Implications of uncertainty: Bayesian modelling of aquatic invasive species spread Corey Chivers & Brian Leung Dept. of Biology, McGill University http://madere.biol.mcgill.ca/cchivers/

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3 Difficulties in forecast modelling of invasive species ● Limited Data – Finite resources – Rare events – Long Distance Dispersal – Large scale phenomena ● Incomplete knowledge of how the processes work – How well is 'reality' captured through our abstraction? ● Stochasticity – Invasion/spread are probabilistic phenomena – Noise and non-determinism

Prediction is very hard, Prediction is very hard, especially about the future especially about the future -Niels Bohr, physicist (1885-1962) -Niels Bohr, physicist (1885-1962)

5 Why does uncertainty matter? ● Rather than a single estimate about a future state of nature, a forecast should be a probability distribution over the range of possible future states. t=0 t=T Probability Density

6 Why does uncertainty matter? t=0 t=T Probability Density Risk =( Probability) x(Consequence)

Outcome A Outcome B Outcome C Outcome E Outcome F Do A Do B Do C Do D Do Nothing t = 0 t = T Time In a changing world, not making a decision has consequences, intended or otherwise.

8 Forecasting aquatic species spread

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10 O O O O = known uninvaded

11 X X X O O O X = known invaded O = known uninvaded

12 X X X O O O ? ? ? ? ? ? ? ? ? X = known invaded O = known uninvaded ? = Unknown ?

13 X X X O O O ? ? ? ? ? ? ? ? ? X = known invaded O = known uninvaded ? = Unknown ?

14 X X X O O O ? ? ? ? ? ? ? ? ? X = known invaded O = known uninvaded ? = Unknown ?

15 X X X O O O ? ? ? ? ? ? ? ? ? X = known invaded O = known uninvaded ? = Unknown ?

16 311/1600 = 19% data coverage

17 Bayesian modelling of dispersal and environmental suitability Non-equilibrium species distribution modelling Q it E it α i GM X i β αi =−log(1−p i ), P i = 1 1+e−z i , z i =β 0 +∑ j=1 E β j X ij . E(Q it ,αi )=1−e−(α i Q it )c , Q it = propagule pressure generated by underlying dispersal network c > 1 indicates an Allee effect α i = habitat suitability X ij = Environmental condition j at site i. β j = Estimated coefficients

18 Environmental Variables: Sodium (mg/L) Potassium (mg/L) Magnesium (mg/L) Calcium (mg/L) Total Phosphorus (μg/L) SiO3 (mg/L as Si) Dissolved Organic Carbon (mg/L) True Colour (TCU) Total inflection point alkalinity (mg/L as CaCO3) Total fixed end point alkalinity to pH 4.5 (mg/L as CaCO3) pH Conductivity @ 25*C (μS/cm) Secchi Depth

19 Results

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21 Validation ● We can evaluate the performance of this model using AUC. (~0.85) ● What we really want to know are probabilities. – Expressions of uncertainty ● Ongoing work into a validation metric which assesses model performance in terms of probability across the entire prediction range. ● Will use 102 new sample points from 2010.

Code available on Github https://github.com/cjbayesian/grav_mod

Thank you Supervisors: Dr. Brian Leung Dr. Elena Bennett Dr. Claire De Mazancourt Dr. Gregor Fussman 300 Lakes Survey Team Lab Mates: Johanna Bradie Paul Edwards Kristina Marie Enciso Andrew Sellers Lidia Della Venezia Erin Gertzen