1. Modeling aquatic invasions and control in a lake system: principles and approaches Alex Potapov Centre for Mathematical Biology, University of Alberta and Lodge Lab, University of Notre Dame http://www.math.ualberta.ca/~apotapov/ Joint work with M. Lewis and D. Finnoff

2. Sea Lamprey Zebra Mussels Motivation: Great Lakes Invasion Rusty crayfish

3. Economic and ecological damage from the invaders Zebra mussels  Clog water pipes and water treatment facilities; yearly cost/facility ~$80,000- 800,000  Ecological damage Sea Lamprey  1 Adult kills about 40 fishes  In 1920s decreased fish harvest in GL about 50 times Spiny waterflea  Collect on fishing equipment, may damage it.  No predators  Consume lots of plankton

4. Zebra mussels spread since 1988 Quickly spreads along rivers, slower between inland lakes 1993 1999 2003

5. Between lakes ZM travel with boats

6. Our goal: modeling and understanding optimal management policies in a system of N lakes Approach: Bioeconomics Clark C.W. Mathematical Bioeconomics. The optimal management of renewable resources. 1990. Van Kooten G. C. and Bulte E. The Economics of Nature, 2000 Integrates: Ecological part (population dynamics and dispersal) Control measures Economic part (costs and benefits) Optimization (analysis techniques) May be implemented in a number of ways

7. Possible model types What details to be included? What kind of model to be chosen? Main choices: Regional scale – Lake scale Deterministic – Stochastic Continuous – Discrete

8. Possible modeling approaches: virtues and shortcomings Model \ scaleRegion resolutionLake resolution Deterministic, continuous simplicity, tractability, analytical results no spatial and population effects Accounts for spatial and population effects Hard to understand a big picture, limitations on system size Stochastic, discrete No advantages over deterministic models More realistic, easier integration with other ecological concepts Serious limitations on system size

9. Model 1: regional scale Lake system description: proportion of invaded lakes p=N I /N Boat and invader traffic:  average traffic between lakes T A  average number of invaders carried by a boat   invader flow into an uninvaded lake w=N I  T A  = pN  T A   probability of a lake to become invaded P I =αwdt  the number of newly invaded lakes dN I =(N-N I )αpN  T A  dt Equation for uncontrolled invasion spread

10. Invader control An infected boat on average carries  invaders Processing (washing) with cost x reduces  to a(x)  Washing a boat after use at an invaded lake: x ( t ) Washing a boat before use at an uninvaded lake: s ( t )

11. Losses due to invasion at an invaded lake: g ($/year) Total costs per year per lake Costs and discounting Optimal control problem: minimize total present cost with discount rate r and terminal cost V T

12. How to solve optimal control problems: Pontryagin Maximum Principle

13. Shadow price Proportion infected lakes Control costs Donor control Recipient control Donor control Example: different types of solution Control costs— proportion infected phase plane Shadow price— proportion infected phase plane

14. Macroscopic model: main results In some cases the optimal control problem can be solved analytically, otherwise can be analyzed by phase plane methods; Good understanding of solution properties and role of different parameters; Invader flow cannot be reduced to zero, invasion can be slowed down, but not stopped; Role of terminal cost: V T =0 leads to no-control strategies for small T.

15. Terminal cost V T and time horizon T V T =0 gives no control at t close to T. “Terminal boost of invasion”. How to estimate terminal cost? Solve optimization problem from T to infinity… Then solution on [0,T] coincides with one for T= 

16. Infinite-horizon problem Solution tends to a steady state Two types of problems: a) Optimal transition to a steady state b) Optimal steady state. Problem can be essentially simplified, little difference at small r : variation problem → optimization problem Nontrivial steady states require eradication of the invader

17. Effects absent in macroscopic model 1. Boat traffic between the lakes may strongly deviate from average 2. Optimal control may vary from lake to lake 3. Population dynamics of some invaders has Allee effect: too small population cannot grow. This may allow to stop the invasion

18. Spatially explicit boat traffic and invader flow Intensity of boat traffic from lake j to lake i : T ij. A boat on average can carry  invaders.

19. Population dynamics with Allee effect No external flow, population goes extinct at small u Weak external flow, w<|F min |, population still goes extinct at small u; Strong external flow, w>|F min |, population grows from any u Allee effect – small populations go extinct.

20. Complete optimal control problem is too complicated. Analytically intractable, only numerical study Optimal asymptotic steady states: considerable simplification. At steady states there is no progress of the invader: therefore this is a problem of optimal invasion stopping How to study?

21. Steady state conditions

22. Example: Fixed configuration of invaded lakes, Optimal spatial control allocation Exponential T ij (d)Power T ij (d)

23. Next step: optimization over different configurations {u i }: where is the best stopping location? Discrete set of configurations, nonlinear function to be optimized: no fast algorithm to solve. Configurations to be tested: Simplifying approaches needed. Clustering?

24. Example: invasion stopping in clustered/nonclustered lake system N identical lakes identically connected N identical lakes with connections forming two clusters. Invasion: C 1 -L 1 -L 2 -C 2

25. Big losses: stop anywhere Small losses: stop only if invaded less than critical # of lakes (Stopping cost) (# lakes invaded)

26. Small losses and two clusters: two critical # of lakes (if invasion proceeds too far, it is optimal to abandon the first cluster and to protect the second) (Stopping cost) (# lakes invaded)

27. Random invasion paths in a gravity model, strong connections are more probable

28. Stochastic model: can account for fluctuations in invader flow and “occasional invasions” The simplest description: each lake can be in invaded/uninvaded state (1 or 0). Description in terms of system state u I, I=1,…,M Each year the state of each uninvaded lake may change with probability depending on incoming invader flow W : p(0→1)=1 – exp(–αW) This allow to calculate transition probabilities P(u I →u J ). They depend on controls at each lake x i at the state u I. Controlled Markov chain or Markov decision process Optimal solution: minimizes total average costs

29. “Curse of dimensionality” NMTime to solve optimal control problem 102 10 =10242—30 min 20 2 20  10 6 ~month 50 2 50  10 15 ~100 mln years Minimizing average costs -> accounting for all possible invasion paths: problem size grows exponentially with N Next step: developing approximate methods of solving this problem

30. Conclusions Spectrum of models allows us to understand different sides of models Simple averaged models show global picture and allows us better setting up more detailed problems Lake-scale models account for local features of population dynamics and transportation Considering of optimal steady states allows considerable simplification: from optimal control to optimization Stochastic problems are more realistic but much harder to solve Acknowledgements ISIS project, (NSF DEB 02-13698) NSERC Collaboration Research Opportunity grant.

31. Examples of relative ecosystem value for different losses g and discount r r = 0.03g = 1.5