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
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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?
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
Individual-Based
Models
Age/Size Structured
Models
Cape Sable
Seaside Sparrow
Snail Kite
White-tailed Deer
Wading Birds
Florida Panther
Fish Functional Groups
Alligators
Radio-telemetry
Tracking Tools
Reptiles and Amphibians
Linked Cell
Models
Lower Trophic Level Components
Vegetation
Process Models
Spatially-Explicit
Species Index Models
Cape Sable
Seaside Sparrow
Long-legged
Wading Birds
Short-legged
Wading Birds
Snail Kite
Abiotic Conditions
Models
High Resolution Topography High Resolution Hydrology
White-tailed Deer
Alligators
Disturbance
© TIEM / University of Tennessee 1999
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
Information Analysis/Control and
Data Representation Layer
Ecological Models
Components Layer
Biotic component
Ecological Modeling Oriented Data Assimilation
Layer
Spatial Information (GIS,
topology, etc)
External Models
(hydrology, climate,
etc)
Abiotic
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
let
be the radius of focus, i, at time, t
and
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 an
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
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.forestryimages.org
LygoMod
•
Finite difference equations
Pi (t  1)  Pi (t)  Pi (t)[1   (1  pij )
I j (t )
]  rRi (t)
I j (t )
]
j
I i (t  1)  I i (t)  Pi (t)[1   (1  pij )
j
(1  r)Ri (t)[1   (1  p 'ij )
I j (t )
]  Ci (t)
j
Ri (t  1)  Ri (t)  (1  r)Ri (t)[1   (1  p 'ij )
j
I j (t )
]  rRi (t)  Ci (t)
LygoMod
• Finite difference
equations
• Epidemiological
form
P(t)[1-(1-pij)I(t)]
rR(t)
P(t)
I(t)
R(t)
(1-r)R(t)[1- (1-p’ij)I(t)]
T(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
• Finite difference
equations
• Epidemiological form
• Space
• Dispersal by spores
• Spatial Heterogeneity
• Cut-and-spray treatment
• Objective function
J   I i (t)Si  c Ci (t)Si
i,t
i,t
Objective is to minimize J
LygoMod
• Finite difference
equations
• Epidemiological form
• Space
• Dispersal by spores
• Spatial Heterogeneity
• Cut-and-spray treatment
• Objective function
• Constraint
 C (t)S
i
i
i
 Budget(t) / CPA
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
– 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
200
0
200
9
Results
• Optimized treatment
• Total number of
infested tree islands
per year
• Range of budgets
Compare optimal treatment with and without space
Without Space
Uninfested
Infested
With Space
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
60
o
40 o
20 o
0
800
Kilometers
1600
120 o
100 o
Source: Hall (1981)
80 o
Current Black Bear Distribution
60
o
40 o
20 o
0
800
Kilometers
1600
120 o
100 o
Source: Pelton and van Manen (1994)
80 o
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
O
P
F
mp=.25 mf =.25
F
P
O
F
F
P
F
F
O
mp=.5 mf =.3
mp=.25 mf =.15

Metapopulation Model
dP
P2
P F2
 rP  r
 m f (1 )r
 hp P
dt
K
K
K
2
2
dF
F
F P
 rF  r
 mp (1 )r
 hf F
dt
K
K K
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
2
P F
mf (1 )r
K K
2
F P
mp (1  )r
K K
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
dO
P2
F2
P
F2 F
P2
 (1 mp )r
 (1 mf )r

mf r

mp r
dt
K
K
K
K
K
K
O = outside population of “nuisance” bears
Case 1: Constant boundaries
J(hp ,hf ) 
T

(O  c p h2p  c f h2f )dt
0
is the objective functional balancing the outside bear
population,O,
with the cost of harvesting, and we

seek the optimal controls
J(h*p ,h*f )  min J(hp ,hf )
h p ,h f
over the control set
{(hp ,hf )  L (0,T) 2 | 0  hp (t)  1,0  hf (t)  1}
with Hamiltonion




H  O  c p h  c f h  1 P  2 F  3 O
2
p
2
f
Case 2: Boundary varying
J(mp ,h f ) 
T

(O  c m m2p  c f h 2f )dt
0
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
J(m*p ,h*f )  min J(mp ,hf )
m p ,h f
over the control set

{(mp ,hf )  L (0,T) 2 | 0  mp (t)  1,0  hf (t)  1}

Under several assumptions: Park and Forest areas the same,
and a square Park region, the optimality equation is
H
P F2
F P2
 2c m mp  ( 2 1)(1   3 )(1 )r
 ( 2   3 )(1 )r
mp
K K
K K
Which allows solution for mp
and a similar equation leads to
a solution for hf
Spatial Control and IndividualBased 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 cubs or females with cubs.
• 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 nonscientific criteria, including personal attacks.
Ignore any of the stakeholders at your peril.