Becky Epanchin-NiellJim Wilen

Prepared for the PREISM workshopERS Washington DC

May 2011

Optimal Management of Established Bioinvasions

Bioinvasions Are:Spatial-dynamic Processes 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

– Forest fires– Floods– Aquifer dynamics– Groundwater contamination– Wildlife movement– Human/animal disease

Questions raised by bioinvasions:

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 Simple small model Build intuition with multiple optimization

“experiments” Identify how space matters with spatial-

dynamic processes Explore how basic bioeconomic parameters

affect the qualitative nature of the solution

Special (Spatial) Modeling Issues

heterogeneity

boundaries

spatial geometry

diffusion process

The model

invadable landinvaded landborder controlclearing

Invasion spread− Cellular automaton model− Approximates reaction-

diffusion Control options

– Spread prevention– Invasion clearing

Min. total costs & damages Simplicity

- 2n*t configurations($d)($b)

($e)

Finding the optimal solution Dynamic problems

Ordinary differential equations End-point conditions—2 point boundary prob.

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

Additional problems Eradicate vs. slow or stop solutions Transversality conditions

2N

5x5 225 33,564,432

Mathematical model

variables parameters

Subject to: damages clearingcosts

spreadprevention

costsCell remains invaded unless cleared

Cell becomes invaded if has invaded neighbor unless prevention applied

Solution approach:

Binary integer programming problem- SCIP (Solving Constraint Integer Programs)

Scaling Solves large-scale problems in seconds/minutes Can perform numerous comparative spatial-

dynamic optimization “experiments”- Cost parameters, discount rate- Invasion and landscape size- Invasion and landscapes shape- Invasion location

Results: Wide range of control approaches

– e.g., eradicate, clear then contain, slow then contain, contain, slow then abandon, abandon

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

Larger delay higher total costs and damages

Total (optimized)

costs & damages

Experiment 1: Initial invasion size

Experiment 2: Landscape size

Finding: Larger landscapes demand greater levels of control

Reason: Larger uninvaded areas Higher potential long-term 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

Central invasions - higher potential long-term damages more control

Invasions near range edge - lower control costs

more control

Central invasions higher costs & damages

Invasion location has ambiguous effect on optimal control effort

t = 0t = 1t = 2t = 3t = 4t = 5t = 6…

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

t = 0t = 1t = 2t = 3t = 4t = 5t = 6…

Spatial control strategies II:1) Reduce the extent of invasion edge2) Protect areas with high potential

damages

t = 0t = 1t = 2t = 3…

Spatial control strategies III:

Again, reduce invasion edge prior to containment.11 7 edges exposed

t = 0t = 1t = 2t = 3t = 4t = 5t = 6…

Spatial control strategies IV:

t = 0t = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8t = 9

Spatial control strategies V:Protect large uninvaded areas

Entire landscape matters

Barrier cost (b) = 50Removal cost (e) = 1500Baseline damages (d) = 1

No control… let spread

If landscape homogeneous

t = 3t = 4t = 6t = 1t = 5t = 7t = 8

Barrier cost (b) = 50Removal cost (e) = 1500Baseline damages (d) = 1High damages (d) = 101

t = 0t = 2

EradicateIf high damage patch in landscape

t = 3t = 4t = 5t = 6t = 7t = 8

Barrier cost (b) = 50Removal cost (e) = 10000Baseline damages (d) = 1High damages (d) = 101

t = 0t = 1t = 2

Slow spread; protect high damage patchIf higher removal costs

t = 9t = 10t = 11t = 12t = 13t = 14t = 15t = 16t = 17…

t = 3t = 4t = 5t = 6t = 7t = 8

Barrier cost (b) = 50Removal cost (e) = 10000Baseline damages (d) = 1High damages (d) = 51

t = 0t = 1t = 2

Slow the spreadIf lower damages in patch

t = 9t = 10t = 11t = 12t = 13…

Summary of control principles High damages, low costs, and low discount rate, higher

optimal control efforts Protect large uninvaded areas

– prevent/delay spread in direction of high potential damages Reduce extent of exposed edge prior to containment

– 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

adjacent to “about to be invaded”

about to be invaded

Offer to “about to beinvaded” cell to induceprevention

•Unilateral management•Bilateral bargaining•Local “club” formation

Border control cost (b)

Outcomes from private control vs. optimal control:

Unilateral management Bilateral bargaining Local “club” coordination

Optimal control

Border control cost (b)

Thank you!!