1. Turbulent Coexistence Heather Berkley, Satoshi Mitarai, Bruce Kendall, David Siegel

2. Species Coexistence Consider two similar species A & B – Species A has a slightly better ability to utilize resources – Recruits compete for limited resources at settlement sites – Spawning timings are separated by weeks Compare cases with i) smooth dispersal kernel & ii) packet model for connectivity – Smooth dispersal kernel: spawning timing does not matter – Packet model: species A & B “catch” different eddies & can settle at different sites

3. Diffusion Case If they are put together, species B becomes extinct, species A thrives On their own, both species can persist Time (years) IC’s: A = 100, B = 100

4. Packet Model Larval settlement as arrival of N packets L = domain size l = eddy size (50 km) T = Spawning time t = eddy turnover rate (14 d) eddy size ( l ) N larval packets

5. Spawning Window Overlap Specify how many days of overlap between spawning times for both species Makes some packets perfectly correlated for both species and others independent Packets will have same settlement locations Species A Spawning Window Species B Spawning Window TIME

6. Parameters Tsp (spawning time) = 30 days for both –Vary amount of overlap Fecundity of Sp.A = 0.5 Fecundity of Sp.B = 0.45 Adult Mortality = 0.09 Run time = 1000 yrs; Patch size = 5 km; Domain size = 500 km; Averaged over 100 simulations Larvae on larvae DD (total # of both sp)

7. 28 days

8. Theory Species A is stronger competitor (f A > f B ) Want to know if the growth rate of Sp. B is positive when rare and Sp. A is at equilibrium Estimate E[R A ] and E[R B ] with Taylor expansion around mean number of settlers –need E[S A ], E[S B ], var(S A ), & cov(S A, S B, )

9. Assume spatially homogenous Adult distribution: Settlers: 1== ∑∑ y B y A DD 0 = = BfSE KfSE BB AAA ][ ][ ()() ()() BAABABA AAAA DDBKffSS DKfS,cov, var 0 22 = =

10. Coexistence Condition Coexistence when: Calculate var(D A ) and cov(D A,D B ) from Packet Model and Flow Simulations

11. cov(D A,D B )

12. Coexistence line calculated with cov(D A,D B ) & var(D A )

13. Equilibrium Pop B/A Coexistence Sp.B extinction

14. Equilibrium Pop B/A Coexistence Sp.B extinction

15. Recruitment Differences Key difference between species is density dependent per-capita recruitment rate, R/N For Species A:

16. At equilibrium, average # Settlers is 50

17. Variability in Settlement If settlement varied between 20 & 80, then the per-capita recruitment rate would be less than if settlement was constant at 50

18. Recruitment Differences For Species B, recruitment is conditional on Sp. A settlers, too: So,

19. Recruitment Differences Do some math, get:  AB is the correlation in dispersal between species Low correlation, 1 st part dominates High correlation, 2 nd part dominates

20. Correlation in Settlement:  = 0 Expected per-capita R rate for Sp B At equilibrium Settlement, R/N for Sp. A > Sp. B

21. Nonlinear averaging leads to R/N Sp. B > Sp. A Expected per-capita R rate for Sp B Correlation in Settlement:  = 0 Variable Settlement between 20 & 80

22. Correlation in Settlement:  = 0.5 Expected per-capita R rate for Sp B

23. Correlation in Settlement:  = 1 Expected per-capita R rate for Sp B

24. Because of nonlinear averaging, variability in settlement will increase the average per-capita recruitment rate for Sp. B above that of Sp A, so Sp. B can invade when rare This advantage becomes weaker as the correlation in dispersal increases Conclusions