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Modeling Colonization of BC Rivers by Feral Atlantic Salmon. 2008 PIMS Mathematical Biology Summer Workshop. Atlantic Salmon (Salmon Salar). Aquacultured species in Northeast Pacific Escapes recorded Feral Atlantic sightings in NE Pacific and Pacific Northwest rivers - PowerPoint PPT Presentation
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Modeling Colonization of BC Rivers by Feral
Atlantic Salmon
2008 PIMS Mathematical Biology Summer Workshop
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)
Ecology & Math
Predict a threshold rate of escape necessary for feral population sustainability
Apply threshold concept to spatial situations Account for stochastic escape events
Aquaculture Feral
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
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
Allee Effect
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
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
Sensitivity of ε to Allee Threshold
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.
di distance from the centre of the cluster of farms to river i
Assign dispersal rates as εdi/(Σi=1→ndi)
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
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
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 KRiver Carrying Capacity
(Adult Steelhead)Distance from Farms (km)
Keogh 910 59.54
Nimpkish 3,600 38.97
Kokish 520 32.28
Tsitika 572 31.10
Eve 897 39.26
Salmon 1200 54.57
Application to One River
m (K) increases
Distancekeogh increases
Distancekeogh decreases
m (K) decreases
1000 reps 10 gens Poisson-
distributed number of escaped spawners at each generation
Application to Six rivers
A Closer View…
94
101200
307
370
795
Next Steps
Rational dispersal mechanism Separate estimation of m from K for rivers Staged, overlapping growth model Biologically-motivated Allee functional form Competition…
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