Dynamics of North American Breeding Bird Populations
Dynamics of North American Breeding Bird Populations Timothy H. Keitt H. Eugene Stanley
SFI WORKING PAPER: 1997-12-089
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
SANTA FE INSTITUTE
Dynamics of North American Breeding Bird Populations Timothy H. Keitt and H. Eugene Stanleyy Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 and Department of Biology, University of New Mexico Albuquerque, NM 87131 USA y Center for Polymer Studies and Department of Physics Boston University, Boston 02215 USA December 12, 1997 The dynamics of biological populations often appear quite complex, exhibiting considerable year-to-year variation in local abundances.1{7 One approach to dealing with ecological complexity is to reduce the system to one or a few species, for which meaningful equations can be written and even solved. Here we explore an alternative approach8 by studying statistical properties of a vastly larger assemblage comprising over 600 species. Specically, we quantitatively analyze one of the most comprehensive data sets available: the North American Breeding Bird Survey, which records annual species abundances over a 31-year period at more than 3,000 survey routes.9, 10 We analyze all the data on an equal footing, and nd features common to other, inanimate systems composed of strongly interacting subunits. For each of the
2 500 routes surveyed in a given year, the North American Breeding
Bird Survey gives Ns(t), the number of birds of a given species s detected on a given route in 1
year t the challenge is to extract information, relevant to population variability, from these quantities. We are interested in the dynamics, so we must compare values of Ns(t) in successive years. We form the quantities
Rs(t) Ns(t + 1)=Ns(t)
(1)
which give the rate of increase or decrease of each species s in each route. Population growth is inherently a multiplicative process, so the distribution of abundance Ns(t) of a species through time is often log-normal or approximately log-normal.11{13 The ratio of two log-normal distributions is also a log-normal, so we might guess that the distribution of
Rs(t) would be log-normal as well. We nd that the distribution of Rs(t) is not log-normal, but is instead a power-law distribution with the tails separated by six orders of magnitude (Fig. 1a), such that
8 >> < R if Rs 1 P (Rs) / > s >: Rs; if Rs 1
(2)
where the exponent = 2. Thus, for the species considered here, there exists no characteristic scale of uctuation in population size, but instead we nd a broad spectrum of growth rates. We may also consider the more commonly used14, 15 logarithm of the ratio of successive abundances rs log Rs. Equation (2) then becomes
P (rs) / exp(;2jrsj) 2
(3)
What is remarkable about Figure 1 is that, despite the fact that the results are taken over a large number of species, the overall distribution can be described by relatively simple mathematical equations. It is also intriguing that the distribution is highly symmetric so that exactly as many species are increasing in abundance as decreasing. In the above analysis, the quantities Rs(t) and rs(t) measure uctuations in population size over a one year interval. It is also possible to analyze the data set for an arbitrary time lag t, where t can take on any value from 1 to 30. We might expect that the variance of the distribution functions corresponding to Figs. 1a will increase with the magnitude of t.16, 17 We nd indeed an increase (Fig. 1c), and moreover this increase is a power law of the form (t)2H , where H is termed the Hurst exponent.18 If the species were uncorrelated, then we would expect H = 1=2. We nd H = 0:14, a value considerably smaller than 1=2 which implies the existence of long-range correlations. Correlation in population uctuations could also arise from correlations in climate variables that inuence reproduction and mortality19 (e.g., the severe 1975{1976 North American winter is thought to have caused large declines of some resident bird species20{22). Fluctuations in population size can lead to local extinctions23, 24 and turnover in local species composition.25, 26 Local extinction dynamics can be studied quantitatively by considering the distribution of \lifetimes" of species within a local region.27 Here we de ne the lifetime of a species as te ; tc , where tc denotes the year in which the species colonized a patch (the rst year the species was recorded within a given route), and te denotes the year that the species became \locally extinct" (the last year the species was recorded within that route). We nd (Fig. 2a) that the distribution of species lifetimes follows a Yule distribution 3
(a power law, modi ed by an exponential cuto)
P ( ) = A ; e;=ch
(4)
where A a normalization constant, the scaling exponent, and ch sets the time scale at which the power-law scaling no longer holds due to the nite size of the data set. We nd
= 1:61 0:02 and ch = 14 years for the breeding bird data. The exponential term in Eq. (4) introduces a characteristic time scale ch = 14 years into the distribution of extinction times. However, we can test that the exponential term is indeed due to nite-size eects by plotting a series of lifetime distributions, each computed within non-overlapping subsets of the time series associated with each survey route (Fig. 2b). The resulting distributions are increasingly truncated by the lengths of the time series. When plotted on rescaled axes (Fig. 2c), the distributions all collapse onto a single power-law curve. Thus, we nd that the power-law distribution is unaected by shortening the time series only the exponential term is aected. The data collapse suggests that the power-law behavior extends beyond the 31-year extent of the data set. The presence of scaling, both in population variability (Fig. 1a) and in local species lifetimes (Fig. 2c), may have implications for understanding population dynamics in general. Scaling of system variables is often observed in physical systems which exhibit cooperative behavior.28{30 Scaling arises in these inanimate systems because each particle interacts directly with a few neighboring particles and, since these neighboring particles interact with their neighborings, interactions can \propagate" long distances|resulting in power-law distributions. Similarly, species in an ecosystem generally interact directly with some (but not 4
all) other species, which in turn interact with other species so that eectively interactions can \propagate."
5
References 1] May, R. M. Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos. Science 186, 645{647 (1974). 2] May, R. M. Deterministic models with chaotic dynamics. Nature 256, 165{166 (1975). 3] Ives, A. R. Chaos in time and space. Nature 353, 214{215 (1991). 4] Turchin, P. Rarity of density dependence or population regulation with lags? Nature
344, 660{663 (1990). 5] Arino, A. and Pimm, S. L. On the nature of population extremes. Evolutionary Ecology
9, 429{443 (1995). 6] Rhodes, C. J. and Anderson, R. M. Power laws governing epidemics in isolated populations. Nature 381, 600{602 (1996). 7] Curnutt, J. L., Pimm, S. L., and Maurer, B. A. Population variablity of sparrows in space and time. OIKOS 76, 131{144 (1996). 8] Brown, J. H. Macroecology. The University of Chicago Press, Chicago (1995). 9] Peterjohn, B. G. The North American breeding bird survey. Birding, Journal of the American Birding Assocation 26 (1994).
10] Sauer, J. R., Hines, J. E., Gough, G., Thomas, I., and Peterjohn, B. G. The North American breeding bird survey results and analysis, version 96.3. Patuxent Wildlife Research Center, Laurel, MD, (1997). http://www.mbr.nbs.gov/bbs/bbs.html. 6
11] Preston, F. W. The canonical distribution of commonness and rarity: Part I. Ecology
43, 185{215 (1962). 12] MacArthur, R. H. On the relative abundance of bird species. Proceedings of the National Academy of Sciences 43, 293{295 (1957).
13] Sugihara, G. Minimal community struture: An explanation of species abundance. The American Naturalist 116, 770{787 (1980).
14] Hassell, M. P., Lawton, J. H., and May, R. M. Patterns of dynamical behavior in single-species populations. Journal of Animal Ecology 45, 471{486 (1976). 15] Royoma, T. Analytical Population Dynamics. Chapman and Hall, London (1992). 16] Connell, J. H. and Sousa, W. P. On the evidence needed to judge ecological stability or persistence. The American Naturalist 121, 789{824 (1983). 17] Pimm, S. L. and Redfearn, A. The variability of population densities. Nature 334, 613{614 (1988). 18] Vicsek, T. Fractal Growth Phenomena. World Scienti c, Singapore, 2 edition (1992). 19] Steele, J. H. A comparison of terrestrial and marine ecological systems. Nature 313, 355{358 (1985). 20] Pitts, T. D. Eastern bluebird populations uctuations in Tennessee during 1970{1979. The Migrant 52, 29{37 (1981).
21] Sauer, J. R. and Droege, S. Recent population trends of the eastern bluebird. Wilson Bulletin 102, 239{252 (1990).
7
22] Sauer, J. R., Pendleton, G. W., and Peterjohn, B. G. Evaluating causes of population change in North American insectivorous songbirds. Conservation Biology 10, 465{478 (1996). 23] Hanski, I. and Gilpin, M. Metapopulation dynamics: Brief history and conceptual domain. Biological Journal of the Linnean Society 42, 3{16 (1991). 24] Hanski, I., Foley, P., and Hassell, M. Random walks in a metapopulation: How much density dependence is necessary for long-term persistence. Journal of Animal Ecology
65, 274{282 (1996). 25] Diamond, J. M. and May, R. M. Species turnover rates on islands: Dependence on cencus interval. Science 197, 266{270 (1977). 26] Schoener, T. W. and Spiller, D. A. High population persistence in a system with high turnover. Nature 330, 474{477 (1987). 27] Pimm, S. L., Diamond, J., Reed, T. M., Russell, G. J., and Verner, J. Times to extinction for small populations of large birds. Proceedings of the National Academy of Sciences 90, 10871{10875 (1993).
28] Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, Oxford (1971). 29] Stanley, H. E. Power laws and universality. Nature 378, 554 (1995).
8
30] Back, C. H., W!ursch, C., Vaterlaus, A., Ramsperger, U., Maier, U., and Pescia, D. Experimental con mation of universality for a phase transition in two dimensions. Nature
378, 597{600 (1995). The support of the Santa Fe Institute is gratefully acknowledged. We thank B. Peterjohn and the Patuxent Wildlife Research Center, United States Department of the Interior, for providing the BBS data in digital form. This manuscript bene ted from the comments of L. Amaral, P. Bak, J. Brown, P. Marquet, B. Maurer, R. May, M. Paczuski, B. Peterjohn, S. Pimm, R. Sol"e, M. Taper, and two anonymous reviewers.
9
Figure Captions Figure 1. (a) Distribution of population growth rates Rs Ns(t +1)=Ns(t) across all species in the entire 31-year data set described in the text. The growth rate Rs is calculated by dividing species abundances in successive years. Abundances are taken as the total number of individuals of a particular species counted within each survey route. (c) Characterization of fractal scaling in population dynamics. The variance of the uctuation distribution E (rs2) increases as a power-law function of time lag t. Deviations from power-law scaling are expected for lags t approaching the 31-year length of the time series. The slope of 0:28 corresponds to a Hurst exponent of 0:14.
Figure 2. (a) Frequency distribution of species lifetimes within local patches over all species. Parameters of the data t are obtained by weighted least-squares. The sum of the distribution is normalized to unity. The lifetime of a species within a patch is the time between colonization and local extinction. Colonization occurs when a species is recorded, but was absent the previous year extinction occurs when a species is absent, but was recorded the previous year. Time series not surveyed in each year between colonization and extinction are not included, as well as time series that begin or end in the rst or last year of the survey data. (b) Finite-size scaling of the lifetime distribution. The distributions are from non-overlapping subsets of the data, broken into shorter and shorter time-series. Number of years in the legend are maximum possible lifetimes between the rst and last year of the individual time-series. (c) Test of data collapse. Axes are rescaled to remove nite-scaling
10
eects. For each nite subset of the data, we compute
F ( ) = A ;P e(;)=ch ch
and plot versus the rescaled time axis =ch .
11
(5)
Figure 1a]
0 −1
log10 P(Rs)
−2 −3 −4 −5 −6 −7
−3
−2
−1
0 log10 Rs
12
1
2
3
Figure 1b]
1.10
2
log10 E(rs )
1.05
1.00
0.95
0.90
0.0
0.5
1.0 log10 Δt
13
1.5
Figure 2a]
0.0
log10 P(τ)
−1.0
−2.0
−3.0
−4.0 0.0
0.5
1.0 log10 τ
14
1.5
Figure 2b]
0.0
28 years 13 years 8 years 5 years 4 years 3 years
log10 P(τ)
−1.0
−2.0
−3.0
−4.0 0.0
0.5
1.0 log10 τ
15
1.5
Figure 2c]
0.0
28 years 13 years 8 years 5 years 4 years 3 years
log10 F(τ)
−1.0
−2.0
−3.0
−4.0 −1.5
−1.0
−0.5 log10 τ/τch
16
0.0
SFI WORKING PAPER: 1997-12-089
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
SANTA FE INSTITUTE
Dynamics of North American Breeding Bird Populations Timothy H. Keitt and H. Eugene Stanleyy Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 and Department of Biology, University of New Mexico Albuquerque, NM 87131 USA y Center for Polymer Studies and Department of Physics Boston University, Boston 02215 USA December 12, 1997 The dynamics of biological populations often appear quite complex, exhibiting considerable year-to-year variation in local abundances.1{7 One approach to dealing with ecological complexity is to reduce the system to one or a few species, for which meaningful equations can be written and even solved. Here we explore an alternative approach8 by studying statistical properties of a vastly larger assemblage comprising over 600 species. Specically, we quantitatively analyze one of the most comprehensive data sets available: the North American Breeding Bird Survey, which records annual species abundances over a 31-year period at more than 3,000 survey routes.9, 10 We analyze all the data on an equal footing, and nd features common to other, inanimate systems composed of strongly interacting subunits. For each of the
2 500 routes surveyed in a given year, the North American Breeding
Bird Survey gives Ns(t), the number of birds of a given species s detected on a given route in 1
year t the challenge is to extract information, relevant to population variability, from these quantities. We are interested in the dynamics, so we must compare values of Ns(t) in successive years. We form the quantities
Rs(t) Ns(t + 1)=Ns(t)
(1)
which give the rate of increase or decrease of each species s in each route. Population growth is inherently a multiplicative process, so the distribution of abundance Ns(t) of a species through time is often log-normal or approximately log-normal.11{13 The ratio of two log-normal distributions is also a log-normal, so we might guess that the distribution of
Rs(t) would be log-normal as well. We nd that the distribution of Rs(t) is not log-normal, but is instead a power-law distribution with the tails separated by six orders of magnitude (Fig. 1a), such that
8 >> < R if Rs 1 P (Rs) / > s >: Rs; if Rs 1
(2)
where the exponent = 2. Thus, for the species considered here, there exists no characteristic scale of uctuation in population size, but instead we nd a broad spectrum of growth rates. We may also consider the more commonly used14, 15 logarithm of the ratio of successive abundances rs log Rs. Equation (2) then becomes
P (rs) / exp(;2jrsj) 2
(3)
What is remarkable about Figure 1 is that, despite the fact that the results are taken over a large number of species, the overall distribution can be described by relatively simple mathematical equations. It is also intriguing that the distribution is highly symmetric so that exactly as many species are increasing in abundance as decreasing. In the above analysis, the quantities Rs(t) and rs(t) measure uctuations in population size over a one year interval. It is also possible to analyze the data set for an arbitrary time lag t, where t can take on any value from 1 to 30. We might expect that the variance of the distribution functions corresponding to Figs. 1a will increase with the magnitude of t.16, 17 We nd indeed an increase (Fig. 1c), and moreover this increase is a power law of the form (t)2H , where H is termed the Hurst exponent.18 If the species were uncorrelated, then we would expect H = 1=2. We nd H = 0:14, a value considerably smaller than 1=2 which implies the existence of long-range correlations. Correlation in population uctuations could also arise from correlations in climate variables that inuence reproduction and mortality19 (e.g., the severe 1975{1976 North American winter is thought to have caused large declines of some resident bird species20{22). Fluctuations in population size can lead to local extinctions23, 24 and turnover in local species composition.25, 26 Local extinction dynamics can be studied quantitatively by considering the distribution of \lifetimes" of species within a local region.27 Here we de ne the lifetime of a species as te ; tc , where tc denotes the year in which the species colonized a patch (the rst year the species was recorded within a given route), and te denotes the year that the species became \locally extinct" (the last year the species was recorded within that route). We nd (Fig. 2a) that the distribution of species lifetimes follows a Yule distribution 3
(a power law, modi ed by an exponential cuto)
P ( ) = A ; e;=ch
(4)
where A a normalization constant, the scaling exponent, and ch sets the time scale at which the power-law scaling no longer holds due to the nite size of the data set. We nd
= 1:61 0:02 and ch = 14 years for the breeding bird data. The exponential term in Eq. (4) introduces a characteristic time scale ch = 14 years into the distribution of extinction times. However, we can test that the exponential term is indeed due to nite-size eects by plotting a series of lifetime distributions, each computed within non-overlapping subsets of the time series associated with each survey route (Fig. 2b). The resulting distributions are increasingly truncated by the lengths of the time series. When plotted on rescaled axes (Fig. 2c), the distributions all collapse onto a single power-law curve. Thus, we nd that the power-law distribution is unaected by shortening the time series only the exponential term is aected. The data collapse suggests that the power-law behavior extends beyond the 31-year extent of the data set. The presence of scaling, both in population variability (Fig. 1a) and in local species lifetimes (Fig. 2c), may have implications for understanding population dynamics in general. Scaling of system variables is often observed in physical systems which exhibit cooperative behavior.28{30 Scaling arises in these inanimate systems because each particle interacts directly with a few neighboring particles and, since these neighboring particles interact with their neighborings, interactions can \propagate" long distances|resulting in power-law distributions. Similarly, species in an ecosystem generally interact directly with some (but not 4
all) other species, which in turn interact with other species so that eectively interactions can \propagate."
5
References 1] May, R. M. Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos. Science 186, 645{647 (1974). 2] May, R. M. Deterministic models with chaotic dynamics. Nature 256, 165{166 (1975). 3] Ives, A. R. Chaos in time and space. Nature 353, 214{215 (1991). 4] Turchin, P. Rarity of density dependence or population regulation with lags? Nature
344, 660{663 (1990). 5] Arino, A. and Pimm, S. L. On the nature of population extremes. Evolutionary Ecology
9, 429{443 (1995). 6] Rhodes, C. J. and Anderson, R. M. Power laws governing epidemics in isolated populations. Nature 381, 600{602 (1996). 7] Curnutt, J. L., Pimm, S. L., and Maurer, B. A. Population variablity of sparrows in space and time. OIKOS 76, 131{144 (1996). 8] Brown, J. H. Macroecology. The University of Chicago Press, Chicago (1995). 9] Peterjohn, B. G. The North American breeding bird survey. Birding, Journal of the American Birding Assocation 26 (1994).
10] Sauer, J. R., Hines, J. E., Gough, G., Thomas, I., and Peterjohn, B. G. The North American breeding bird survey results and analysis, version 96.3. Patuxent Wildlife Research Center, Laurel, MD, (1997). http://www.mbr.nbs.gov/bbs/bbs.html. 6
11] Preston, F. W. The canonical distribution of commonness and rarity: Part I. Ecology
43, 185{215 (1962). 12] MacArthur, R. H. On the relative abundance of bird species. Proceedings of the National Academy of Sciences 43, 293{295 (1957).
13] Sugihara, G. Minimal community struture: An explanation of species abundance. The American Naturalist 116, 770{787 (1980).
14] Hassell, M. P., Lawton, J. H., and May, R. M. Patterns of dynamical behavior in single-species populations. Journal of Animal Ecology 45, 471{486 (1976). 15] Royoma, T. Analytical Population Dynamics. Chapman and Hall, London (1992). 16] Connell, J. H. and Sousa, W. P. On the evidence needed to judge ecological stability or persistence. The American Naturalist 121, 789{824 (1983). 17] Pimm, S. L. and Redfearn, A. The variability of population densities. Nature 334, 613{614 (1988). 18] Vicsek, T. Fractal Growth Phenomena. World Scienti c, Singapore, 2 edition (1992). 19] Steele, J. H. A comparison of terrestrial and marine ecological systems. Nature 313, 355{358 (1985). 20] Pitts, T. D. Eastern bluebird populations uctuations in Tennessee during 1970{1979. The Migrant 52, 29{37 (1981).
21] Sauer, J. R. and Droege, S. Recent population trends of the eastern bluebird. Wilson Bulletin 102, 239{252 (1990).
7
22] Sauer, J. R., Pendleton, G. W., and Peterjohn, B. G. Evaluating causes of population change in North American insectivorous songbirds. Conservation Biology 10, 465{478 (1996). 23] Hanski, I. and Gilpin, M. Metapopulation dynamics: Brief history and conceptual domain. Biological Journal of the Linnean Society 42, 3{16 (1991). 24] Hanski, I., Foley, P., and Hassell, M. Random walks in a metapopulation: How much density dependence is necessary for long-term persistence. Journal of Animal Ecology
65, 274{282 (1996). 25] Diamond, J. M. and May, R. M. Species turnover rates on islands: Dependence on cencus interval. Science 197, 266{270 (1977). 26] Schoener, T. W. and Spiller, D. A. High population persistence in a system with high turnover. Nature 330, 474{477 (1987). 27] Pimm, S. L., Diamond, J., Reed, T. M., Russell, G. J., and Verner, J. Times to extinction for small populations of large birds. Proceedings of the National Academy of Sciences 90, 10871{10875 (1993).
28] Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, Oxford (1971). 29] Stanley, H. E. Power laws and universality. Nature 378, 554 (1995).
8
30] Back, C. H., W!ursch, C., Vaterlaus, A., Ramsperger, U., Maier, U., and Pescia, D. Experimental con mation of universality for a phase transition in two dimensions. Nature
378, 597{600 (1995). The support of the Santa Fe Institute is gratefully acknowledged. We thank B. Peterjohn and the Patuxent Wildlife Research Center, United States Department of the Interior, for providing the BBS data in digital form. This manuscript bene ted from the comments of L. Amaral, P. Bak, J. Brown, P. Marquet, B. Maurer, R. May, M. Paczuski, B. Peterjohn, S. Pimm, R. Sol"e, M. Taper, and two anonymous reviewers.
9
Figure Captions Figure 1. (a) Distribution of population growth rates Rs Ns(t +1)=Ns(t) across all species in the entire 31-year data set described in the text. The growth rate Rs is calculated by dividing species abundances in successive years. Abundances are taken as the total number of individuals of a particular species counted within each survey route. (c) Characterization of fractal scaling in population dynamics. The variance of the uctuation distribution E (rs2) increases as a power-law function of time lag t. Deviations from power-law scaling are expected for lags t approaching the 31-year length of the time series. The slope of 0:28 corresponds to a Hurst exponent of 0:14.
Figure 2. (a) Frequency distribution of species lifetimes within local patches over all species. Parameters of the data t are obtained by weighted least-squares. The sum of the distribution is normalized to unity. The lifetime of a species within a patch is the time between colonization and local extinction. Colonization occurs when a species is recorded, but was absent the previous year extinction occurs when a species is absent, but was recorded the previous year. Time series not surveyed in each year between colonization and extinction are not included, as well as time series that begin or end in the rst or last year of the survey data. (b) Finite-size scaling of the lifetime distribution. The distributions are from non-overlapping subsets of the data, broken into shorter and shorter time-series. Number of years in the legend are maximum possible lifetimes between the rst and last year of the individual time-series. (c) Test of data collapse. Axes are rescaled to remove nite-scaling
10
eects. For each nite subset of the data, we compute
F ( ) = A ;P e(;)=ch ch
and plot versus the rescaled time axis =ch .
11
(5)
Figure 1a]
0 −1
log10 P(Rs)
−2 −3 −4 −5 −6 −7
−3
−2
−1
0 log10 Rs
12
1
2
3
Figure 1b]
1.10
2
log10 E(rs )
1.05
1.00
0.95
0.90
0.0
0.5
1.0 log10 Δt
13
1.5
Figure 2a]
0.0
log10 P(τ)
−1.0
−2.0
−3.0
−4.0 0.0
0.5
1.0 log10 τ
14
1.5
Figure 2b]
0.0
28 years 13 years 8 years 5 years 4 years 3 years
log10 P(τ)
−1.0
−2.0
−3.0
−4.0 0.0
0.5
1.0 log10 τ
15
1.5
Figure 2c]
0.0
28 years 13 years 8 years 5 years 4 years 3 years
log10 F(τ)
−1.0
−2.0
−3.0
−4.0 −1.5
−1.0
−0.5 log10 τ/τch
16
0.0
