Author Posting. © Ecological Society of America, 2015.  This article is posted here by permission of Ecological Society of America for personal use, not for redistribution.  The definitive version was published in Ecology 96 (2015): 2027-2028, doi:10.1890/14-1900.1

Solow, Andrew R.

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Ecology, 96(7), 2015, pp. 2027–2028 2015 by the Ecological Society of AmericaRiver dolphins can act as populationtrend indicators in degradedfreshwater systems: commentANDREW R. SOLOW1The record of historical sightings of a species providesa basis for inference about its population status. Inmany cases, however, it is not possible to reconstruct areliable sighting record. In an interesting paper, Turveyet al. (2012) used the recollections of the most recentsightings by a number of individuals in local ﬁshingcommunities to compare population declines amongfour species in the Yangtze River, the idea being that themost recent sighting is more memorable than earlierones. Brieﬂy, Turvey et al. (2012) found that theempirical distributions of the most recent sightings ofthree of these species had similar declining upper tails,suggesting a common pattern of population decline, butfound no such decline for the remaining species eventhough its population was known to be declining. Thisraises a general question about the relationship betweenthe distribution of the most recent sightings and theoverall distribution of sightings. The purpose of thiscomment is to address some aspects of this question and,in particular, to show that, even in simple situations, thisrelationship is somewhat complicated.Let the random variables T1, T2, . . . , Tn be thesighting times for a single individual over the observa-tion period (0,T ). These sightings are assumed to arisefrom a Poisson process with unknown rate function b(t). 0 that is directly related to population size. It is astandard statistical result that, conditional on theirnumber n, these sightings represent independent obser-vations from a distribution with probability densityfunction (pdf)f ðtÞ ¼ bðtÞZ T0bðuÞdu0  t  T ð1Þ(e.g., Cox and Lewis 1966) with cumulative distributionfunction (cdf) F(t). LetTðnÞ ¼ max Tj; j ¼ 1; 2; . . . ; n  ð2Þbe the most recent sighting time for this individual.It is another standard statistical result that the pdf ofT(n) isgtðnÞ¼ n3Fn1tðnÞf ðtðnÞÞ ð3Þ(e.g., David and Nagaraja 2003).Consider now a number m of independent sightingrecords arising from this model, all of which contain nsightings. The collection of the most recent sightingsextracted from these records is precisely a randomsample of size m from the distribution with pdf given inEq. 3. The question is how the behavior of the pdf g ofthe most recent sighting times is related to the pdf f ofoverall sighting times, which, by assumption, is directlyrelated to population size. To be more speciﬁc, I willfocus here on the relationship between the signs of thederivative of g and f both evaluated at T. From Eq. 3g0ðTÞ ¼ n3 f 0ðTÞ þ nðn 1Þf 2ðTÞ: ð4ÞTwo general points arise. First, the only term on theright-hand side of Eq. 4 that can be negative is f 0(T ). Itfollows that, if g0(T ) is negative, then f 0(T ) must also benegative and, by assumption, the population is decliningat the end of the observation period. Second, theconverse is not true: it is possible for f 0(T ) to benegative so that the population is declining at the end ofthe observation period, but g0(T ) to be positive so thatthe pdf of the most recent sightings is increasing.Moreover, provided f 0(T ) is ﬁnite and f (T ) is positive,this is bound to occur for large enough n.It is instructive to consider some examples. Supposethat the size of a population is constant over theobservation period so that, conditional on n, thesightings by each individual are uniformly distributedover (0, T ). For convenience, here and following, I willtake T¼ 1 so that, in the uniform case, f (t)¼ 1, 0  t 1. The pdf of the most recent sighting isgtðnÞ¼ n3 tn1ðnÞ ð5Þwhich, provided n . 1, increases with t(n) with g0(T ) ¼n(n  1). That is, for a constant population size, thedistribution of the most recent sighting actually increas-es with time.Suppose next that the sighting rate declines linearly overthe observation period at rate b. The pdf of sighting time isf ðtÞ ¼ 1þ b2  b3 t ð6Þwith 0  b  2 where the upper bound ensures that thesighting rate is positive over the observation period. In thiscaseg0 ðTÞ ¼ nbþ nðn 1Þ 1 b2 2ð7ÞManuscript received 5 October 2014; revised 16 December2014; accepted 22 December 2014. Corresponding Editor: B. E.Kendall.1 Woods Hole Oceanographic Institution, Woods Hole,Massachusetts 02543 USA. E-mail: asolow@whoi.eduJuly 2015 2027COMMENTSwhich can be shown to be positive ifb,2ðn ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2n 1p Þn 1 : ð8ÞSo, for example, if n¼ 5, then g0(T ) . 0 if b , 1. Toput this into context, if b ¼ 1, the sighting rate declinesby two-thirds over the observation period. For large n,the right-hand side of Eq. 8 approaches 2 so that g0(T ). 0 for all values of b.As a ﬁnal example, suppose the sighting rate declinesexponentially at rate b. In this case, conditional on n, thesighting times follow a truncated exponential distribu-tion with pdff ðtÞ ¼ b expðbtÞ1 expðbÞ 0  t  1: ð9ÞIt is straightforward to show for this model that ispositive ifexpðbÞ. n1: ð10ÞThe quantity on the left-hand side of Eq. 10 is theratio of the sighting rate at the end of the observationperiod to the sighting rate at the beginning. So, forexample, if n¼5, the pdf of the most recent sighting timeincreases as long as this ratio is greater than 0.2. Forlarge n, the right-hand side of Eq. 10 approaches 0 sothat g0(T ) is again positive for all values of b.In the much more realistic case where the numbersn1, n2, . . . , nm of sightings in the different records aredifferent, the pdf g of the most recent sightings is amixture of pdf’s each of the form in Eq. 3gtðnÞ¼ f ðtðnÞÞmXmj¼1njFnj1ðtðnÞÞ ð11Þand it is straightforward to show thatg0ðTÞ ¼ nf 0ðTÞ þ nðn 1Þf 2ðTÞ ð12Þwhere the over bar indicates the average. As before, ifg0(T ) , 0, then f 0(T ) , 0, but not the converse. Specificresults such as those above about the sign of g0(T ) aremore complicated and depend on both the average andspread of the sighting numbers. Briefly taking a broaderview, if each of the most recent sightings is paired withthe overall number of sightings, then it would bepossible to fit a parametric model of f and to test, forexample, the null hypothesis of a common f among acollection of populations.The main result of this comment has been that thebehavior of the record of most recent sightings of apopulation depends on both the underlying populationtrend and the numbers of overall sightings by differentobservers. Returning to the paper of Turvey et al.(2012), this suggests that, without further assumptionsabout these overall sighting numbers, the similar rates ofdecline in most recent sightings among three of theYangtze species need not imply similar rates ofpopulation decline. By the same token, the absence ofa decline in most recent sighting rate for the remainingspecies need not imply a different rate of populationdecline.AcknowledgmentsHelpful comments from Bruce Kendall, Samuel Turvey, andan anonymous reviewer are acknowledged with gratitude.Literature citedCox, D. R., and P. A. W. Lewis. 1966. The statistical analysis ofseries of events. Methuen, London, UK.David, H. A., and H. N. Nagaraja. 2003. Order statistics. JohnWiley & Sons, Hoboken, New Jersey, USA.Turvey, S. T., C. L. Risley, L. A. Barrett, Y. Hao, and D.Wang. 2012. River dolphins can act as population trendindicators in degraded freshwater systems. PLoS ONE7:e37902.COMMENTS2028 Ecology, Vol. 96, No. 7

River dolphins can act as population trend indicators in degraded freshwater systems : comment

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River dolphins can act as population trend indicators in degraded freshwater systems : comment

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