This document introduces spatial occupancy models for analyzing metapopulation viability over time. It discusses motivations like estimating extinction risk over 100 years and how hydrology and connectivity affect risk. Occupancy data from multiple sites and years are presented. A spatial occupancy model is described that accounts for colonization between sites based on distance. Results show local extinction is lower in permanent wetlands and colonization increases over time as connectivity increases. Modeling predicts reintroducing populations in certain areas can significantly reduce long-term extinction risk for the metapopulation.

1. Spatial occupancy models for
metapopulation viability
analysis
Richard Chandler, Erin Muths, Brent Sigafus, Cecil Schwalbe,
Christopher Jarchow, and Blake Hossack

2. Motivating example
Chandler et al. Spatial occupancy models 2 / 13

3. Motivating questions
(1) What is extinction risk over the next 100
years?
Chandler et al. Spatial occupancy models 3 / 13

4. Motivating questions
(1) What is extinction risk over the next 100
years?
(2) How do hydrology and connectivity aect
extinction risk?
Chandler et al. Spatial occupancy models 3 / 13

5. Motivating questions
(1) What is extinction risk over the next 100
years?
(2) How do hydrology and connectivity aect
extinction risk?
(3) What is the best strategy for increasing
viability?
Chandler et al. Spatial occupancy models 3 / 13

6. Motivating data
Year
2007 2008 . . . 2013
Site 1 2 3 1 2 3 . . . 1 2 3
1 0 1 1 0 0 0 . . . 1 0 1
2 0 0 0 0 0 0 . . . 0 0 0
3 1 1 0 . . . 0 0 1
...
...
...
...
...
...
...
...
...
...
...
41 0 1 1 0 1 0 . . . 0 0 0
Chandler et al. Spatial occupancy models 4 / 13

7. Motivating data
Year
2007 2008 . . . 2013
Site 1 2 3 1 2 3 . . . 1 2 3
1 0 1 1 0 0 0 . . . 1 0 1
2 0 0 0 0 0 0 . . . 0 0 0
3 1 1 0 . . . 0 0 1
...
...
...
...
...
...
...
...
...
...
...
41 0 1 1 0 1 0 . . . 0 0 0
42 . . .
...
...
...
...
...
...
...
...
...
...
...
273 . . .
Chandler et al. Spatial occupancy models 4 / 13

8. Motivating data
Year
2007 2008 . . . 2013
Site 1 2 3 1 2 3 . . . 1 2 3
1 0 1 1 0 0 0 . . . 1 0 1
2 0 0 0 0 0 0 . . . 0 0 0
3 1 1 0 . . . 0 0 1
...
...
...
...
...
...
...
...
...
...
...
41 0 1 1 0 1 0 . . . 0 0 0
42 . . .
...
...
...
...
...
...
...
...
...
...
...
273 . . .
Plus, coordinates and covariates for each site
Chandler et al. Spatial occupancy models 4 / 13

9. Metapopulation theory
Basic elements
• Dispersal-based colonization function
• Rescue eect
• Correlated extinction
Chandler et al. Spatial occupancy models 5 / 13

10. Metapopulation theory
Basic elements
• Dispersal-based colonization function
• Rescue eect
• Correlated extinction
Missing elements
• Observation model
Chandler et al. Spatial occupancy models 5 / 13

11. Metapopulation theory
Basic elements
• Dispersal-based colonization function
• Rescue eect
• Correlated extinction
Missing elements
• Observation model
MacKenzie et al. (2003) occupancy models
provided the latter, but not the former
Chandler et al. Spatial occupancy models 5 / 13

12. Standard dynamic occupancy model
Initial occupancy
zi,1 ∼ Bern(ψ)
Colonization and extinction
zi,k ∼ Bern(µi,k)
µi,k = (1 − zi,k)γ + zi,k(1 − ε)
Detection
yi,j,k ∼ Bern(zi,k × p)
Chandler et al. Spatial occupancy models 6 / 13

13. Standard dynamic occupancy model
Initial occupancy
zi,1 ∼ Bern(ψ)
Colonization and extinction
zi,k ∼ Bern(µi,k)
µi,k = (1 − zi,k)γ + zi,k(1 − ε)
Detection
yi,j,k ∼ Bern(zi,k × p)
Useful, but doesn't allow for
metapopulation extinction
Chandler et al. Spatial occupancy models 6 / 13

14. A spatial occupancy model
Probability that site i is colonized by ≥ 1 individual from site m
γ(xi, xm) = γ0 exp(− xi − xm
2
/(2σ2
))zm,k−1
Chandler et al. Spatial occupancy models 7 / 13

15. A spatial occupancy model
Probability that site i is colonized by ≥ 1 individual from site m
γ(xi, xm) = γ0 exp(− xi − xm
2
/(2σ2
))zm,k−1
Probability that site i is colonized by ≥ 1 individual from any site
γi,k−1 = 1 −
M
m=1
1 − γ(xi, xm)
Chandler et al. Spatial occupancy models 7 / 13

16. A spatial occupancy model
Probability that site i is colonized by ≥ 1 individual from site m
γ(xi, xm) = γ0 exp(− xi − xm
2
/(2σ2
))zm,k−1
Probability that site i is colonized by ≥ 1 individual from any site
γi,k−1 = 1 −
M
m=1
1 − γ(xi, xm)
Hence:
• Metapopulation extinction is possible
• Useful for PVA, connectivity planning
Chandler et al. Spatial occupancy models 7 / 13

17. Results Local extinction and hydroperiod
q
q
q
0.00.20.40.60.81.0
Localextinctionprobability(ε)
Intermittent Semi−permanent Permanent
Chandler et al. Spatial occupancy models 8 / 13

18. Results Colonization and connectivity
Chandler et al. Spatial occupancy models 9 / 13

19. Results Colonization and connectivity
2008 2009 2010
2011 2012 2013
2014 2015 2016
2017 2018 2019
2020 2021 2022
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Chandler et al. Spatial occupancy models 9 / 13

20. Results Proportion of sites occupied
q
q
q
q
q q q q q q q q q q q q
2010 2015 2020
0.00.20.40.60.81.0
Year
Proportionofsitesoccupied
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 10 / 13

21. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

22. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

23. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

24. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

25. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

26. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

27. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

28. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

29. Extinction risk after new reintroductions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 km
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
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
qq
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
q qqqq
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq q q
q
q
q
qqq qq
q
q
qq
q q
q
q
q
q
q
q qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
qq
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
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
2020 2040 2060 2080 2100
0.00.10.20.30.40.50.6
Year
Extinctionrisk
Chandler et al. Spatial occupancy models 11 / 13

30. Future directions
• Abundance-based formulation
Chandler et al. Spatial occupancy models 12 / 13

31. Future directions
• Abundance-based formulation
• Landscape resistance to movement
Chandler et al. Spatial occupancy models 12 / 13

32. Future directions
• Abundance-based formulation
• Landscape resistance to movement
• Undiscovered sites
Chandler et al. Spatial occupancy models 12 / 13

33. Thanks