ellison (1996) provides a detailed discussion on the advantages of Bayesian inference in ecological research and decision making.

METHODS

To study the impact of entanglement rates, I had to predict the dynamic of population size into the future. I therefore began by assuming a deterministic exponential model for population growth

(1) 1 (1 t)

r t

t N e h

where r is the population growth rate, Nt the population size at the beginning of year t,

and ht the fraction of the population dying in year t as a result of entanglement.

The stranding-rate data in each year t, denoted It, was related to population size Nt by the observation model

(2) Y

t t qN e

I =

where q is a stranding coefficient and Y a Gaussian random variable with mean zero and standard deviation σ. The values of t range from t = 1 (for 1979) to t = 20 (for 1998); values It are missing for t = 10 to t = 13 (i.e., years 1988 to 1991).

Because 1994 is the year for which additional information is available on incidental catch, it was taken as the starting year for the population growth model. Letting C16 denote the incidental catch in 1994 and assuming for the moment that C16 and h16 are known, I defined the population size at the beginning of that year,

(3) 16 16 16 _h C N =

Starting with N16, population growth was projected into the future by means of Eq. 1 or into the past after Eq. 1 was rearranged to yield

(4) ) 1 ( 1 t r t t h e N N − = −

I further assumed that changes in entanglement rates were proportional to the changes in indicative fishing effort before 1994 and have remained constant ever since. This assumption is acceptable in view of the fisheries history and is summarized as follows: (5) ⎩ ⎨ ⎧ > < = − 16 for 16 for 16 16 16 t h t h h t t β

After analyzing the various indicative-effort measures calculated for different target species and fishing gear by Pinedo and Polacheck (1999), I fixed β = 0.715 to reflect the approximate average annual increase of 72% in effort between 1979 and 1994.

Prior distributions.—Five parameters need estimation in the model: r, h15, C15, σ and q. With the exception of r, all are nonnegative, and h15 is further restricted to values between 0 and 1. To avoid computational difficulties, I decided to work with appropriate transformations, so the components of the five-dimensional parameter ψ are defined as follows:

(6) ) 10 ln( ) ln( ) ln( ) ( logit 3 5 2 4 16 3 16 2 1 ⋅ = = = = = q C h r ψ σ ψ ψ ψ ψ

For each of these transformed parameters, I defined a Gaussian prior distribution with mean and standard deviation (µi, τi) i = 1 to 5. These prior hyperparameters were

chosen as follows:

Monte Carlo simulations for Lotka's equation were performed by Secchi (1999) under different scenarios and produced a range of possible distributions for r. My choice of (µ1 = -0.11, τ1 = 0.124) is centered at the mean of his 'most realistic' scenario, allowing about 5% of values to be more extreme than any observed value over all scenarios that were considered.

Secchi (1999) gave rough estimates of the possible ranges for h16 on the basis of density estimates of stock IIa and documentation available in the region regarding incidental catches. A beta distribution with parameters 15 and 286 (i.e., mode and mean of 0.047 and 0.050, respectively) spreads well over the range of reported plausible values. After generating 1000 values from this beta distribution, I found that the logit-transformed values followed an approximate Gaussian distribution with parameters (µ2 = -2.97, τ2 = 0.263).

Kinas and Secchi (1998) presented estimates of C16, the total number of franciscanas incidentally killed in 1994 in the same region for which stranding rates are available. Depending on whether the assumed model for numbers of dolphins killed per trip is Poisson or geometric, the bootstrap estimates of C16 follow approximate lognormal distributions with mean ± standard deviation of 5.52 ± 0.138 and 5.01 ± 0.148, respectively. A logarithmic transformation of simulations from a random mixture of these distributions was used to produce the parameters I used in the prior (µ3 = 5.27, τ3 = 0.291).

Variance σ2 could not be larger than the overall variance observed in ln(It). This

value of 0.398 was taken as the mean of a lognormal distribution. A linear regression of ln(It) with respect to t had residual variance of 0.297; this value was taken as the mode.

Combining the two resulted in a Gaussian prior for ψ4 with parameters (µ4 = -1.02, τ4 = 0.443), a very noninformative prior for this parameter.

It was hardest to get a prior distribution for q because about this parameter we have the least information. On the basis of extreme (lowest and highest) values of C16 and h16 and a preliminary estimate of abundance in 1996, I was able to set its order of magnitude. I found the range from 0.5 to 1.5 to be appropriate and sufficiently noninformative for q × 103. Using a lognormal distribution with mean 1.0 and mode 0.8 for q × 103, I established the prior hyperparameters (µ5 = -0.075, τ5 = 0.386) for ln(q), which allows for about 17% of values outside the given range.

Likelihood function.—Let u = {Ut = ln(It); t = 1, . . . ,9 and 14, . . . ,20} denote the

log-transformed stranding rates data on which the posterior distribution of the parameter ψ is to be conditioned. On the basis of Eq. 2, the kernel of the log-likelihood function is

(7) =− −

∑

where the sum is over all values t listed in the data set u, and

(8) ln( 10 3 )

t t = q⋅ ⋅N

−

η

denotes the expected value of Ut.

Posterior distribution.—I implemented a Metropolis-Hastings (MH) algorithm (Gelman

following steps:

Step one. I started at the mode (µ1, µ2, µ3, µ4, µ5) of the prior distribution p(ψ) defined by a multivariate Gaussian distribution with diagonal covariance matrix Τ = diag(τ1, τ2, τ3, τ4, τ5) and, using a simplex algorithm, searched for the approximate location of the posterior mode, by minimizing the score function S(ψ)

(9) S(ψ)=−

(

)

which is proportional to the log-posterior probability function.

Step two. The outcome from step one was used as the starting value in a more refined Newton-Raphson optimization. New iterations resulted in a new minimum that was reached at the point ψˆ . This procedure also provided an estimated covariance matrix (10) 1 ) ( ˆ _{= −} − Σ H

where H is the 5-×-5 matrix of second derivatives of the log-posterior density evaluated at the mode ψˆ .

Step three. Using the estimates ψˆ and Σˆ as location and scale parameters in a five-dimensional Student's distribution with nine degrees of freedom (to allow for heavy tails), I sampled to obtain six starting points for the MH procedure.

Step four. I ran each of the six series for 20,000 points and kept the last 2000. In this way I obtained 6 × 2000 = 12,000 points as my final posterior sample of ψ.

Checking for convergence and for model fit.—I ran various series in step four to check

for convergence of the Markov Chain. Following Gelman et al. (1998), I examined convergence by calculating R ; i = 1, . . . , 5 and determining whether these values _i

were close enough to one (values below 1.2 are generally acceptable). For each parameter ψi, the value Ri is defined as

(11) W B n W n n R 1 1 ₊ − =

where B and W denote between- and within-sequence variances for the last n = 2000 points in each of the J = 6 sequences. Formulas for B and W (Gelman et al. 1998: 331) are those from ordinary one-way analysis of variance.

The model was checked by comparison of the data to the posterior predictive distribution by way of the χ2 discrepancy T(u,ψ) described by Gelman et al. (1998),

(12) =

∑

(

)

The adequacy of the model (prior density, likelihood, or both) is suspicious if the posterior probability of obtaining a discrepancy at least as extreme as the observed

T(u,ψ) is small.

Population viability analysis.—To study the impact of current fisheries practices on the

traditional measure used in population viability analysis (PVA) is the time until extinction or, alternatively, the probability of extinction within a fixed time window. Further, I found it appropriate to define a population to be at 'quasi collapse' whenever it reached 10% of its current size, and I calculated the probability of quasi collapse for franciscanas within the next 30 years. For each of the 12,000 sample points from the posterior p(ψ|u) by way of the MH procedure, I projected the population model Eq. 1 into the future, recording the time (in years) until quasi collapse and the time until extinction in each case. In this way I incorporated all posterior uncertainty about population parameters into the analysis. I repeated the analysis for three scenarios: (1) current entanglement-mortality rates, (2) rates half their current level, and (3) no entanglement mortality.

RESULTS

The values for √Ri calculated according to Eq. 11 for each of the posterior

parameters resulted in values close enough to 1 (≤ 1.034) to suggest convergence. The 12,000 points are therefore assumed to be a representative sample from the posterior distribution p(ψ|u).

To verify the ability of the model to adjust the stranding data, I use these 12,000 points to calculate the probability that a discrepancy T(u,ψ) would be at least as extreme as the observed one. The probability that T(upred, ψ) ≥ T(u,ψ) was estimated from the proportion of cases satisfying the result. Because upred are random variables, the final proportion can differ somewhat in different trials. I generated various replicates and calculated an average probability of 0.526 (SE = 0.002).

The most important results of the present paper are the descriptive summaries of the posterior distribution for important parameters of the model given in Table 2. The probability that the stock is currently decreasing (i.e., Pr{intrinsic growth rate r < 0.0}) is 0.99 with mean intrinsic growth rate r estimated as -0.053. Uncertainty about the stock size in the study area is high, however, as indicated by the wide 95% posterior credibility interval for the population in year 1994, with lower and upper limits at 2688 and 8290 animals, respectively.

In the context of a population-viability analysis (PVA), I evaluated the performance of the population into the future under different protective measures. Times to reach quasi collapse and extinction are summarized in Table 3 and Fig. 2. The expected time until quasi collapse of about 25 years, if current levels of entanglement mortality are maintained, increases to 34 and to 54 years if these levels are respectively halved or reduced to zero. In comparison, the expected times until extinction are longer but follow the same pattern. The larger difference between mean and median, however, reflects increased asymmetry in times due to increased probabilities for large values. The probabilities of reaching quasi collapse in 30 years were 0.854 under current entanglement mortality rates, 0.496 if rates were reduced to half the current values, and 0.125 if entanglement mortality was eliminated completely.

DISCUSSION

The practical difference between the Bayesian analysis used here and more traditional approaches is the possibility of using probability distributions not only for observed data but also to describe uncertainties of parameters about which we wish to learn. In Bayesian terms, the prior probability distribution incorporates the information about these parameters available from other sources, and the time series of observed stranding data is modeled with the likelihood function. The posterior probability distribution, which is proportional to the product of prior and likelihood, is the resulting

probabilistic description of the unknown parameters conditional on observed data. It is the cornerstone in Bayesian inference. Therefore, the sample of 12,000 values of ψ taken from this distribution describes the relative plausibility of different parameter combinations so that in the final risk assessment (PVA) a wide range of parameter possibilities is weighted accordingly.

To check the fit of the model to data, I used a generalized goodness-of-fit χ2 statistic to compare simulated samples drawn from the posterior predictive distribution to the observed data. It is a generalization in the sense that this discrepancy measure depends not only on the data, as classical measurements do, but on the parameter as well. As pointed out by Gelman et al. (1998), in contrast to the classical approach, Bayesian model checking does not require special methods to handle nuisance parameters like q and σ2_{. By using posterior simulations, we implicitly are averaging} over all parameters in the model. A model is suspect if the tail probability of the discrepancy measure is near 0 or 1. The value calculated here (0.526) indicates that the observed data could have arisen by chance under the proposed model. It can further be seen in Fig. 3 that the fit of the expected stranding rates, when the parameter vector ψ is taken as the posterior mean, reflects adequately the general trend in the data.

The value of information contained in the observed stranding data for the reduction in uncertainties about model parameters can be evaluated from the extent to which the posterior distribution changes location and gains precision in comparison to prior specifications. From Table 4 it is clear that all five parameters gain in precision because the ratio between posterior and prior standard deviations SD(ψi|u)/SD(ψi) < 1 in

all cases. In particular, the relative reduction was most substantial for ψ1 = r, the intrinsic growth rate, with posterior standard deviation about 17% of the prior value. Important changes with respect to the location are given in the third column of Table 4, where the posterior means of ψ2 and ψ4 are shown to be reduced by 3 and 3.5 standard deviations, whereas those of ψ1 and ψ5 increased by 2.9 and 3.8 standard deviations, respectively. The least-affected mean was for ψ3, the logarithm of numbers incidentally killed in 1994, C16.

The population's intrinsic growth rate and incidental kills are modeled as separate effects: r and h. The high probability that r is negative results in a positive probability of extinction even if incidental kill could be stopped completely (h = 0). The value of r < 0 can reflect some biological unknown factor or be a consequence of the simplistic model, which does not take account of density dependence. In either case this interesting and most important question must be investigated further.

Population-viability analysis (PVA) provides estimates of important quantities in conservation biology, such as minimum viable population and expected time to extinction (Gilpin and Soulé, 1986). Ludwig (1999) identified and examined three possible defects in the method: (1) lack of precision of estimates, (2) sensitivity to model assumptions, and (3) lack of attention to important factors influencing extinction. In his conclusions he claims that a proper PVA should include estimates of likely ranges in parameter estimates and quantities regarding extinction.

The usefulness of a Bayesian approach to assess a population's viability or related probabilities was pointed out by Ludwig (1996) and applied in a decision analysis by Taylor et al. (1996) to classify the status of spectacled eiders. Performing a PVA using the posterior sample for the population parameters can incorporate various types of uncertainties and parameter correlations into the calculations of probabilities of extinction and expected time windows until the occurrence of some event.

Because uncertainty about abundance of the franciscana stock is high, I found it difficult to give an absolute minimum target stock size. Therefore I define 'quasi

collapse' as a target stock size in relation to current abundance. The concept is similar to Ludwig’s (1996) definition of 'quasi extinction', differing only in the percentages with respect to initial size—he uses 1%, whereas I chose a more conservative 10%. However, the results obtained here agree with his claim that parameter uncertainties have the effect of reduced precision in the estimates. Expected times to quasi collapse and to extinction have wide confidence intervals and long right tails (positive asymmetry) (Table 3), so excessive confidence should not be placed in these estimates.

The inclusion of demographic stochasticity and the risk of environmental catastrophes may affect the results by reducing the expected times until quasi collapse or extinction (Mangel and Tier, 1993). Simultaneously, a reduced stock might favor higher immigration rates and thus act on the expected times in the opposite direction.

Preliminary studies (Secchi, 1999) indicate that males may differ in spatial distribution from females and young, the latter tending to stay in shallower waters. Furthermore, work currently in progress seems to indicate that fisheries concentrating on different fish species are associated with different-entanglement mortality rates and that dolphin catches are aggregate. The possible increased risks to the stock caused by aggregated catches are pointed out by Mangel (1993). Models able to quantify the interplay of the components listed in the previous paragraph, as well as to incorporate spatial and sexual discrimination, are desirable and should be used when more data become available.

Despite the simplicity of the current model, however, and all the uncertainties in the estimates, some questions relevant to conservation biology can be readily answered because of the Bayesian framework in which the analysis was performed. First, to the question ‘what is the probability that the stock is currently decreasing?’ my results indicate that the answer is 'dangerously high (0.99)'.

CONCLUSION

The Bayesian analysis is an adequate framework for construction of a probabilistic model to describe uncertainties for parameters about which one wishes to learn. In this framework, relevant questions for conservation biology of franciscana can be answered through inference about the probability of stock quasi collapse. The stranding data provide an adequate fit to the model, indicating that they are plausible under this model and assumptions, but the lack of the data needed to fit alternative models precludes sensitivity analysis at the present stage.

Exclusions of demographic and environmental stochasticities are limitations of the current model. They may be causing positive bias in expected times until quasi collapse or extinction. An alternative age-specific model that allows for spatial structure and that includes compensatory mechanisms in its demography is necessary to quantify the impact of management actions that could be designed to protect franciscanas.

Much room for improvement remains in the quantitative modeling of the probability of stock collapse, mainly with the inclusion of density dependence in the dynamic of population size, but the overall result already indicates clearly that current entanglement-mortality rates are dangerously high and that protective measures are needed.

ACKNOWLEDGMENTS

I thank Eduardo Secchi, whose concern with the franciscana dolphin brought the problem to my attention. Three anonymous reviewers provided valuable suggestions and comments that greatly improved the manuscript.

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ADDRESS: Departamento de Matemática, Fundação Universidade Federal do Rio

Grande, C.P. 474, Rio Grande, RS, Brazil, 96201-900; E-mail <dmtkinas@super.furg.br>.

Table 1: Stranding rates (numbers/100 km) for the period September–December,

1979–1998, for the region in which gillnet fisheries take place and standardized indicative average effort. (1) From Pinedo and Polacheck (1999). (2) Adapted from Haimovici (1997).

Year Stranding rate (1)

Effort (2) Year Stranding rate (1) Effort (2) 1979 20.2 1.00 1989 — 1.50 1980 9.9 1.13 1990 — 1.71 1981 23.8 1.64 1991 — 2.42 1982 14.2 1.45 1992 3.7 2.54 1983 4.1 1.73 1993 8.1 2.41 1984 3.8 2.24 1994 4.8 4.19 1985 6.9 1.94 1995 2.5 — 1986 16.0 1.52 1996 6.3 — 1987 8.0 3.05 1997 6.2 — 1988 — 2.71 1998 4.5 —

Table 2: Summary statistics for the posterior distribution for some parameters of the

model (SD = standard deviation).

95% Credibility Interval

Parameter Median Mean SD Lower Upper

R -0.053 -0.053 0.021 -0.095 -0.010 h16 0.046 0.047 0.011 0.029 0.070 C16 214 222 57.6 133 351 N16 4655 4868 1420.0 2688 8290 q × 103 1.086 1.145 0.346 0.627 1.986 σ2 u 0.329 0.346 0.104 0.191 0.597

Table 3: Estimated probability of quasi-collapse within 30 years (column 3) and

summary statistics for time until quasi-collapse (QC) and extinction (ET) as derived from 12000 points sampled from the five-parameter posterior distribution for ψ.

Protective 95% cred. interval

Policy Target Pr(TQC≤30) Median Mean SD Lower Upper QC 0.854 23.0 24.7 7.0 16 41 Current (h) ET − 71.0 75.2 24.4 44 133 QC 0.496 31.0 34.0 17.9 20 66 Half (h/2) ET − 93.0 102.1 41.5 54 205 QC 0.125 44.0 53.9 39.1 25 151 None (h=0) ET − 132.0 151.4 72.1 69 352

Table 4.: Ratios between posterior and prior standard deviations and standardized

difference between posterior and prior expectations for the five parameters of the model.

Parameter sd(ψi| y)/ sd(ψi) (E(ψi| y)- E(ψi))/ sd(ψi|y)

ψ1 = r 0.169 2.86

ψ2 = logit(h16) 0.905 -3.00

ψ3 = ln(C16) 0.863 0.40

ψ4 = ln(σ2) 0.754 -3.52

Fig. 1: Observed stranding rates (numbers/100 km) and average standardized indicative

effort (equivalent days at sea by industrial trawling vessels). Obs.: Indicative effort was calculated for various target species (corvina and pescada) and for other and pair trawl, standardized and averaged.

Fig. 2: Observed stranding rates (numbers/100 km) and the prediction from the model

when parameters in the model are set at its posterior mode obtained from 12,000 simulations.

Fig. 3: Cumulative probability that population will reach 'quasi collapse' under three

scenarios: current catch rate (hcur); half of current catch rate (h/2), and zero catch rate (h

= 0). Distribution result from 12,000 simulations with the joint posterior distribution for the five-parameter model.