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Extinction

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Today’s topics 1 Introduction 2 Deterministic Models 3 Stochastic Models 4 Allee effects

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Extinction • Many believe that humans are causing the 6th mass extinction event Introduction Deterministic Models Stochastic Models Allee effects 3 / 15

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Extinction • Many believe that humans are causing the 6th mass extinction event • At least 1000 species have gone extinct over past 500 years Introduction Deterministic Models Stochastic Models Allee effects 3 / 15

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Extinction • Many believe that humans are causing the 6th mass extinction event • At least 1000 species have gone extinct over past 500 years • Extinction rate unknown, but may be 100-1000 times higher than during the past 25 million years Introduction Deterministic Models Stochastic Models Allee effects 3 / 15

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Extinction • Many believe that humans are causing the 6th mass extinction event • At least 1000 species have gone extinct over past 500 years • Extinction rate unknown, but may be 100-1000 times higher than during the past 25 million years • Almost 200 bird species have gone extinct since 1500 Introduction Deterministic Models Stochastic Models Allee effects 3 / 15

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Extinction • Many believe that humans are causing the 6th mass extinction event • At least 1000 species have gone extinct over past 500 years • Extinction rate unknown, but may be 100-1000 times higher than during the past 25 million years • Almost 200 bird species have gone extinct since 1500 • Avian extinctions in GA: passenger pigeon, Bachman’s warbler, ivory-billed woodpecker, Carolina parakeet Introduction Deterministic Models Stochastic Models Allee effects 3 / 15

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Extinction • Many believe that humans are causing the 6th mass extinction event • At least 1000 species have gone extinct over past 500 years • Extinction rate unknown, but may be 100-1000 times higher than during the past 25 million years • Almost 200 bird species have gone extinct since 1500 • Avian extinctions in GA: passenger pigeon, Bachman’s warbler, ivory-billed woodpecker, Carolina parakeet https://vimeo.com/42592260 Introduction Deterministic Models Stochastic Models Allee effects 3 / 15

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Correlates of extinction risk Species with high extinction risk often have: Introduction Deterministic Models Stochastic Models Allee effects 4 / 15

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Correlates of extinction risk Species with high extinction risk often have: • Small range • Low population size • Limited dispersal ability • Low population growth rate Introduction Deterministic Models Stochastic Models Allee effects 4 / 15

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Deterministic models For a model with no random variation, time to extinction can be calculated easily Introduction Deterministic Models Stochastic Models Allee effects 5 / 15

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Deterministic models For a model with no random variation, time to extinction can be calculated easily But how should we define extinction? Introduction Deterministic Models Stochastic Models Allee effects 5 / 15

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Deterministic models For a model with no random variation, time to extinction can be calculated easily But how should we define extinction? Time to quasi-extinction (Te ) is the time it takes for a population to reach an extinction threshold beyond which it is doomed Introduction Deterministic Models Stochastic Models Allee effects 5 / 15

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Deterministic models For a model with no random variation, time to extinction can be calculated easily But how should we define extinction? Time to quasi-extinction (Te ) is the time it takes for a population to reach an extinction threshold beyond which it is doomed Threshold usually based on genetic considerations, Allee effects, etc. . . Introduction Deterministic Models Stochastic Models Allee effects 5 / 15

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Geometric growth example q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 10 20 30 40 50 0 10 20 30 40 50 Time Population size N=2 Time to quasi−extinction Te = 30 Introduction Deterministic Models Stochastic Models Allee effects 6 / 15

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Extinction risk In addition to time to extinction (Te ), we are now interested in extinction risk Introduction Deterministic Models Stochastic Models Allee effects 7 / 15

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Extinction risk In addition to time to extinction (Te ), we are now interested in extinction risk Extinction risk is the probability that a species goes extinct in some time period Introduction Deterministic Models Stochastic Models Allee effects 7 / 15

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Extinction risk In addition to time to extinction (Te ), we are now interested in extinction risk Extinction risk is the probability that a species goes extinct in some time period For a stochastic model, extinction risk can be calculated as the proportion of simulations in which the population goes extinct. Introduction Deterministic Models Stochastic Models Allee effects 7 / 15

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Extinction risk In addition to time to extinction (Te ), we are now interested in extinction risk Extinction risk is the probability that a species goes extinct in some time period For a stochastic model, extinction risk can be calculated as the proportion of simulations in which the population goes extinct. Calculating extinction risk requires a specification of the time horizon of interest Introduction Deterministic Models Stochastic Models Allee effects 7 / 15

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Logistic growth with stochastic carrying capacity Nt+1 = Nt + Nt rmax (1 − Nt /Kt ) where Kt ∼ Normal( ¯ K, σ2 e ) Introduction Deterministic Models Stochastic Models Allee effects 8 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 400 0 5 10 15 20 0 50 100 150 100 Simulations Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 9 / 15

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Extinction risk Zero extinctions in 100 simulations, hence extinction risk is (approximately) zero over the 20 year time horizon Introduction Deterministic Models Stochastic Models Allee effects 10 / 15

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Extinction risk Zero extinctions in 100 simulations, hence extinction risk is (approximately) zero over the 20 year time horizon Assumptions • We have the correct model • We know the parameters with certainty Introduction Deterministic Models Stochastic Models Allee effects 10 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) q Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 Time Population size (N) q Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Logistic example, rmax = 0.3, ¯ K = 100, σ2 e = 1600 0 5 10 15 20 0 50 100 150 17 extinctions in 100 simulations Time Population size (N) q q q q q q q q q q q q q q q q q Introduction Deterministic Models Stochastic Models Allee effects 11 / 15

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Allee effects Normally, population growth rates increase as the population decreases (negative correlation) Introduction Deterministic Models Stochastic Models Allee effects 12 / 15

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Allee effects Normally, population growth rates increase as the population decreases (negative correlation) The Allee effect is the phenomenon of positive correlation between population growth rates and population size Introduction Deterministic Models Stochastic Models Allee effects 12 / 15

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Allee effects Normally, population growth rates increase as the population decreases (negative correlation) The Allee effect is the phenomenon of positive correlation between population growth rates and population size Mechanisms • Finding a mate becomes difficult • Social systems collapse • Inbreeding depression • etc. . . Introduction Deterministic Models Stochastic Models Allee effects 12 / 15

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Allee effects Normally, population growth rates increase as the population decreases (negative correlation) The Allee effect is the phenomenon of positive correlation between population growth rates and population size Mechanisms • Finding a mate becomes difficult • Social systems collapse • Inbreeding depression • etc. . . Allee effects can greatly increase extinction risk for small populations Introduction Deterministic Models Stochastic Models Allee effects 12 / 15

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Allee effects 0 20 40 60 80 100 0.06 0.08 0.10 0.12 0.14 No density−dependence Population size (N) Population growth rate (r) 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Standard density−dependence Population size (N) Population growth rate (r) 0 20 40 60 80 100 −0.020 −0.015 −0.010 −0.005 0.000 0.005 0.010 Allee effect Population size (N) Population growth rate (r) Introduction Deterministic Models Stochastic Models Allee effects 13 / 15

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Summary Humans have increased extinction rates dramatically Models allow us to predict time to extinction and extinction risk Models can be used to assess effects of management actions on extinction risk Introduction Deterministic Models Stochastic Models Allee effects 14 / 15

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Assignment Read pages 27–31 in Conroy and Carroll Introduction Deterministic Models Stochastic Models Allee effects 15 / 15