Applied Mathematical Ecology/ Ecological Modelling

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Applied Mathematical Ecology/Ecological Modelling

• Dr Hugh Possingham
• The University of Queensland
• (Professor of Mathematics and Professor of
  Ecology)
• AMSI Winter School 2004

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Overview

• Ecology and mathematics
• Mathematics to design reserve systems
• Mathematics to manage fire
• Mathematics to manage populations
• Mathematics to manage and learn simultaneously
• Optimisation, Markov chains

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Take home messages

• Do enough to solve the problem
• What is interesting is not always important, what
  is important is not always interesting
• Unusual dynamic behaviour may well be just that -
  unusual
• The solution to our problems in science is not
  always to make more and more complex models.
• Reductionism vs Holism.

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Optimal Reserve System Design

• Hugh Possingham and Ian Ball (Australian
  Antarctic Division) and others

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History of reserve design

• Recreation
• What is left over
• Special features
• SLOSS and Island biogeography
• CAR reserve systems (Gap analysis)
• The minimum set problem

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The minimum set problemHow do we get an
efficient comprehensive reserve system

• Minimise the cost of the reserve system
• Subject to the constraints that all
  biodiversity targets are met
• New age problems - add in spatial considerations,
  like total boundary length

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Example Problem
1 Find the smallest number of sites that
represents all species
The data matrix - A
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Algorithms to solve the reserve system design
problem

• Wild guess
• Heuristics
• Mathematical Programming
• Heuristic algorithms
• Simulated annealing
• Genetic Algorithms

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Heuristics

• Richness algorithms
• Rarity algorithms
• Neither work so well with bigger data sets,
  especially where space is an issue

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ILP formulation
Minimise
Subject to
if the site is in the reserve system
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Simulated annealingand Genetic Algorithms
We could evolve a good solution to the problem
treating a reserve system like a piece of DNA.
Fitness is a combination of number of sites plus
a penalty for missing species. Fitness -
number of sites - 2xmissing species If sites cost
1 and there is a 2 point penalty for missing a
species then in problem 1 the fitness of the
system A,B,D - 3 - 2 - 5
Which is not as fit as A,B - 2 - 2 - 4
or A,B,C - 3
- 3 With best solution C,E - 2 - 0
- 2
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GAs Breeding a reserve system
2 4 7 8 20 25 28 cost 7 3
7 8 10 11 12 cost
6 ... babies 2 4 7 10 11 12
infeasible 3 7 8 20 25 28
cost 6 ...
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Simulated annealing
A genetic algorithm with no recombination, only
point mutations and a population size of
1. Selection process allows a decrease in
fitness at the start of the process Relies on
speed and placing constraints in the objective
function
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Objectives and constraints

• Typical constraints are to meet a variety of
  conservation targets eg 30 of each habitat
  type or enough area for 2000 elephants (not just
  get one occurrence)
• Typical objectives are to satisfy the constraints
  while minimising the total cost (which may be
  area, actual cost, management cost, cost of
  rehabilitation)
• Objectives and constraints are somewhat
  interchangeable

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Spatial problems

• There is more to the cost of a reserve system
  than its area
• Boundary length and shape are important
• Other rules about minimising boundary length,
  cost of land, forgone development opportunties,
  minimum reserve size, issues of adequacy

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Boundary Length Problem Non-linear IP problem
Minimise
Subject to
if the site is in the reserve system
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Example 1 The GBR

• Divided in to hexagons
• 70 different bioregions (reef and non-reef)
• 13,000 planning units
• What is an appropriate target?
• What are the costs?
• Replication and
• minimum reserve size
• www.ecology.uq.edu.au/marxan.htm

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The GBR process

• Determine optimal system based on ecological
  principles alone
• For low targets there are many many options
• Introduce socio-economic data
• Special places, targets, industry goals,
  community aspirations
• Delivered decision support by providing options

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The consequences of not planning

• The South Australian dilemma of 18 reserves (4
  by area), 9 add little to the goal of
  comprehensiveness (Stewart et al in press), they
  are effectively useless in the context of a well
  defined problem even if targets are 50 of every
  feature type!
• Complimentarity is the key
• The whole is more
• than the sum of the parts

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Effect of South Australias existing marine
reserves
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But reserve systems arenot built in a day

• Idea of irreplaceability introduced to deal with
  the notion that when some sites are lost they are
  more (or less) irreplaceable (Pressey 1994).
• The irreplaceability of a site is a measure of
  the fraction of all reserve systems options lost
  if that particular site is lost

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Example 3 Identifying Irreplaceable Areas
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Future/General issues

• Problems are largely problem definition not
  algorithmic
• Issues are mainly ones of communication
• What is a model, algorithm, or problem?
• Many complexities can be added
• More complex spatial rules
• Zoning
• Etc etc.
• Dynamic reserve selection

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• Optimal Fire Management
• for biodiversity conservation
• Hugh Possingham, Shane Richards, James Tizard and
  Jemery Day
• The University of Queensland/Adelaide
• NCEAS - Santa Barbara

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What is decision theory?

• Set a clear objective
• Define decision variables - what do you control?
• Define system dynamics including state variables
  and constraints

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The problem

• How should I manage fire in Ngarkat Conservation
  Park - South Australia?
• What scale?
• What biodiversity?
• How is it managed now?
• What is the objective?

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Vegetation

• Dry sandplain heath (like chapparal) - 300mm,
  winter rainfall
• Little heterogeneity in soil type or topography -
  poor soils
• Diverse shrub layer with some mallee
• Key species - Banksia, Callitris, Melaleuca,
  Leptospermum, Hibbertia, Eucalyptus

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Habitat
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Assume three successional states
fire, f
late
mid
1/sm
1/se
early
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Ngarkat Conservation Park
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Nationally threatened bird species

• Slender-billed Thornbill - early
• Rufous Fieldwren - early
• Red-lored Whistler - mid
• Mallee Emu-wren - mid/late
• Malleefowl - late
• Western Whipbird - late

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Vegetation dynamicsTransition probability from
j early to i early
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Fire model
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Fire transition matrix and Succession transition
matrix are combined to generate state
dynamicsBUT

• Succession Markovian
• Fire model naive

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The optimization problem

• Objective - 20 each stage
• State space - of park in each successional
  stage
• Control variable - given the current state of
  park should you do nothing,fight fires, start
  fires?
• System dynamics determined by transition matrices

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Solution method

• Stochastic dynamic programming (SDP)
• Optimal solution without simulation but can be
  hard to determine
• Only works with a relatively small state space -
  (Nx(N1))/2

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Conclusion

• Decision is state-dependent - there is no simple
  rule
• Costs may be important
• The decision theory framework allows us to
  address the problem and find a solution
• Details - Richards, Possingham and Tizard (1999)
  - Ecological Applications

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Where are we going?

• Rules of thumb - depend on the intervals between
  successional states and fire frequency (Day)
• Spatial version (Day)
• More detailed vegetation and animal population
  models

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Eradicate, Exploit, Conserve

Decision Theory
Pure Ecological Theory

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How to manage a metapopulation

• Michael Westphal (UC Berkeley),
• Drew Tyre (U Nebraska), Scott Field (UQ)
• Can we make metapopulation theory useful?

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Specifically how to reconstruct habitat for a
small metapopulation

• Part of general problem of optimal landscape
  design the dynamics of how to reconstruct
  landscapes
• Minimising the extinction probability of one part
  of the Mount Lofty Ranges Emu-wren population.
• Metapopulation dynamics based on Stochastoc Patch
  Occupancy Model (SPOM) of Day and Possingham
  (1995)
• Optimisation using Stochastic Dynamic Programming
  (SDP) see Possingham (1996)

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The Mount Lofty Ranges, South Australia
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MLR Southern Emu Wren

• Small passerine (Australian malurid)
• Very weak flyer
• Restricted to swamps/fens
• Listed as Critically Endangered subspecies
• About 450 left hard to see or hear
• Has a recovery team (flagship)

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The Cleland Gully Metapopulation basically
isolated Figure shows options Where should we
revegetate now, and in the future? Does it
depend on the state of the metapopulation?
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Stochastic Patch Occupancy Model(Day and
Possingham, 1995)
State at time, t, (0,1,0,0,1,0)
Intermediate states
Extinction process
(0,1,0,0,1,0)
(0,1,0,0,0,0)
(0,0,0,0,1,0)
Colonization process
State at time, t1, (0,1,1,0,1,0)
Plus fire
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The SPOM

• A lot of population states, 2n, where n is the
  number of patches. The transition matrix is 2n
  by 2n in size (128 by 128 in this case).
• A chain binomial model (Possingham 96, 97 Hill
  and Caswell 2001?)
• SPOM has recolonisation and local extinction
  where functional forms and parameterization
  follow Moilanen and Hanski
• Overall transition matrix, a combination of
  extinction and recolonization, depends on the
  landscape state, a consequence of past
  restoration activities

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Decision theory steps

• Set objective (minimize extinction prob)
• Define state variables (population and landscape
  states) and control variables (options for
  restoration)
• Describe state dynamics the SPOM
• Set constraints (one action per 5 years)
• Solve in this case SDP

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Control options (one per 5 years, about 1ha
reveg) E5 largest patch bigger, can do 6
times E2 most connected patch bigger, 6
times C5 connect largest patch C2 connect
patches1,2,3 E7 make new patch DN do nothing
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Management trajectories1 only largest patch
occupied
C5
E5
E5
E5
E5
E5
E5
E7
DN
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Management trajectories2 all patches occupied
E5
E5
E2
E2
C5
C2
E5
E2
E5
E5
E7
DN
E5
E5
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Take home message

• Metapopulation state matters
• Actions justifiable but no clear sweeping
  generalisation, no simple rule of thumb!
• Previous work has assumed that landscape and
  population dynamics are uncoupled. This paper
  represents the first spatially explicit optimal
  landscape design for a threatened species.

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Computational Problems

• The huge state space population state space is
  2N where N is the number of patches. The
  landscape state space is all the possible
  landscape states!
• Solution aggregation of state space? Rules of
  thumb tested via simulation?

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Other applications of decision theory to
population managementand conservation

• Optimal metapopulation management (Possingham 96,
  97 Haight et al 2002)
• Optimal fire management (Possingham and Tuck 98,
  Richards et al 99, McCarthy et al. 01)
• Optimal biocontrol release (Shea and Possingham,
  00)
• Optimal landscape reconstruction (Westphal et
  al., submitted)
• Optimal captive breeding management (Tenhumberg
  et al, to submit)
• Optimal weed management (Moore and Possingham, to
  submit)
• Decision theory and PVA/conservation (Possingham
  et al. 01, 02 book chapters) The Business of
  Biodiversity
• Optimal Reserve System Design, MARXAN, TNC
  (several papers)

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Optimal translocation strategies
Brigitte Tenhumberg, Drew Tyre (U Nebraska),
Katriona Shea (Penn State)

• Consider the Arabian Oryx Oryx leucoryx if we
  know how many are in the wild, and in a zoo, and
  we know birth and death rates in the zoo and the
  wild, how many should we translocate to or from
  the wild to maximise persistence of the wild
  population

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Oryx problem
Growth rate R 0.85 Capacity 50
Growth rate R 1.3 Capacity 20
??
Zoo Population
Wild Population
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Result base parameters
R release, mainly when population in zoo is
near capacity C capture, mainly when zoo
population small, capture entire wild population
when this would roughly fill the zoo
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If zoo growth rate changes, results change but
for a new species we wont know R in the zoo
Enter active adaptive management, Management
with a plan for learning
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Active adaptive management
Cindy Hauser, Petra Kuhnert, Katriona Shea, Tony
Pople, Niclas Jonzen (Lund)

• Management of uncertain stochastic systems with a
  plan for learning
• How do you trade-off the need to optimally manage
  a system with the information gain you need to
  manage that same system

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Toy fish problem
Unharvested
Harvested
Secure
Secure
Harvest, Yes or no?
Fragile
Fragile
?????
?????
Collapsed
Collapsed
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• The best decision depends on our current state of
  knowledge which is a function of the number of
  times the stock has recovered and the number of
  times it hasnt
• Use Bayes formula to update a Beta prior for the
  probability of recovery. This means that the
  state space is now the stock state and the
  parameters of the Beta distribution
• Stochastic dynamic programming is used to
  determine the optimal state-dependent decision
• Cindy is now applying to kangaroos with a large
  population state space

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Active adaptive monitoring the problem of the
Swedish lynx Lynx lynx
Henrik Andren (SLU), Anna Daniel (SLU), Cindy
Hauser, Tony Pople

• The toy fish problem assumes that we know the
  size of the fish stock. Now assume that we do
  know the system dynamics, but we have to spend
  money monitoring to determine the population size
  which then determines the harvesting strategy
• How much money do we spend monitoring and is
  optimal monitoring state dependent??

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Information for Swedish lynx problem

• Population size (N) is number of Lynx family
  units
• Compensation cost per N 20,000 SK, higher if N
  gt 200
• Cost of current monitoring program
• Political cost of N falling below 50
  100,000,000 SK
• Fixed harvest strategy 15 if N gt 80, 0
  otherwise
• Growth rate R normally distributed (mean 1.17)
• Monitoring strategies
• M1 cost 2,000,000 SK and generates N with a SD
  of 0.1N
• M2 cost ? SK generates N with a SD of 0.3N
• (M0 no count, cost 0, estimate based on last
  year and mean growth rate

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Results should monitoring bestate-dependent?
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Utility ? 106
40
10
50
200
Number of Lynx family units (N)
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Applied Theoretical Ecologist Dreaming

• Optimal Harvesting
• Optimal Monitoring
• Optimal Learning

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New approaches to the evolution of complex
ecological systems kangaroo population dynamics
What is important is not always interesting, What
is interesting is not always important

• PIs Hugh Possingham, Gordon Grigg,
• Stuart Phinn, Clive McAlpine
• PDFs Tony Pople, Niclas Jonzén,
• Brigitte Tenhumberg
• PhDs Cindy Hauser, Norbert Menke
• Money ARC Linkage, UQ, Environment Australia,
  DEH (SA), EPA (QLD), MDBC, Packer Tanning

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Overview

1. Background and History
2. Visualisation of the patterns
3. The confrontation of models with data
4. Why model prediction, utility or understanding?
5. The evolutionary impact of harvesting a just
   so story
6. Optimal adaptive monitoring
7. Learning while managing a new discipline -
   applied theoretical ecology

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1 Background and History

• Data collected from 1978
• Kangaroo quotas, 15 of the estimates
• Previous mathematical modeling, single spatial
  scale with a short time series
• Few other population studies on a large scale
  locusts, phytoplankton
• Harvesting theory typically for fish only

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Data collection Fixed-wing Survey
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2 Visualisation of the patterns

• With complex ecological systems visualising the
  data can be an important part of understanding
  and theorising
• Aside from kangaroo numbers we have
• rainfall data
• National Digitised Vegetation Index (NDVI,
  satellite) data
• sheep data
• pasture biomass models, and
• harvest data

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Animation of Kangaroo survey data
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Temporal patterns at a whole region scale
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20
Growth Rate
15
Vegetation Index
Scaled measure
10
Kangaroo Numbers
5
0
1980
1985
1990
1995
2000
2005
Year
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3 The confrontation of model with data

• Is rainfall a good surrogate for resources?
• What is the most plausible time lag?
• How does density dependence work if at all?
• Do sheep compete with kangaroos?
• Are there environmental correlations between
  regions?

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South Australia main management zones
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The competing models

• Ratio model (theoretical support)
• Growth rate is determined by an abstract function
  of rainfall and harvest
• Growth rate 0.55 1.55.exp( 0.08.RAIN / Dt)
  harvest rate
• (plausible but abstract)
• Interactive model (Caughley data hungry)
• (rainfall ? pasture ? kangaroos)
• (more plausible but complex)

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Ratio Model
Northeast Pastoral Zone
Population size
Year
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Interactive model
Population size
Year
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A more complex statistical model

• The model with nested effects of
• density dependence bN
• rainfall R
• sheep S
• harvesting H, and
• correlated environmental variability, E

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We dont know as much as we thought

• Use Akaikes information criteria to select the
  most parsimonious model
• Best model, 50 support, suggests
• There is strong density dependence
• Harvesting matters, BUT
• Kangaroos eat sheep
• There are correlations between the regions not
  explained by rainfall

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Why model prediction understanding or utility?

• Prediction forecasting the future accurately
• Understanding increase in knowledge, easy to
  explain, mechanisms
• Utility making good management decisions, who
  cares if we understand

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Optimal Harvest Strategy?
Mean net harvest per year
Percentage harvest
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Learning, monitoring and managing

• Management ultimately needs robust predictive
  models, but which model?
• Can you monitor and manage to increase the rate
  at which you refine your model choice?
• For example to learn more maybe we should vary
  the harvesting and monitoring active adaptive
  monitoring/management

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Monitoring and managing
500
0.5
Cost
Probability of collapse
450
0.4
400
Probability of collapse
Cost of monitoring ?1000
0.2
350
0.1
300
0.0
250
0
1
2
3
4
5
Infrequent
Annual
Period of monitoring (years)
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Conclusion

• A diversity of novel methodologies
• Visualisation
• Simulation models
• Statistical models
• Process models
• Analysis in space and time
• An emphasis on confronting alternative models
  with data
• Applied Theoretical Ecology new field and
  approach?

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Take home messages

• Do enough to solve the problem you can put a
  nail in a wall with a frying pan but frying pans
  are better for cooking
• What is interesting is not always important, what
  is important is not always interesting
• There are several reasons why one might want to
  construct a model
• The solution to our problems in science is not
  always to make more and more complex models.
• Reductionism vs Holism
• The complex systems band wagon
• Philosophy and ethics why do you do what you
  do?