Modeling Colonization of BC Rivers by Feral Atlantic Salmon

1. Modeling Colonization of BC Rivers by Feral Atlantic Salmon 2008 PIMS Mathematical Biology Summer Workshop

2. Atlantic Salmon(Salmon Salar) • Aquacultured species in Northeast Pacific • Escapes recorded • Feral Atlantic sightings in NE Pacific and Pacific Northwest rivers • Habitat use, life history point to competition with native Steelhead (O. Mykiss)

3. Aquaculture Feral Ecology & Math • Predict a threshold rate of escape necessary for feral population sustainability • Apply threshold concept to spatial situations • Account for stochastic escape events

4. Assumptions • Allee Effect in Atlantic reproduction • No hybridization with native populations • No competition, even though it’s ecologically important • Probabilistic colonization of rivers determined by distance from farm • Sex ratio of escapees is even • Surpassing the Allee threshold is establishment • Non-overlapping generations

5. Modeling the Allee Effect • xt+1 = (k+m)(xt)2/(xt + Km) • xt := number of Atlantic salmon at time t • K := carrying capacity • m := Allee threshold • For xt < m, the population will crash • For xt > m, the population will grow to the carrying capacity

6. Allee Effect

7. Including Immigration • Assume a constant rate of immigration of escaped fish (we will allow for stochasticity later). • Model: • xt+1 = (K+m)(xt)2/(xt + Km) + ε • ε := the amount of escaped salmon entering the population

8. Allee Growth with Immigration • When immigration ε exceeds threshold τ, only one stable state, corresponding to carrying capacity K • For ε > τ, where τ => f (x) = x and f ’(x) = 1, single equilibrium

9. Sensitivity of ε to Allee Threshold

10. Applying the Immigration Model across Space • Consider fish farm(s) located near rivers in space • εamount of fish escaping a cluster of farms in each time period. • di distance from the centre of the cluster of farms to river i • Assign dispersal rates as εdi/(Σi=1→ndi)

11. Spatial Model with Immigration • xr,t+1 = (K+m)(xr,t)2/(xr,t + Km) + ε/di(Σi=1→n1/di) • Distribution of escapees allows for an larger ε before without colonization • Stochasticity: ε - stochastic variable with Poisson distribution

12. Real World Scenario North East Vancouver Island • Six Steelhead Rivers: Keogh, Nimpkish, Kokish, Tsitika, Eve, Salmon • Each K estimated (for Steelhead) by British Columbia Conservation Foundation • Intensive Aquaculture in Broughton Archipelago

13. Parameters • Distances estimated from Broughton center via Google Earth • K set equal to Steelhead estimates per BCCF http://www.bccf.com/steelhead/watersheds.htm • m set at 10% of K

14. Application to One River • 1000 reps • 10 gens • Poisson-distributed number of escaped spawners at each generation m (K) decreases m (K) increases Distancekeogh increases Distancekeogh decreases

15. Application to Six rivers

16. A Closer View… 795 94 307 370 200 101

17. Next Steps • Rational dispersal mechanism • Separate estimation of m from K for rivers • Staged, overlapping growth model • Biologically-motivated Allee functional form • Competition…

18. Acknowledgements • Frank Hilker & Peter Molnar for formal guidance and lots of their time • Lou Gross & Mark Lewis for free agent advising • Gerda De Vries, Cecilia Hutchinson and all who participated in the PIMS Summer Workshop