Becky Epanchin-Niell Jim Wilen Prepared for the PREISM workshop ERS Washington DC May 2011 Optimal Management of Established Bioinvasions

Bioinvasions Are: Spatial-dynamic Processes n Spatial-dynamic processes are driven by dynamics at a point and diffusion between points n Generate patterns that evolve over both space and time n Some other examples –Forest fires –Floods –Aquifer dynamics –Groundwater contamination –Wildlife movement –Human/animal disease

Questions raised by bioinvasions: n How does uncontrolled invasion spread? n Intensity and timing of optimal controls when and how much control? n Spatial strategies for control  Where should control be applied? n Effect of spatial characteristics of the invasion and landscape on optimal control n Externalities, institutions, and reasons for intervention

Modeling Optimal Bioinvasion Control with Explicit Space n Simple small model n Build intuition with multiple optimization “experiments” n Identify how space matters with spatial- dynamic processes n 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 land invaded land border control clearing n Invasion spread − Cellular automaton model − Approximates reaction- diffusion n Control options –Spread prevention –Invasion clearing n Min. total costs & damages n Simplicity - 2 n*t configurations ($d) ($b) ($e)

Finding the optimal solution n Dynamic problems  Ordinary differential equations  End-point conditions—2 point boundary prob. n 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

Mathematical model variables parameters Subject to: damages clearing costs spread prevention costs Cell remains invaded unless cleared Cell becomes invaded if has invaded neighbor unless prevention applied

Solution approach: n Binary integer programming problem - SCIP (Solving Constraint Integer Programs) n Scaling n Solves large-scale problems in seconds/minutes n Can perform numerous comparative spatial- dynamic optimization “experiments” - Cost parameters, discount rate - Invasion and landscape size - Invasion and landscapes shape - Invasion location

Results: n Wide range of control approaches –e.g., eradicate, clear then contain, slow then contain, contain, slow then abandon, abandon n If clearing is optimal, it is initiated immediately n Landscape & invasion geometry important n 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

n 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 n Central invasions - higher potential long-term damages  more control n Invasions near range edge - lower control costs  more control Central invasions  higher costs & damages Invasion location has ambiguous effect on optimal control effort

t = 0 t = 1t = 2t = 3t = 4t = 5 t = 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 = 0 t = 1t = 2t = 3t = 4t = 5 t = 6… Spatial control strategies II: 1)Reduce the extent of invasion edge 2)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 = 0 t = 1t = 2t = 3t = 4t = 5 t = 6… Spatial control strategies IV:

t = 0 t = 1t = 2t = 3t = 4t = 5 t = 6 t = 7 t = 8 t = 9 Spatial control strategies V: Protect large uninvaded areas Entire landscape matters

Barrier cost (b) = 50 Removal cost (e) = 1500 Baseline damages (d) = 1 No control… let spread If landscape homogeneous 

t = 3t = 4 t = 6t = 1 t = 5 t = 7t = 8 Barrier cost (b) = 50 Removal cost (e) = 1500 Baseline damages (d) = 1 High damages (d) = 101 t = 0 t = 2 Eradicate If high damage patch in landscape 

t = 3t = 4t = 5 t = 6t = 7t = 8 Barrier cost (b) = 50 Removal cost (e) = Baseline damages (d) = 1 High damages (d) = 101 t = 0 t = 1t = 2 Slow spread; protect high damage patch If higher removal costs  t = 9 t = 10 t = 11 t = 12 t = 13 t = 14 t = 15t = 16 t = 17…

t = 3t = 4t = 5 t = 6t = 7t = 8 Barrier cost (b) = 50 Removal cost (e) = Baseline damages (d) = 1 High damages (d) = 51 t = 0 t = 1t = 2 Slow the spread If lower damages in patch  t = 9 t = 10 t = 11 t = 12 t = 13…

Summary of control principles n High damages, low costs, and low discount rate,  higher optimal control efforts n Protect large uninvaded areas –prevent/delay spread in direction of high potential damages n Reduce extent of exposed edge prior to containment –employ landscape features –alter shape of invasion (spread, removal) n Entire invasion landscape matters n Geometry matters (initial invasion, landscape) n 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 be invaded” cell to induce prevention Unilateral management Bilateral bargaining Local “club” formation

Clearing cost ( e ) Border control cost (b) Outcomes from private control vs. optimal control: Unilateral managementBilateral bargainingLocal “club” coordination Optimal control Border control cost (b) Clearing cost ( e )

Thank you!!