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