- Email: [email protected]
On inference about the introduction time of an introduced species with an application to the pine marten on Mull
Biological Conservation 159 (2013) 4–6
Contents lists available at SciVerse ScienceDirect
Biological Conservation journal homepage: www.elsevier.com/locate/biocon
Short communication
On inference about the introduction time of an introduced species with an application to the pine marten on Mull Andrew Solow a,⇑, Sugoto Roy b, Christina Bell c, Jo Milborrow b, David Roberts d a
Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Food and Environment Research Agency, Sand Hutton, York YO41 1LZ, UK c Scottish Natural Heritage, Cameron House, Albany Street, Oban, Argyll PA34 4AE, UK d Durrell Institute of Conservation and Ecology, University of Kent, Canterbury, Kent CT2 7NR, UK b
a r t i c l e
i n f o
Article history: Received 24 September 2012 Received in revised form 13 December 2012 Accepted 17 December 2012 Available online 13 January 2013 Keywords: Isle of Mull Pine marten (Martes martes) Sighting record Species introduction
a b s t r a c t Although there is general agreement that the pine marten was recently introduced to the Isle of Mull, there is uncertainty about when. According to a persistent local rumor, the species was clandestinely introduced in 1986. A novel statistical approach to inference about introduction time based on a sighting record is described and applied to assess the plausibility of this rumor. The results indicate that the sighting record is consistent with the rumored introduction time. Inference about the introduction time is useful in the management of introduced species for the light it can shed on the mode of introduction, post-introduction population growth and spatial dispersal, and the potential role of the introduced species in other observed ecological changes. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Reconstructing the history of the introduction of a non-native species to a particular location or habitat is of interest to both science and management. A case in point is the pine marten (Martes martes) on the Isle of Mull in the Inner Hebrides off the west coast of Scotland (Fig. 1). The pine marten is a member of the mustelid family that also includes minks, otters, badgers, and weasels. The species is native to forested regions of mainland Scotland and northern England and Wales, although its numbers have been reduced over the centuries by hunting and habitat loss (Langley and Alden, 1977; Proulx et al., 2004). The pine marten is semiarboreal and among its prey are nesting birds so that its current presence and potential future spread on Mull has raised concern about the impact on the island’s diverse avian fauna. Motivated in part by such concern, Scottish Natural Heritage recently funded a study of the history (and future prospects) of the pine marten on Mull. While the species is now established on Mull, no documentary evidence of its presence on the island exists prior to 1996. It was not listed in the Old Statistical Account for Scotland (Sinclair, 1800) or in later surveys of the island (Barrett-Hamilton and Hinton, 1913; Delany, 1961), nor is there a specimen from Mull in any of the major national museums or university collections.
⇑ Corresponding author. Tel.: +1 508 289 2746. E-mail address: [email protected] (A. Solow). 0006-3207/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.biocon.2012.12.026
Although there is general agreement that the pine marten was recently introduced – or possibly reintroduced after a long absence – to Mull, there is uncertainty about when and how. This uncertainty is magnified by the fact that even now the presence on Mull of this rare and elusive species is known only through occasional chance sightings. According to a persistent local rumor, a clandestine introduction of newly weaned juveniles occurred in 1986. Here, we modify and apply a method for inference about extinction time to test the plausibility of this rumor based on the sighting record of the pine marten on Mull. The results suggest that the sighting record is indeed consistent with the rumored introduction in 1986. This paper focuses on the specific problem of assessing the plausibility of a rumored introduction time of the pine marten on Mull. However, the methods described here can be used for more general inference about introduction time based on a sighting record. The importance of information about introduction time to the management of introduced species is discussed in the final section.
2. Methods Statistical inference about the introduction time of a non-native species from a record of its sightings is the mirror image of inference about the extinction time of an extinct species based on its sightings (Solow, 2005). Here, the term sighting is not restricted to visual sightings of individuals, but can include other forms of evidence of
5
A. Solow et al. / Biological Conservation 159 (2013) 4–6
specified introduction time. The observed significance level (or p value) for the uniformly most powerful unbiased test of Ho is:
p ¼ 2 minðGðt n Þ=Gðso Þ; 1 Gðt n Þ=Gðso ÞÞ
ð3Þ
Although it is of less interest here, the minimum variance unbiased estimate under the truncated exponential model is:
s^ ¼ tn þ nðn 1Þ
P½t=tn j¼1
Gðt n Þ n1 ðt jtn Þn2 ð1Þj1 j1
ð4Þ
As noted, inference about s depends on the behavior of the post-introduction sighting rate. For this reason, it is important to have a way to assess the adequacy of the truncated exponential model. This can be done by plotting the empirical cumulative dis^ ¼ #ftj 6 tg=n and the theoretitribution of the sighting times FðtÞ cal cumulative distribution: Fig. 1. Location of the Isle of Mull.
FðtÞ ¼ ð1 expðktÞÞ=ð1 expðksÞÞ
the presence of a species such as tracks, hair, and fecal matter (Stanley and Royle, 2005). The central question in inference about extinction is how far after the last sighting of a species must we go before safely concluding that it is gone. The question for introduction is how far before the first sighting of a species must we go before safely concluding that it was not yet present. The answer to this question depends on the behavior of the post-introduction sighting rate. If this rate, which depends on factors such as the size of the population and sighting effort, is low, then even a relatively long period without sightings does not imply that the species was absent. Conversely, if this rate is high, then even a relatively short period without sightings implies that it was absent. The basic statistical model underlying inference about extinction is that the sighting record follows a possibly non-stationary Poisson process with a rate function that is positive before extinction and 0 thereafter. The details of statistical inference about extinction time depend on what is assumed about this rate function. Here, we use a straight forward modification of a method for inference about extinction time in a declining population (Solow, 1993) to test the null hypothesis that the pine marten was introduced to Mull in 1986. Let 0 < t1 < t2 < < tn < s be the ordered sighting times of an introduced species with time measured backward from the present, so that t1 corresponds to the most recent sighting and tn to the earliest, and where s denotes the unknown time of introduction before which the sighting rate must be 0. We assume that these sighting times follow a non-stationary Poisson process with a rate function that increases exponentially at rate k following introduction at time s. This model is appropriate for a population that grows exponentially following introduction with a sighting rate that is proportional to population size. It should be emphasized, however, that it can arise in other ways. It follows from the properties of the Poisson process that, conditional on their number n, these sighting times represent an ordered sample from a truncated exponential distribution with probability density function:
f ðtÞ ¼ k expðktÞ=ð1 expðksÞÞ 0 6 t 6 s
ð1Þ
where k is an unknown parameter that governs the growth of the sighting rate over time. P Following Solow (1993), let T ¼ nj¼1 tj and define the function:
GðxÞ ¼ 1
½T=x X n ð1Þj1 ð1 jx=TÞn1 j j¼1
ð2Þ
and [T/x] is the integer part of T/x. Interest here centers primarily on testing a null hypothesis of the form Ho: s = so where so is a
ð5Þ
against t using the values of interest of k and s. This plot can be supplemented with pointwise upper and lower a/2-quantiles of ^ FðtÞ estimated by simulating sighting records of size n from F(t). When, as here, interest centers on an hypothesized value of s, a convenient value of k to use in constructing F(t) is found by solving:
1 s ¼ T=n k expðksÞ 1
ð6Þ
(Al-Athari, 2008). Inadequacies in the fitted model are indicated ^ by systematic departures of FðtÞ from the fitted distribution function and particularly from excursions outside the sampling bounds. 3. Results The record of sightings of the pine marten on Mull is given in Table 1. This record was compiled as part of the study supported by Scottish Natural Heritage of the history of this species on Mull. We took the end of the observation period to be February 2012 and therefore omitted the sighting in that month – its corresponding sighting time of 0 being an artefact – leaving a total of n = 11 sightings. The corresponding values of tj measured in months are also given in Table 1. As noted, the individuals rumored to have been introduced in 1986 were newly weaned juveniles. Based on the life history of the pine marten (Mead, 1994), we took the hypothesized month of introduction to be September 1986. This corresponds to a value of so of 305. The p value based on Eq. (2) is 0.44. By conventional standards, this result is not significant and the null hypothesis that the rumored introduction time is true cannot be rejected. This re-
Table 1 Sighting dates of the pine marten on Mull. Also reported are the sighting times tj in months prior to February 2012. Sighting number
Date
Time
1 2 3 4 5 6 7 8 9 10 11 12
July 1996 April 2000 January 2007 March 2008 May 2008 May 2010 June 2010 September 2010 October 2010 February 2011 March 2011 February 2012
187 142 61 47 45 21 20 17 16 12 11 0
6
A. Solow et al. / Biological Conservation 159 (2013) 4–6
CUMULATIVE PROBABILITY
1
early part of the record and is therefore more consistent with an even earlier introduction, although it is still not possible to reject the rumored introduction time of September 1986.
0.8
4. Discussions 0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
t (MONTHS) Fig. 2. Empirical cumulative distribution of sighting time (solid) along with theoretical distribution (dashed) and upper and lower 0.95 sampling bounds (dotted).
sult is insensitive to the month in 1986 of the hypothesized introduction. To assess the adequacy of the truncated exponential model on which this result is based, in Fig. 2, the empirical cumulative distribution of the sighting times is plotted along with the theoretical cumulative distribution function (5) with s = 305 and k = 0.019. This latter value corresponds to the moment estimate of k given in (6) for the sighting record in Table 1 when s = 305. Also shown in Fig. 2 are point-wise 0.95 sampling bounds for the empirical cumulative distribution function. These bounds were constructed from 1000 simulated samples of size 11 from the fitted truncated exponential distribution. The empirical cumulative distribution function generally follows the theoretical function and lies wholly within the sampling bounds, indicating no systematic departure from the truncated exponential model. As noted, interest here centers on assessing the plausibility of the rumored introduction in 1986. However, it is also instructive to take a step back and simply estimate the introduction time using ^ ¼ 328:3, corresponding to (4). For the sighting record in Table 1, s an estimated introduction time of August 1984. In light of the sparseness of the sighting record, this is not far from the rumored introduction in September 1986 and does not militate against it. Finally, the sightings in July 1996 and April 2000 were not based on physical material and their validity is therefore open to question. We repeated the analysis omitting each of these sightings separately and both together. The greatest impact occurs when the sighting in April 2000 is omitted, in which case the p value is reduced to 0.14. Roughly speaking, the omission of this sighting leaves a record consistent with an even lower sighting rate in the
The analysis described in this paper was aimed at testing a specific hypothesis about the introduction time of the pine marten to Mull. To our knowledge, this paper is the first to describe a formal statistical approach to inference about introduction time based on a record of sightings. The general approach outlined here can be applied more widely to address questions about introduction time. Introduced species can pose significant ecological and economic problems and information about their introduction times is important for managing these problem in at least three ways. First, as in the case of the pine marten on Mull, this information can contribute to an understanding of the mode of introduction. This can support measures to reduce the likelihood of further introductions. Second, by establishing an initial condition, information about introduction time contributes to an understanding of the temporal and spatial dynamics of an introduced species. Such an understanding is valuable in the design of control or eradication programs. Finally, information about introduction time can be useful in assessing the potential role of an introduced species in other ecological changes. Acknowledgment The authors thank Andrew Kitchener and an anonymous reviewer for their helpful comments. References Al-Athari, F.M., 2008. Estimation of the mean of truncated exponential distribution. J. Maths. Stats. 4, 284–288. Barrett-Hamilton, G.E., Hinton, M.A.C., 1913. On a collection of mammals from the inner hebrides. Proc. Zool. Soc. Lond. 1913, 821–839. Delany, M.J., 1961. The ecological distribution of small mammals in Northwest Scotland. Proc. Zool. Soc. Lond. 137, 107–126. Langley, P.J.W., Yalden, D.W., 1977. The decline of the rare carnivores in Great Britain during the nineteenth century. Mammal Rev. 7, 95–116. Mead, R., 1994. Reproduction in Martes. In: Buskirk, S.W., Harestad, A.S., Raphael, M.G., Powell, R.A. (Eds.), Martens, Sables, and Fishers: Biology and Conservation. Cornell University Press, Ithaca, pp. 404–422. Proulx, G. et al., 2004. World distribution and status of the genus Martes in 2000. In: Harrison, D.J., Fuller, A.K., Proulx, G. (Eds.), Martens and Fishers (Martes) in Human-altered Environments An International Perspective. Springer, New York, pp. 21–76. Sinclair, J. 1800. Old Statistical Account for Scotland 1791–1799. Solow, A.R., 1993. Inferring extinction in a declining population. J. Math. Biol. 32, 79–82. Solow, A.R., 2005. Inferring extinction from a sighting record. Math. Biosci. 195, 47– 55. Stanley, T.R., Royle, J.A., 2005. Estimating site occupancy and abundance using indirect detection indices. J. Wild. Mgmt. 69, 874–883.
Contents lists available at SciVerse ScienceDirect
Biological Conservation journal homepage: www.elsevier.com/locate/biocon
Short communication
On inference about the introduction time of an introduced species with an application to the pine marten on Mull Andrew Solow a,⇑, Sugoto Roy b, Christina Bell c, Jo Milborrow b, David Roberts d a
Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Food and Environment Research Agency, Sand Hutton, York YO41 1LZ, UK c Scottish Natural Heritage, Cameron House, Albany Street, Oban, Argyll PA34 4AE, UK d Durrell Institute of Conservation and Ecology, University of Kent, Canterbury, Kent CT2 7NR, UK b
a r t i c l e
i n f o
Article history: Received 24 September 2012 Received in revised form 13 December 2012 Accepted 17 December 2012 Available online 13 January 2013 Keywords: Isle of Mull Pine marten (Martes martes) Sighting record Species introduction
a b s t r a c t Although there is general agreement that the pine marten was recently introduced to the Isle of Mull, there is uncertainty about when. According to a persistent local rumor, the species was clandestinely introduced in 1986. A novel statistical approach to inference about introduction time based on a sighting record is described and applied to assess the plausibility of this rumor. The results indicate that the sighting record is consistent with the rumored introduction time. Inference about the introduction time is useful in the management of introduced species for the light it can shed on the mode of introduction, post-introduction population growth and spatial dispersal, and the potential role of the introduced species in other observed ecological changes. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Reconstructing the history of the introduction of a non-native species to a particular location or habitat is of interest to both science and management. A case in point is the pine marten (Martes martes) on the Isle of Mull in the Inner Hebrides off the west coast of Scotland (Fig. 1). The pine marten is a member of the mustelid family that also includes minks, otters, badgers, and weasels. The species is native to forested regions of mainland Scotland and northern England and Wales, although its numbers have been reduced over the centuries by hunting and habitat loss (Langley and Alden, 1977; Proulx et al., 2004). The pine marten is semiarboreal and among its prey are nesting birds so that its current presence and potential future spread on Mull has raised concern about the impact on the island’s diverse avian fauna. Motivated in part by such concern, Scottish Natural Heritage recently funded a study of the history (and future prospects) of the pine marten on Mull. While the species is now established on Mull, no documentary evidence of its presence on the island exists prior to 1996. It was not listed in the Old Statistical Account for Scotland (Sinclair, 1800) or in later surveys of the island (Barrett-Hamilton and Hinton, 1913; Delany, 1961), nor is there a specimen from Mull in any of the major national museums or university collections.
⇑ Corresponding author. Tel.: +1 508 289 2746. E-mail address: [email protected] (A. Solow). 0006-3207/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.biocon.2012.12.026
Although there is general agreement that the pine marten was recently introduced – or possibly reintroduced after a long absence – to Mull, there is uncertainty about when and how. This uncertainty is magnified by the fact that even now the presence on Mull of this rare and elusive species is known only through occasional chance sightings. According to a persistent local rumor, a clandestine introduction of newly weaned juveniles occurred in 1986. Here, we modify and apply a method for inference about extinction time to test the plausibility of this rumor based on the sighting record of the pine marten on Mull. The results suggest that the sighting record is indeed consistent with the rumored introduction in 1986. This paper focuses on the specific problem of assessing the plausibility of a rumored introduction time of the pine marten on Mull. However, the methods described here can be used for more general inference about introduction time based on a sighting record. The importance of information about introduction time to the management of introduced species is discussed in the final section.
2. Methods Statistical inference about the introduction time of a non-native species from a record of its sightings is the mirror image of inference about the extinction time of an extinct species based on its sightings (Solow, 2005). Here, the term sighting is not restricted to visual sightings of individuals, but can include other forms of evidence of
5
A. Solow et al. / Biological Conservation 159 (2013) 4–6
specified introduction time. The observed significance level (or p value) for the uniformly most powerful unbiased test of Ho is:
p ¼ 2 minðGðt n Þ=Gðso Þ; 1 Gðt n Þ=Gðso ÞÞ
ð3Þ
Although it is of less interest here, the minimum variance unbiased estimate under the truncated exponential model is:
s^ ¼ tn þ nðn 1Þ
P½t=tn j¼1
Gðt n Þ n1 ðt jtn Þn2 ð1Þj1 j1
ð4Þ
As noted, inference about s depends on the behavior of the post-introduction sighting rate. For this reason, it is important to have a way to assess the adequacy of the truncated exponential model. This can be done by plotting the empirical cumulative dis^ ¼ #ftj 6 tg=n and the theoretitribution of the sighting times FðtÞ cal cumulative distribution: Fig. 1. Location of the Isle of Mull.
FðtÞ ¼ ð1 expðktÞÞ=ð1 expðksÞÞ
the presence of a species such as tracks, hair, and fecal matter (Stanley and Royle, 2005). The central question in inference about extinction is how far after the last sighting of a species must we go before safely concluding that it is gone. The question for introduction is how far before the first sighting of a species must we go before safely concluding that it was not yet present. The answer to this question depends on the behavior of the post-introduction sighting rate. If this rate, which depends on factors such as the size of the population and sighting effort, is low, then even a relatively long period without sightings does not imply that the species was absent. Conversely, if this rate is high, then even a relatively short period without sightings implies that it was absent. The basic statistical model underlying inference about extinction is that the sighting record follows a possibly non-stationary Poisson process with a rate function that is positive before extinction and 0 thereafter. The details of statistical inference about extinction time depend on what is assumed about this rate function. Here, we use a straight forward modification of a method for inference about extinction time in a declining population (Solow, 1993) to test the null hypothesis that the pine marten was introduced to Mull in 1986. Let 0 < t1 < t2 < < tn < s be the ordered sighting times of an introduced species with time measured backward from the present, so that t1 corresponds to the most recent sighting and tn to the earliest, and where s denotes the unknown time of introduction before which the sighting rate must be 0. We assume that these sighting times follow a non-stationary Poisson process with a rate function that increases exponentially at rate k following introduction at time s. This model is appropriate for a population that grows exponentially following introduction with a sighting rate that is proportional to population size. It should be emphasized, however, that it can arise in other ways. It follows from the properties of the Poisson process that, conditional on their number n, these sighting times represent an ordered sample from a truncated exponential distribution with probability density function:
f ðtÞ ¼ k expðktÞ=ð1 expðksÞÞ 0 6 t 6 s
ð1Þ
where k is an unknown parameter that governs the growth of the sighting rate over time. P Following Solow (1993), let T ¼ nj¼1 tj and define the function:
GðxÞ ¼ 1
½T=x X n ð1Þj1 ð1 jx=TÞn1 j j¼1
ð2Þ
and [T/x] is the integer part of T/x. Interest here centers primarily on testing a null hypothesis of the form Ho: s = so where so is a
ð5Þ
against t using the values of interest of k and s. This plot can be supplemented with pointwise upper and lower a/2-quantiles of ^ FðtÞ estimated by simulating sighting records of size n from F(t). When, as here, interest centers on an hypothesized value of s, a convenient value of k to use in constructing F(t) is found by solving:
1 s ¼ T=n k expðksÞ 1
ð6Þ
(Al-Athari, 2008). Inadequacies in the fitted model are indicated ^ by systematic departures of FðtÞ from the fitted distribution function and particularly from excursions outside the sampling bounds. 3. Results The record of sightings of the pine marten on Mull is given in Table 1. This record was compiled as part of the study supported by Scottish Natural Heritage of the history of this species on Mull. We took the end of the observation period to be February 2012 and therefore omitted the sighting in that month – its corresponding sighting time of 0 being an artefact – leaving a total of n = 11 sightings. The corresponding values of tj measured in months are also given in Table 1. As noted, the individuals rumored to have been introduced in 1986 were newly weaned juveniles. Based on the life history of the pine marten (Mead, 1994), we took the hypothesized month of introduction to be September 1986. This corresponds to a value of so of 305. The p value based on Eq. (2) is 0.44. By conventional standards, this result is not significant and the null hypothesis that the rumored introduction time is true cannot be rejected. This re-
Table 1 Sighting dates of the pine marten on Mull. Also reported are the sighting times tj in months prior to February 2012. Sighting number
Date
Time
1 2 3 4 5 6 7 8 9 10 11 12
July 1996 April 2000 January 2007 March 2008 May 2008 May 2010 June 2010 September 2010 October 2010 February 2011 March 2011 February 2012
187 142 61 47 45 21 20 17 16 12 11 0
6
A. Solow et al. / Biological Conservation 159 (2013) 4–6
CUMULATIVE PROBABILITY
1
early part of the record and is therefore more consistent with an even earlier introduction, although it is still not possible to reject the rumored introduction time of September 1986.
0.8
4. Discussions 0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
t (MONTHS) Fig. 2. Empirical cumulative distribution of sighting time (solid) along with theoretical distribution (dashed) and upper and lower 0.95 sampling bounds (dotted).
sult is insensitive to the month in 1986 of the hypothesized introduction. To assess the adequacy of the truncated exponential model on which this result is based, in Fig. 2, the empirical cumulative distribution of the sighting times is plotted along with the theoretical cumulative distribution function (5) with s = 305 and k = 0.019. This latter value corresponds to the moment estimate of k given in (6) for the sighting record in Table 1 when s = 305. Also shown in Fig. 2 are point-wise 0.95 sampling bounds for the empirical cumulative distribution function. These bounds were constructed from 1000 simulated samples of size 11 from the fitted truncated exponential distribution. The empirical cumulative distribution function generally follows the theoretical function and lies wholly within the sampling bounds, indicating no systematic departure from the truncated exponential model. As noted, interest here centers on assessing the plausibility of the rumored introduction in 1986. However, it is also instructive to take a step back and simply estimate the introduction time using ^ ¼ 328:3, corresponding to (4). For the sighting record in Table 1, s an estimated introduction time of August 1984. In light of the sparseness of the sighting record, this is not far from the rumored introduction in September 1986 and does not militate against it. Finally, the sightings in July 1996 and April 2000 were not based on physical material and their validity is therefore open to question. We repeated the analysis omitting each of these sightings separately and both together. The greatest impact occurs when the sighting in April 2000 is omitted, in which case the p value is reduced to 0.14. Roughly speaking, the omission of this sighting leaves a record consistent with an even lower sighting rate in the
The analysis described in this paper was aimed at testing a specific hypothesis about the introduction time of the pine marten to Mull. To our knowledge, this paper is the first to describe a formal statistical approach to inference about introduction time based on a record of sightings. The general approach outlined here can be applied more widely to address questions about introduction time. Introduced species can pose significant ecological and economic problems and information about their introduction times is important for managing these problem in at least three ways. First, as in the case of the pine marten on Mull, this information can contribute to an understanding of the mode of introduction. This can support measures to reduce the likelihood of further introductions. Second, by establishing an initial condition, information about introduction time contributes to an understanding of the temporal and spatial dynamics of an introduced species. Such an understanding is valuable in the design of control or eradication programs. Finally, information about introduction time can be useful in assessing the potential role of an introduced species in other ecological changes. Acknowledgment The authors thank Andrew Kitchener and an anonymous reviewer for their helpful comments. References Al-Athari, F.M., 2008. Estimation of the mean of truncated exponential distribution. J. Maths. Stats. 4, 284–288. Barrett-Hamilton, G.E., Hinton, M.A.C., 1913. On a collection of mammals from the inner hebrides. Proc. Zool. Soc. Lond. 1913, 821–839. Delany, M.J., 1961. The ecological distribution of small mammals in Northwest Scotland. Proc. Zool. Soc. Lond. 137, 107–126. Langley, P.J.W., Yalden, D.W., 1977. The decline of the rare carnivores in Great Britain during the nineteenth century. Mammal Rev. 7, 95–116. Mead, R., 1994. Reproduction in Martes. In: Buskirk, S.W., Harestad, A.S., Raphael, M.G., Powell, R.A. (Eds.), Martens, Sables, and Fishers: Biology and Conservation. Cornell University Press, Ithaca, pp. 404–422. Proulx, G. et al., 2004. World distribution and status of the genus Martes in 2000. In: Harrison, D.J., Fuller, A.K., Proulx, G. (Eds.), Martens and Fishers (Martes) in Human-altered Environments An International Perspective. Springer, New York, pp. 21–76. Sinclair, J. 1800. Old Statistical Account for Scotland 1791–1799. Solow, A.R., 1993. Inferring extinction in a declining population. J. Math. Biol. 32, 79–82. Solow, A.R., 2005. Inferring extinction from a sighting record. Math. Biosci. 195, 47– 55. Stanley, T.R., Royle, J.A., 2005. Estimating site occupancy and abundance using indirect detection indices. J. Wild. Mgmt. 69, 874–883.