Managing Natural Resources: Mathematics meets Politics, Greed and the Army Corps of Engineers Louis J. Gross The Institute for Environmental Modeling Departments of Ecology and Evolutionary Biology and Mathematics University of Tennessee ATLSS.org

IEEE Computing in Science and Engineering (2007) 9:40-48

Overview Background on natural resource management Everglades restoration and ATLSS (Across Trophic Level System Simulation) Invasive species management Optimal control for generic focus/satellite spread Spatially-explicit management of Old World Climbing Fern Wildlife management -the black bear case Optimal control in a simple metapopulation case Individual-based models and preserve placement Some lessons for model use in resource management Supported by NSF Awards DMS-0110920, DEB-0219269 and IIS-0427471

Problems in Natural Resource Management Harvest management Wildlife management including hunting regulations and preserve design Water regulation Invasive species management Wildfire management Biodiversity and conservation planning

What is challenging about natural resource management? Involves complex interactions between between humans and natural systems Often includes multiple scales of space and time Multiple stakeholders with differing objectives Monetary consequences can be considerable

How can mathematics and modeling assist in solving problems in resource management? Provide tools to project dynamic response of systems to alternative management Estimate the “best” way to manage systems - optimal control Provide means to account for spatial changes in systems and link models with geographic information systems (GIS) and decision support tools that natural system managers and policy makers need. Consider methods to account for multiple criteria and differing opinions of the variety of stakeholders Game theory is one possible methodology

Wildfire Control

Wet Season: May-October Dry Season: November-April Photos: South Florida Water Management District

Collaborators on ATLSS

Everglades natural system management requires decisions on short time periods about what water flows to allow where and over longer planning horizons how to modify the control structures to allow for appropriate controls to be applied. This is very difficult! The control objectives are unclear and differ with different stakeholders. Natural system components are poorly understood. The scales of operation of the physical system models are coarse.

So what have we done? http://atlss.org Developed a multimodel (ATLSS - Across Trophic Level System Simulation) to link the physical and biotic components. Compare the dynamic impacts of alternative hydrologic plans on various biotic components spatially. Let different stakeholders make their own assessments of the appropriate ranking of alternatives. http://atlss.org

Across Trophic Level System Simulation ATLSS STRUCTURE Across Trophic Level System Simulation Model Type Snail Kite White-tailed Deer Individual-Based Models Cape Sable Seaside Sparrow Radio-telemetry Tracking Tools Wading Birds Florida Panther Age/Size Structured Models Fish Functional Groups Alligators Reptiles and Amphibians Linked Cell Models Lower Trophic Level Components Vegetation Process Models Cape Sable Seaside Sparrow Spatially-Explicit Species Index Models Long-legged Wading Birds Short-legged Wading Birds White-tailed Deer Alligators Snail Kite Abiotic Conditions Models High Resolution Topography High Resolution Hydrology Disturbance © TIEM / University of Tennessee 1999

To deal with his we use a relative assessment protocol This is not optimal control, but rather is a scenario analysis in which we compare and contrast the impacts of alternative management plans But there are numerous uncertainties due to incomplete knowledge of system processes and species dynamics, stochasticity in model inputs (e.g. weather patterns, mortality, etc.), complex model structure, and inaccuracy in the measurement of parameters To deal with his we use a relative assessment protocol

Relative assessment i = 1, 2, …, k indicate a collection alternative scenarios, defined by a collection of anthropogenic and environmental factors that affect the modeled ecosystem or a particular model configuration. P represents a particular configuration of the model, including the values of model parameters, specific assumptions and functional forms used. Ej, j = 1, 2, …, n are anthropogenic and environmental factors (rainfall, fire events, etc.) Mi (P, E1, …,En ) for i = 1, 2, …, k are the output of a model which projects the response of natural system components under scenario i with model parameters P and environmental inputs Ej. R ( M1 (P, E1, …,En ), …, Mk (P, E1, …,En ) ) is a ranking of the scenarios based upon evaluation criteria and utilizing the results of the models A relative assessment is robust to a particular variation in model parameters and/or environmental inputs if the variation does not change the ranking R. When the model results Mi are recomputed based upon a particular variation in the Ej and P, the rank order of the models in R does not change.

ATLSS grid-service module ATLSS Model Interface Wang et al. 2005. A grid service module for natural resource managers. IEEE Internet Computing 9:35-41

GEM: Grid-based Ecological Modeling, http://www.tiem.utk.edu/gem Ecological Modeling Oriented Data Assimilation Layer Ecological Models Components Layer Information Analysis/Control and Data Representation Layer Spatial Information (GIS, topology, etc) External Models (hydrology, climate, etc) Abiotic component Biotic component GEM: Grid-based Ecological Modeling, http://www.tiem.utk.edu/gem Wang, D., M. W. Berry, E. A. Carr, L. J. Gross. Towards Ecosystem Modeling on Computing Grids, Computing in Science and Engineering Vol. 13, No. 1, pp55-76, 2005

Invasives: control of foci of infection Suzanne Lenhart and Andrew Whittle

Introduction An invasive infestation (plants) consisting of a larger source population (main focus) and several smaller surrounding populations (satellites) Given limitations (cost) on control efforts, where should these efforts be concentrated and how should they vary over time?

Previous Work In a paper which has been the main reference for natural resource managers tackling invasion problems, Moody and Mack (1988) considered this problem and set up two simple control scenarios: control the main focus or control the satellites (assume all invasions are circular) Control is applied initially; either all satellites are destroyed (permanently) or an area, equivalent to the area of the satellites, is removed from the main focus The infestation grows and conclusions are drawn based on the total area covered by the invasive under each scenario

Modifications of Moody and Mack They used two possible scenarios: we consider all possible controls (amounts removed from each focus) rather than choosing to control or not to control each focus They assumed equilibrium and single time control: we consider a control problem on a finite time horizon with control applied at discrete intervals We form an optimal control problem to find the most economical solution

Mathematical Model Let be the radius of focus, i, at time, t and let be the radius removed. Then where k and are growth parameters for the foci.

The Objective Our objective is to minimize the area of infestation of n foci at the end of the control period of T years and the cost of the control used. Therefore where C is the cost per unit area of the invasive and B is a cost for the control (balancing coefficient).

Results B = 15 B = 5

Total Costs

Finite Resource Constraints The objective is to minimize the final area subject to the financial constraint where M is the total cost of control over the time period

Results Unconstrained B=50 Constrained B=50

Conclusions Optimal solutions suggest it’s better to control both the main focus and the satellites (different from Moody & Mack!) Control strategy is sensitive to the balancing coefficient B Similar results are obtained with different numbers of satellites

Scott M. Duke-Sylvester Spatial control of an invasive plant in a heterogeneous landscape : Lygodium microphyllum in South Florida Scott M. Duke-Sylvester

Motivation Find treatment plans that optimize the use of finite resources to control invasive species Apply this to Lygodium in Loxahatchee Evaluate the applicability of genetic algorithms for solving such problems

Loxahatchee WCA-2 WCA-3 BCNP ENP

Loxahatchee 57324 ha Sheet flow of water from N to S. 2.4-3.7 m of organic soils Large areas of open slough, wet prairies and sawgrass strands Thousands of tree islands

Tree Islands Size : 0.1 to 125 ha 85% < 0.13 ha Slightly higher elevation than surrounding marsh.

Lygodium microphyllum Old world climbing fern Native ranges : Africa to SE Asia/Australia (Pemberton, et. al) Introduced to Florida : prior to 1958 (Nauman and Austin, 1978) Negatively impacts both flora and fauna

Photo by :Peggy Greb, USDA Agricultural Research Service, www Photo by :Peggy Greb, USDA Agricultural Research Service, www.forestryimages.org

LygoMod Finite difference equations

LygoMod P(t)[1-(1-pij)I(t)] rR(t) P(t) R(t) I(t) T(t) Finite difference equations Epidemiological form P(t)[1-(1-pij)I(t)] rR(t) P(t) I(t) R(t) T(t) (1-r)R(t)[1- (1-p’ij)I(t)]

LygoMod Finite difference equations Epidemiological form Space

LygoMod Finite difference equations Epidemiological form Space 23x37 grid 1x1 km cells

LygoMod Finite difference equations Epidemiological form Space Dispersal by spores

LygoMod Finite difference equations Epidemiological form Space Dispersal by spores Spatial Heterogeneity Number of tree islands

LygoMod Finite difference equations Epidemiological form Space Dispersal by spores Spatial Heterogeneity Mean tree island size

LygoMod Finite difference equations Epidemiological form Space Dispersal by spores Spatial Heterogeneity Cut-and-spray treatment

LygoMod Objective is to minimize J Finite difference equations Epidemiological form Space Dispersal by spores Spatial Heterogeneity Cut-and-spray treatment Objective function Objective is to minimize J

LygoMod Finite difference equations Epidemiological form Space Dispersal by spores Spatial Heterogeneity Cut-and-spray treatment Objective function Constraint

LygoMod Finite difference equations Epidemiological form Space Dispersal by spores Spatial Heterogeneity Cut-and-spray treatment Objective function Constraint Genetic Algorithm

Steps in a classic GA Start with a randomly selected set (population) of individuals (treatment plans) Compute fitness for each individual: Dynamics Objective functional Check stopping criteria Reproduction : Form a new set of individuals by randomly selecting from current set Selection is biased by fitness Randomly cross individuals Randomly mutate individuals Return

Results No treatment Optimized treatment Quarantine treatment Genetic Algorithm With and without space Quarantine treatment Based on a recommended approach to treating invasive, non-native, plants in south Florida. Treat small peripheral infestations first, followed by larger more severely infested areas.

No Treatment 2000 2009

Results Optimized treatment Total number of infested tree islands per year Range of budgets

Compare optimal treatment with and without space With Space Uninfested Infested Recovering Treated

Results Optimal control with limited resources Total Treatment Effort

Results Quarantine treatment $2.8M/year

American Black Bear (Ursus americanus)

Rene Salinas and Suzanne Lenhart Salinas, R., S. Lenhart and L. Gross. 2005. Control of a metapopulation harvesting model for black bears. Natural Resource Modeling 18:307-321

Historic Black Bear Distribution 100 o 60 40 120 80 20 800 1600 Kilometers Source: Hall (1981)

Current Black Bear Distribution 100 o 60 40 120 80 20 800 Kilometers 1600 Source: Pelton and van Manen (1994)

Natural History Great Smoky Mountains National Park (GSMNP) and the surrounding national forests were established in the mid- 1930’s. By then the black bear population had lost 80% of its range. Since the 1930’s, harvesting restrictions have helped establish the black bear population as one of the most dense in North America.

Current Issues The human population surrounding the GSMNP has also grown over the last 70 years. Nuisance bear activity is a major problem all along the Appalachian range. With the increase in bear-human encounters, the likelihood of harmful encounters also increases.

Control of a Bear Metapopulation Harvesting Model Derive a simple model to evaluate options for bear management at regional spatial extent Provide methods to incorporate trade-off between bear population maintenance and reduction of nuisance activity Evaluate harvesting and spatial design as controls in an optimization framework

Some Model Assumptions Two populations (Park and Forest) as well as an Outside population for emigrants with movement between regions depending upon connectivity and population sizes No population structure or mast dynamics (fixed growth rate and carrying capacity) Case 1: Static spatial layout, derive optimal harvest policy over relatively short time-frame Case 2: Harvesting is allowable only in Forest, not Park, derive optimal harvest policy and optimal connectivity between Park and Forest with fixed equal areas in Park and Forest

P F O P F O mp=.25 mf =.25 P F O mp=.5 mf =.3 mp=.25 mf =.15

Metapopulation Model P, F = bear numbers in Park, Forest K, r, hi = carrying capacity, growth rate, harvest rate mp, mf = connectivities, proportion of Park border connected to Forest region and vice-versa

Metapopulation Model represent density-dependent immigration from one population (P or F) to the other, so only a portion of the emigrating population enters the other region O = outside population of “nuisance” bears

Case 1: Constant boundaries is the objective functional balancing the outside bear population,O, with the cost of harvesting, and we seek the optimal controls over the control set with Hamiltonion

Case 2: Boundary varying is the objective functional balancing the outside bear population, O, with a varying boundary between the Park and Forest, and we seek the optimal controls over the control set

Under several assumptions: Park and Forest areas the same, and a square Park region, the optimality equation is Which allows solution for mp and a similar equation leads to a solution for hf

Spatial Control and Individual-Based Models Spatial components Reserve design Habitat conditions Resource availability Individual-Based Models Model space explicitly Account for differences between individuals Model movement explicitly Test demographic forcing

State Variables Age Sex Location Denning Estrus Mating Status Cubs

Harvesting Tennessee Season Dec. 1- Dec. 14 No cubs or females with cubs. North Carolina Season Oct.14 - Nov. 21 and Dec. 14 - Dec. 31 No harvesting in GSMNP or bear sanctuaries. One bear limit per calendar year

BASE scenario during a good mast year. BASE scenario during a poor mast year. ALT2 scenario during the same poor mast year.

Model Comparisons to Empirical Data IBM = solid lines, Empirical = dotted lines Error bars indicate 95% CIs for IBM. (Except in graph A (upper left) where large error bars are from Jolly-Seber Empirical estimates.)

Alternative Harvesting Strategies We analyzed various alternative strategies to determine the impact on: Total Population across the entire region. Potential Encounters, measured as the total number of bears in areas with greater than 10 people/km2, summed over all days for each year. Alternative strategies are implemented from 2003 through 2022.

Results for Each Alternative

Lessons - Doing the Modeling: Work closely with those with long experience in the system being modeled and use their experience to determine key species, guild and trophic functional groups on which to focus. Moderate the above based first on the availability of data to construct reasonable models, and secondly on the difficulty of constructing and calibrating the models. Don't try to do it all at once - start small - but have a long-term plan for what you wish to include overall, given time and funding.

Lessons - Doing the Modeling: Leave room for multiple approaches: don't limit your options. In the face of limited or inappropriate data, use this as an opportunity to encourage further empirical investigations of key components of the system. Build flexibility in as much as possible. Be flexible about what counts as success.

Lessons - Personnel Matters: Build a quality team who respect each others abilities and won't second guess each other, but who accept criticism in a collegial manner. Keep some part of the team out of the day-to-day political fray. Be persistent, and have at least one member of the team who is totally dedicated to the project and willing to stake their future on it. Do whatever you can to maintain continuity in the source of long-term support for the project.

Lessons - Interacting with Stakeholders: Constantly communicate with stakeholders. Regularly explain the objectives of your modeling effort, as well as the limitations, to stakeholders. Be prepared to do this over and over for the same people, and do not get frustrated when they forget what you are doing and why. Be prepared to regularly defend the scientific validity of your approach.

Lessons - Interacting with Stakeholders: Don't limit your approach because one stakeholder/funding agency wants you to. Be prepared for criticism based upon non-scientific criteria, including personal attacks. Ignore any of the stakeholders at your peril.