Optimal Management of
Established Bioinvasions
Becky Epanchin-Niell
Jim Wilen
Prepared for the PREISM workshop
ERS Washington DC
May 2011
Bioinvasions Are:
Spatial-dynamic Processes
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Spatial-dynamic processes are driven by dynamics
at a point and diffusion between points
Generate patterns that evolve over both space and
time
Some other examples
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Forest fires
Floods
Aquifer dynamics
Groundwater contamination
Wildlife movement
Human/animal disease
Questions raised by bioinvasions:
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How does uncontrolled invasion spread?
Intensity and timing of optimal controls
• when and how much control?
Spatial strategies for control
 Where should control be applied?
Effect of spatial characteristics of the invasion
and landscape on optimal control
Externalities, institutions, and reasons for
intervention
Modeling Optimal Bioinvasion
Control with Explicit Space
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Simple small model
Build intuition with multiple optimization
“experiments”
Identify how space matters with spatialdynamic processes
Explore how basic bioeconomic parameters
affect the qualitative nature of the solution
Special (Spatial) Modeling Issues
boundaries
heterogeneity
spatial
geometry
diffusion
process
The model
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Invasion spread
−
−
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Cellular automaton model
Approximates reactiondiffusion
Control options
– Spread prevention
– Invasion clearing
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Min. total costs & damages
Simplicity
- 2n*t configurations
invadable land
invaded land ($d)
border control ($b)
clearing ($e)
Finding the optimal solution
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Dynamic problems
 Ordinary differential equations
 End-point conditions—2 point boundary prob.
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Spatial-dynamic problems
 Partial differential equations
 End-points---infinite dimension spatial bound.
 difficult/impossible to analytically solve
Finding the optimal solution
Dynamic programming solutions
 Backwards recursion
 Curse of dimensionality amplified
 Number of states/period 2 N
5x5  2 25  33,564, 432
 Additional problems
 Eradicate vs. slow or stop solutions
 Transversality conditions
Mathematical model
spread
damages clearing
costs prevention
Cell remains invadedcosts
unless cleared
Subject to:
Cell becomes invaded if has invaded
neighbor unless prevention applied
variables
parameters
Solution approach:
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Binary integer programming problem
- SCIP (Solving Constraint Integer Programs)
Scaling
Solves large-scale problems in seconds/minutes
Can perform numerous comparative spatialdynamic optimization “experiments”
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Cost parameters, discount rate
Invasion and landscape size
Invasion and landscapes shape
Invasion location
Results:
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Wide range of control approaches
– e.g., eradicate, clear then contain, slow then contain,
contain, slow then abandon, abandon
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If clearing is optimal, it is initiated immediately
Landscape & invasion geometry important
Spatial strategies for control
– prevent/delay spread in direction of high potential
damages
– reduce extent of exposed edge prior to containment
– whole landscape matters
Experiment 1: Initial invasion size
Finding: Larger invasion
decreases optimal control
Reason: Larger invasion 
higher control costs & less
uninvaded area to protect
Invasion size = Control delay
Experiment 1: Initial invasion size
Total
(optimized)
costs &
damages
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Larger delay
 higher total costs and damages
Experiment 2: Landscape size
Finding: Larger landscapes demand greater levels of
control
Reason: Larger uninvaded areas  Higher potential longterm damages
Experiment 3: Landscape shape
Finding: Higher optimal control effort in more compact
landscapes
Reason: Damages accrue faster  Higher long-term
potential damages in more compact landscapes
Experiment 4: Invasion location
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Central invasions
- higher potential long-term damages
 more control
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Invasions near range edge
- lower control costs
 more control
Invasion location has ambiguous effect on optimal
control effort
Central invasions  higher costs & damages
Spatial control strategies I:
1) Prevent spread in direction of
high potential long-term
damages
2) Reduce the extent of invasion
edge prior to containment
0
t = 36…
1
2
4
5
Spatial control strategies II:
1) Reduce the extent of invasion edge
2) Protect areas with high potential
damages
0
t = 36…
1
2
4
5
Spatial control strategies III:
t = 3…
0
1
2
Again, reduce invasion edge prior to containment.
11  7 edges exposed
Spatial control strategies IV:
0
t = 36…
1
2
4
5
Spatial control strategies V:
Protect large
uninvaded areas
Entire landscape
matters
078
t = 396
1
2
4
5
If landscape homogeneous
No control… let spread
Barrier cost (b) = 50
Removal cost (e) = 1500
Baseline damages (d) = 1
If high damage patch in landscape 
Eradicate
Barrier cost (b) = 50
Removal cost (e) = 1500
Baseline damages (d) = 1
High damages (d) = 101
t=8
632
7
1
0
4
5
If higher removal costs 
Slow spread; protect high damage patch
Barrier cost (b) = 50
Removal cost (e) = 10000
Baseline damages (d) = 1
High damages (d) = 101
t=8
6311
7
0
4
5
1
2
10
14
12
15
16
917…
13
If lower damages in patch
Slow the spread
Barrier cost (b) = 50
Removal cost (e) = 10000
Baseline damages (d) = 1
High damages (d) = 51
t=8
6311
7
0
4
5
1
2
10
12
9
13…
Summary of control principles
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High damages, low costs, and low discount rate,
 higher optimal control efforts
Protect large uninvaded areas
–
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Reduce extent of exposed edge prior to
containment
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–
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prevent/delay spread in direction of high potential
damages
employ landscape features
alter shape of invasion (spread, removal)
Entire invasion landscape matters
Geometry matters (initial invasion, landscape)
Control sequences/placement can be complex
Modeling multi-manager
landscapes
invaded
about to be invaded
adjacent to
“about to be invaded”
Offer to “about to be
invaded” cell to induce
prevention
•Unilateral management
•Bilateral bargaining
•Local “club” formation
Outcomes from private control
vs. optimal control:
Bilateral bargaining
Clearing cost (e)
Unilateral management
Border control cost (b)
Clearing cost (e)
Optimal control
Border control cost (b)
Local “club” coordination
Thank you!!