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

2. 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) Atlantic Salmon (Salmon Salar)

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

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 x t+1 = (k+m)(x t )2/(x t + Km) x t := number of Atlantic salmon at time t K := carrying capacity m := Allee threshold For x t < m, the population will crash For x t > 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: x t+1 = (K+m)(x t )2/(x t + Km) + ε ε := the amount of escaped salmon entering the population

8. 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 Allee Growth with Immigration

9. Sensitivity of ε to Allee Threshold

10. Applying the Immigration Model across Space Consider fish farm(s) located near rivers in space ε amount of fish escaping a cluster of farms in each time period. d i distance from the centre of the cluster of farms to river i Assign dispersal rates as εd i /(Σ i=1→n d i )

11. Spatial Model with Immigration x r,t+1 = (K+m)(x r,t )2/(x r,t + Km) + ε/d i (Σ i=1→n 1/d i ) 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 RiverCarrying Capacity (Adult Steelhead) Distance from Farms (km) Keogh91059.54 Nimpkish3,60038.97 Kokish52032.28 Tsitika57231.10 Eve89739.26 Salmon120054.57

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

15. Application to Six rivers

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

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