Main Points. Note: Tutorial #1, help session, and due date have all been pushed back 1 class period. Will send Tutorial #1 by end of day.

Transcription

1 1) Demographic PVA -- age- and stage-structured matrices Main Points Pre-reading: Thursday 21 September: Marris Tuesday 26 September: NA Note: Tutorial #1, help session, and due date have all been pushed back 1 class period. Will send Tutorial #1 by end of day. Terms: age-structured (Leslie) matrix, stage-structured (Lefkovitch) matrix 1

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3 Hirola Declines Coincided With Elephant Extirpation Ali et al J. Appl. Ecol. 3

4 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) From the data, calculate λ for each time step: N t+1 /N t = λ t 4

5 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) From the data, calculate λ for each time step: N t+1 /N t = λ t this will give you a vector of observed lambdas. 11,5/16, =.72 5

6 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Draw randomly from vector of observed λ's, and simulate N 2, N 3, N 4, and so forth, until a desired year t: N 1 = N λ 6

7 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Draw randomly from vector of observed λ's, and simulate N 2, N 3, N 4, and so forth, until a desired year t: N 1 = N λ drawn at random from vector of observed λ's 7

8 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Draw randomly from vector of observed λ's, and simulate N 2, N 3, N 4, and so forth, until a desired year t: N 1 = N λ starting population size from observed data 8

9 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Draw randomly from vector of observed λ's, and simulate N 2, N 3, N 4, and so forth, until a desired year t: N b1 = N λ bootstrap projection for population size after 1 year 9

10 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Draw randomly from vector of observed λ's, and simulate N 2, N 3, N 4, and so forth, until a desired year t: N b1 = N λ N b2 = N b1 λ N b3 = N b2 λ... N bt = N bt-1 λ 1

11 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Draw randomly from vector of observed λ's, and simulate N 2, N 3, N 4, and so forth, until a desired year t: N b1 = N λ N b2 = N b1 λ N b3 = N b2 λ... N bt = N bt-1 λ for each N b, λ is drawn at random from the observed vector 11

12 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Repeat many times (1s or 1s) to generate a distribution of projected population sizes t years into the future. 12

13 Bootstrapping to Conduct Count-Based PVA 1 years n = 23 95% CI = ) Calculate summary statistics on distribution (e.g., mean, variance, standard deviation, CI, etc). 13

14 Bootstrapping to Conduct Count-Based PVA 1 years population size = 23 95% CI = years pop size = 19 95% CI = years pop size = 14

15 Tree Cover Has Increased ~25% Since Mid-198s Extent of tree cover 1985 Extent of tree cover 212 Ali et al J. Appl. Ecol.

16 Resource selection coefficient + SE Hirola Avoid Trees Ali et al J. Appl. Ecol. 16

17 Matrix Modeling to Conduct Demographic PVA Most populations in nature are age- or stagestructured; that is, their vital rates depend on age or stage Age and stage-structured matrices account for this variability 17

18 Matrix Modeling to Conduct Demographic PVA Leslie matrix = projection matrix of age-dependent survival rates and fecundities >2 yr trees seedlings 15-2 yr trees yr trees saplings 5-1 yr trees

19 Matrix Modeling to Conduct Demographic PVA 6 5 >2 yr trees 15-2 yr trees 1 4 seedlings 1-15 yr trees saplings 5-1 yr trees 2 3 G 21 F 3 F 4 F 5 F 6 G 32 G 43 G 54 G 65 S 66 19

20 Matrix Modeling to Conduct Demographic PVA A n n 1 ~ ~ F 3 F 4 F 5 F 6 n t n t+1 G 21 n 1t n 1t+1 G 32 G 43 * n 2t n 3t = n 2t+1 n 3t+1 G 54 n 4t n 4t+1 G 65 S 66 n 5t n 5t+1 6 x 6 matrix 6 x 1 vector 6 x 1 vector 2

21 Matrix Modeling to Conduct Demographic PVA Lefkovitch matrix = projection matrix of stagedependent transitions and fecundities >4 m adults seedlings 3-4 m adults m trees saplings 1-2 m trees

22 Matrix Modeling to Conduct Demographic PVA 6 5 >4 m adults 3-4 m adults 1 4 seedlings 2-3 m trees saplings 1-2 m trees 2 3 G 21 F 3 F 4 F 5 F 6 S 22 R 23 G 32 S 33 R 34 G 43 S 44 R 45 G 54 S 55 R 56 G 65 S 66 22

23 Matrix Modeling to Conduct Demographic PVA A n n 1 ~ ~ F 1 G 21 F 2 F 3 * G 32 S 33 n t n 1t = n 2t F 1 *n t + F 2 *n 1t + F 3 *n 2t G 21 *n t + S 22 *n 1t + G 23 *n 2t = G 31 *n t + G 32 *n 1t + S 33 *n 2t n t+1 n 1t+1 n 2t+1 23

24 Matrix Modeling to Conduct Demographic PVA A n n 1 ~ ~ * = = / km 2 A n 1 ~ n 2 ~ * = = / km

25 Discussion Q: we discussed how to incorporate stochasticity into count-based PVA. How might we incorporate stochasticity into demographic PVA? 6 5 >4 m adults 3-4 m adults 1 4 seedlings 2-3 m trees saplings 1-2 m trees 2 3 G 21 F 3 F 4 F 5 F 6 S 22 G 32 S 33 R 23 G 43 S 44 R 34 G 54 S 55 R 45 G 65 S 66 25

26 N N Annual population growth rate (λ) λ t = N t+1 /N t Annual rates of population growth calculated from abundance data and/or data on vital rates (i.e., birth rate, death rate, growth (recruitment) rate, etc) Correlations in vital rates through time without correlations with correlations t t 26

27 N N Discussion Q: Why does this happen? without correlations with correlations t t 27

28 Example With Correlations in Vital Rates 1.8 A n n * ~ ~ = λ = 55/3 = 1.83 A n n * ~ ~ = λ = 5.5/3 =.18 28

29 Example Without Correlations in Vital Rates 1.8 A n n * ~ ~ = λ = 37.9/3 = 1.26 A n n * ~ ~ = λ = 22.6/3 =.75 29