Control of a Lake Network Invasion: Bioeconomics Approach Alex Potapov, Mark Lewis, and David Finnoff* Centre for Mathematical Biology, University of Alberta. *Department of Economics and Finance, University of Wyoming

Sea Lamprey Zebra Mussels Great Lakes Invasion by Alien Species Rusty crayfish

Economic Impact: Average Cost to Control Zebra Mussels by Plant Type as of 1995: Hydroelectric facilities $83,000. Fossil fuel generating facilities $145,000. Drinking water treatment facilities $214,000. Nuclear power plants $822,000.

Damage to Recreation, Change in Ecosystems Beaches covered by shells, smell, cleared water, less sport fish

Crusty boats…

Zebra mussel spread

Rusty Crayfish in North America

How do invaders spread? To the Great lakes – various ways, mainly by ships in ballast water. Within the lake system – naturally To land lakes and between them – mainly through fishing and boating equipment. Prevention – equipment washing…

20 researchers from 5 universities: D. Lodge and Gary Lamberti (U Notre Dame), M. Lewis (U Alberta), H. MacIsaac (U Windsor), J. Shogren and D. Finnoff (U Wyoming), Brian Leung (McGill) 5 year project Collaborative project between biologists, economists and mathematicians

Clark C.W. Mathematical Bioeconomics. The optimal management of renewable resources Van Kooten G. C. and Bulte E. The Economics of Nature, Main idea: Model an ecological system as a dynamical system Include human activity and costs/benefits Determine the optimal harvesting/management via optimal control theory Analysis using optimal control theory

Invader dynamics + costs/benefits optimization (integrative bioeconomic models) Population level Dispersal IntroductionTransportation Prevention/ Control Costs/Benefits Optimization Expenses Losses/changes

General invasion model with control Model includes dynamics of the invader in the lake u i, possible controls, minimization of costs (or maximizing benefits) Minimize costs or maximize benefits

Macroscopic model for invasion spread Invasion is described in terms of proportion of infected lakes p=N I /N. Invader propagules are transported from lake to lake by boats (intensity A 1 ), probability of survival A 2, increase in number of infected lakes  t=N  p during  t is (N–N I )N I  tA 1 A 2,

Invader Control Prevention effort at infected and uninfected lakes: x and s (effort/lake/time). Probability of propagule escaping treatment at infected lake is a  1, and at uninfected lake is b  1. Washing efficiency 1–a, and 1–b. Assume effects of two successive prevention treatments are independent: a(x 1 +x 2 )=a(x 1 )a(x 2 ) Dynamic equation for proportion of lakes invaded:

Costs Invasion cost: g ($/lake/time) – decrease in benefits or increase in costs Prevention cost: w x at invaded lakes w s at uninvaded lakes Total invasion cost/lake:

Discounting and optimality Total cost during time interval 0  t  T: Optimal control problem: minimize J by choosing x(t) and s(t) 0  t  T Cost functional Discounting function Dynamical equation for proportion of lakes invaded p(t) (optimization constraints)

Maximum Principle Goal: maximize H (Hamiltonian) Dynamical equation for shadow price  t) with terminal condition Dynamical equation for proportion of lakes invaded p(t) with initial condition Optimality (max in x, s) conditions at any 0  t  T

Optimality conditions Three types of control 1. Donor control 2. Recipient control 3. No control

Non-overlapping control regions x-control s-control No control The current value Hamiltonian H is maximized by x=x *, s=s * Finish here at time t=T, p=p e Start here at time t=0 When there is no discounting (r=0), solution can be calculated analytically from p0p0

Terminal time specifies optimal trajectory pepe pepe pepe 1 Proportion infected lakes Shadow price Donor controlRecipient control Proportion infected lakes Terminal time T T T T small: No control T intermediate:Donor, then No control T large: Donor, then Recipient then No control For any given T, there exists and optimal trajectory

Shadow price Proportion infected lakes Control costs Donor controlRecipient control Donor control Two different phase plane representations (p-  plane, control-p plane) Control costs— proportion infected phase plane Shadow price— proportion infected phase plane

Donor controlRecipient control Shadow price Proportion infected lakes Donor control Recipient control Proportion infected lakes Control costs Solid line: No discounting (solution is calculated analytically) Dashed line: With discounting (solution must be calculated numerically) Effect of the discount rate

Outcomes with and without discounting Proportion infected lakes Control levels Without discounting With discounting

Control efficiency k=k 1 =k 2 is varied. Thick solid — x(t), thick dashed — s(t), thin solid — p(t), thin dashed — uncontrolled p(t), A=1, p 0 =0.3, g=0.5, r=0, T=50,. No control is optimal

Conclusions-1 We can delay invasion but not stop it. Goal is to delay invasion so as to increase net benefit from a bioeconomic perspective. Problem can be analyzed using phase plane methods. Three main strategies for controlling invaders: Donor control, recipient control, no control. Switching occurs between strategies as the invasion progresses. Short (e.g., political) time horizons can yield no control as optimal. Control strategies are sensitive to discounting. Discounting reduces early investment in control and allows invasion to progress quickly.

Model extension: eradication C is linear in h,  bang-bang control: h=0 or h=h max.

Controls in the phase plane

New kind of solution: complete eradication If we eradicate invader by some moment t 1, then for t>t 1 there are no losses and no costs. New formulation: free terminal time, fixed end state p=0, and hence s=0. Different boundary condition

Variety of solutions: isochrones view New effect: several locally optimal solutions. Complete eradication is the optimum only for big enough T. p0p0 Isochrone with appropriate T Isochrone = set of all initial states (p,  ) such that  (T)=0 Beginning of optimal trajectory Beginning of suboptimal trajectory Eradication is optimal Complete eradication trajectory

Terminal value: beyond the control horizon At t = T the ecosystem remains and still can bring benefits, must have some value V ( p e ). Then it is necessary optimize cost+terminal value. Let a system with invasion level p under controls x(t) produces benefits with a rate W(p,x), then we need How to define V T ( p(T) )? No agreement on this at present.

Invariant terminal value Let us define V through infinite horizon problem. p(0)=p 0. Define Value = present cost of maximum future benefits under optimal management Then solution of a finite time horizon T optimal control problem with terminal cost V(p e ) coincides with x  (t) on (0,T) (x(t) does not depend on T). Can be formulated in terms of minimizing future costs under

A solution of an infinite-horizon problem (IHP) ends at an invariant set of the dynamical system. Theorem. Let the solution of IHP {x  (t),p  (t)} exists and is unique for each p 0 =p(0) and the corresponding invariant end- state. Then optimal control x T (t) for finite-horizon problem with the terminal value V(p(T)) and the same p 0 x T (t)=x  (t) on (0,T). Either x T (t)=x  (t), 0<t<T, p(T)=p  (T), or a contradiction

Proof: suppose p(T)  p  (T), then V T >V , then x  (t) is not optimal V T =V , then x  (t) is not unique V T <V , then x(t) is not optimal  x T (t)=x  (t), 0<t<T (optimality principle)

Example: no eradication r=0.01r=0.07

Example: with eradication r=0.01 r=0.03 r=0.10 Complete eradication No eradication at the end Optimal trajectory Suboptimal trajectory

Implications of terminal value for the problem with explicit spatial dependence Optimal control problem – system of 2N equations; Infinite-horizon problem – only steady states are important; at small discount – look for the best steady state; May be a considerable simplification: first study steady states, then choose a best way to them

Accounting for Allee effect Allee effect – population cannot grow at low density Cannot be integrated into the macroscopic model Single lake description

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 with external flow

Explicit spatial model with Allee effect Optimal invasion stopping: find optimal spatial controls distribution that keeps flow below critical at uninvaded lakes We can look for the optimal place to stop the invasion

Example: Linearly ordered lakes, B ij =B (|i–j|) Numerical solution gives spatial distribution of controls B ij =B 0 exp(–  |i–j|) B ij =B 0 / (1+(  |i–j|) 2 )

Conclusions-2 Eradication of the invader can make the problem of finding optimal control more complicated and gives new strategies; Terminal value through infinite-horizon problem reduces analysis to steady states and trajectories leading to them – a considerable simplification of analysis, especially for high- dimensional problems, + more transparent management recommendations; Allee effect allows to stop invasion without eradication; accounting for the terminal value leads to the natural problem of optimal invasion stopping

Acknowledgements ISIS project, NSF DEB NSERC Collaboration Research Opportunity grant.. References A.B. Potapov, M.A. Lewis, D.C. Finoff. Optimal Control of Biological Invasions in Lake Networks. Journal of Economic Dynamics and Control, 2005 (submitted). D.C. Finoff, M.A. Lewis, A.B. Potapov. Optimal Control of Biological Invasions in Lake Networks., 2005 (in preparation). A.B. Potapov, M.A. Lewis. Optimal Spatial Control of Invasions with Allee Effect., 2005 (in preparation).

Influence of invasion losses per lake g on the optimal control policy

Influence of control time horizon T on the optimal control policy

Influence of initial proportion of infected lakes  on the optimal control policy

Influence of discounting rate r on the optimal control policy