Control of a Lake Network Invasion: Bioeconomics Approach

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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
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Great Lakes Invasion by Alien Species
Rusty crayfish
Zebra Mussels
Sea Lamprey
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Economic Impact
Average Cost to Control Zebra Mussels by Plant
Type as of 1995Hydroelectric facilities
83,000.Fossil fuel generating facilities
145,000.Drinking water treatment facilities
214,000.Nuclear power plants 822,000.
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Damage to Recreation, Change in Ecosystems
Beaches covered by shells, smell, cleared water,
less sport fish
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Crusty boats
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Zebra mussel spread
2003
1988
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Rusty Crayfish in North America
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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

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• 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
• http//www.math.ualberta.ca/mathbio/ISIS

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Analysis using optimal control theory

• Clark C.W. Mathematical Bioeconomics. The
  optimal management of renewable resources. 1990.
• Van Kooten G. C. and Bulte E. The Economics of
  Nature, 2000 .
• 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

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Invader dynamics costs/benefits optimization
(integrative bioeconomic models)
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General invasion model with control
Model includes dynamics of the invader in the
lake ui, possible controls, minimization of costs
(or maximizing benefits)
Minimize costs or maximize benefits
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Macroscopic model for invasion spread
Invasion is described in terms of proportion of
infected lakes pNI/N. Invader propagules are
transported from lake to lake by boats (intensity
A1), probability of survival A2, increase in
number of infected lakes ?tN?p during ?t is
(NNI)NI?tA1A2,
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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 1a, and 1b. Assume effects of two
successive prevention treatments are independent
a(x1x2)a(x1)a(x2 )
Dynamic equation for proportion of lakes invaded
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Costs
Invasion cost g (/lake/time) decrease in
benefits or increase in costs Prevention cost wx
at invaded lakes ws
at uninvaded lakes Total invasion cost/lake
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Discounting and optimality
Total cost during time interval 0 ? t ? T
Cost functional
Discounting function
Dynamical equation for proportion of lakes
invaded p(t) (optimization constraints)
Optimal control problem minimize J by choosing
x(t) and s(t) 0 ? t ? T
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Maximum Principle
Goal maximize H (Hamiltonian)
Dynamical equation for shadow price m(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
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Optimality conditions
Three types of control 1. Donor control 2.
Recipient control 3. No control
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Non-overlapping control regions
x-control
s-control
No control
p0
Finish here at time tT, ppe
The current value Hamiltonian H is maximized by
xx, ss
Start here at time t0
When there is no discounting (r0), solution can
be calculated analytically from
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Terminal time specifies optimal trajectory
Proportion infected lakes
T small No control T intermediateDonor, then No
control T large Donor, then Recipient then No
control
Shadow price
Donor control
Recipient control
Terminal time
For any given T, there exists and optimal
trajectory
T
T
T
pe
pe
pe
1
Proportion infected lakes
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Two different phase plane representations (p-?
plane, control-p plane)
Proportion infected lakes
Shadow priceproportion infected phase plane
Shadow price
Recipient control
Donor control
Donor control
Recipient control
Control costsproportion infected phase plane
Control costs
Proportion infected lakes
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Effect of the discount rate
Proportion infected lakes
Solid line No discounting (solution is
calculated analytically) Dashed line With
discounting (solution must be calculated
numerically)
Shadow price
Donor control
Recipient control
Recipient control
Donor control
Control costs
Proportion infected lakes
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Outcomes with and without discounting
Without discounting
Proportion infected lakes
Control levels
With discounting
Proportion infected lakes
Control levels
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No control is optimal
Control efficiency kk1k2 is varied. Thick solid
x(t), thick dashed s(t), thin solid p(t),
thin dashed uncontrolled p(t), A1, p00.3,
g0.5, r0, T50,.
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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.

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Model extension eradication
C is linear in h, ? bang-bang control h0 or
hhmax.
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Controls in the phase plane
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New kind of solution complete eradication
If we eradicate invader by some moment t1, then
for tgtt1 there are no losses and no costs. New
formulation free terminal time, fixed end state
p0, and hence s0. Different boundary condition
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Variety of solutions isochrones view
Isochrone set of all initial states (p,?) such
that ?(T)0
p0
Complete eradication trajectory
Beginning of optimal trajectory Beginning of
suboptimal trajectory Eradication is optimal
Isochrone with appropriate T
New effect several locally optimal
solutions. Complete eradication is the optimum
only for big enough T.
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Terminal value beyond the control horizon
At tT the ecosystem remains and still can bring
benefits, must have some value V(pe). Then it is
necessary optimize costterminal 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 VT(p(T))? No agreement on this at
present.
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Invariant terminal value
Let us define V through infinite horizon
problem. p(0)p0. 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(pe) coincides with x?(t) on
(0,T) (x(t) does not depend on T). Can be
formulated in terms of minimizing future costs
under
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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
p0p(0) and the corresponding invariant
end-state. Then optimal control xT(t) for
finite-horizon problem with the terminal value
V(p(T)) and the same p0 xT(t)x?(t) on (0,T).
Either xT(t)x?(t), 0lttltT, p(T)p?(T), or a
contradiction
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• Proof suppose p(T)?p?(T), then
• VTgtV?, then x?(t) is not optimal
• VTV?, then x?(t) is not unique
• VTltV?, then x(t) is not optimal
• ? xT(t)x?(t), 0lttltT (optimality principle)

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Example no eradication
r0.01
r0.07
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Example with eradication
No eradication at the end
r0.03
r0.01
Complete eradication
Optimal trajectory Suboptimal trajectory
r0.10
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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

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Accounting for Allee effect

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

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Allee effect with external flow
Weak external flow, wltFmin, population still
goes extinct at small u
Strong external flow, wgtFmin, population grows
from any u
No external flow, population goes extinct at
small u
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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
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Example Linearly ordered lakes, BijB
(ij) Numerical solution gives spatial
distribution of controls
BijB0exp(?ij)
BijB0/ (1(?ij)2)
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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

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Acknowledgements

• ISIS project, NSF DEB 02-13698
• 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).
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Influence of invasion losses per lake g on the
optimal control policy
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Influence of control time horizon T on the
optimal control policy
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Influence of initial proportion of infected lakes
? on the optimal control policy
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Influence of discounting rate r on the optimal
control policy