1. 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

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

3. 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 , DEB and
IIS

4. 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

5. 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

6. 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

8. Wildfire Control

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

16. Collaborators on ATLSS

17. 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.

18. 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.

19. 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

21. 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

22. 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.

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

27. 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,
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

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

29. 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?

30. 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

31. 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

32. 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.

33. 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).

34. Results
B = 15
B = 5

35. Total Costs

36. 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

37. Results
Unconstrained
B=50
Constrained
B=50

38. 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

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

40. 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

41. Loxahatchee
WCA-2
WCA-3
BCNP
ENP

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

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

44. 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

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

46. LygoMod
Finite difference equations

47. 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)]

48. LygoMod
Finite difference equations
Epidemiological form
Space

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

50. LygoMod Finite difference equations Epidemiological form Space
Dispersal by spores

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

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

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

54. 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

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

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

57. 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

58. 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.

59. No Treatment
2000
2009

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

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

63. Results
Optimal control with limited resources
Total Treatment Effort

64. Results
Quarantine treatment
$2.8M/year

65. American Black Bear (Ursus americanus)

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

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

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

69. 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.

70. 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.

71. 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

72. 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

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

74. 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

75. 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

76. 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

77. 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

78. 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

81. 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

82. State Variables
Age
Sex
Location
Denning
Estrus
Mating Status
Cubs

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

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

87. 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.)

88. 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.

90. Results for Each Alternative

91. 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.

92. 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.

93. 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.

94. 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.

95. 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.