Applied Mathematical Ecology/Ecological Modelling

Dr Hugh Possingham

The University of Queensland

(Professor of Mathematics and Professor of Ecology)

AMSI Winter School 2004

• 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

Overview

• 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.

Take home messages

Optimal Reserve System Design

Hugh Possingham and Ian Ball (Australian Antarctic Division) and

others

History of reserve design

• Recreation

• What is left over

• Special features

• SLOSS and Island biogeography

• CAR reserve systems (Gap analysis)

• The minimum set problem

13,000 planning units

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

Example Problem 1Find the smallest number of sites that represents all species

The data matrix - A

SITESPECIES A B C D E F G H RangePowerful Owl 1 1 1 1 1 1 0 1 7Red Goshawk 1 1 1 1 0 0 0 1 5Olive Whistler 1 1 0 1 1 1 0 0 5Albert's Lyrebird 1 1 1 0 0 0 1 1 5Coxen's Fig Parrot 1 1 1 1 0 0 1 0 5Diamond Firetail 1 0 0 0 1 1 1 0 4Black-b Button-quail 1 0 1 1 0 0 0 0 3Eastern Bristlebird 1 1 1 0 0 0 0 0 3Rufous Scrub-bird 0 1 0 0 1 0 0 0 2Ground Parrot 0 0 1 0 0 0 0 0 1Site Richness 8 7 7 5 4 3 3 3 40

Algorithms to solve the reserve system design problem

• Wild guess

• Heuristics

• Mathematical Programming

• Heuristic algorithms– Simulated annealing– Genetic Algorithms

Heuristics

• Richness algorithms

• Rarity algorithms

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

ILP formulation

1

m

jj

x

1

1n

ij jj

a x

1, ,i m

, 0,1ij ja x

Minimise

Subject to

1jx if the site is in the reserve system

Problem 2: Select the smallest set of the 12 sites below that conserves all species

SITE RaritySPECIES A B C D E F G H I J K L Rng ScorePowerful Owl 1 0 1 1 0 1 1 1 0 1 1 1 9 0.11Red Goshawk 1 0 0 1 1 1 1 1 1 0 1 1 9 0.11Pale Yellow Robin 1 1 1 1 1 0 0 1 1 0 0 1 8 0.13Albert's Lyrebird 1 1 0 1 1 1 1 0 1 0 0 0 7 0.14Coxen's Fig Parrot 1 0 1 1 0 0 0 0 0 0 1 1 5 0.20Diamond Firetail 0 0 1 1 0 0 1 1 0 0 1 0 5 0.20Black-b Button-quail 0 1 0 1 1 1 0 0 0 1 0 0 5 0.20Eastern Bristlebird 1 1 0 0 1 0 1 0 1 0 0 0 5 0.20Rufous Scrub-bird 1 0 0 0 0 0 1 0 0 1 1 0 4 0.25Black-throated Finch 1 0 1 1 0 0 0 1 0 0 0 0 4 0.25Star Finch 1 1 1 0 0 1 0 0 0 0 0 0 4 0.25Sthn Emu wren 1 0 1 0 0 1 0 0 0 0 0 0 3 0.33Glossy Back Cockatoo 0 1 0 0 1 0 0 0 1 0 0 0 3 0.33Diamond Firetail 0 1 0 0 0 1 0 0 0 1 0 0 3 0.33Black-chinned Honeyeater 0 1 0 0 0 0 0 0 0 1 0 0 2 0.50Riflebird 0 0 1 0 1 0 0 0 0 0 0 0 2 0.50Olive Whistler 0 0 0 0 0 0 0 1 1 0 0 0 2 0.50Ground Parrot 0 1 0 0 0 0 0 0 0 1 0 0 2 0.50Site Richness 10 9 8 8 7 7 6 6 6 6 5 4 82Rarity Score 2 2.6 2 1.3 1.6 1.5 1 1.3 1.4 1.9 0.9 0.5

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 = - 3With best solution {C,E} = - 2 - 0 = - 2

GAs: Breeding a reserve system

2 4 7 8 20 25 28 cost 7 3 7 8 10 11 12 cost 6...

babies2 4 7 10 11 12 infeasible3 7 8 20 25 28 cost 6 ...

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 theobjective function

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

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

1 1 1

m m m

j j jk j kj j k

c x BLM b x x

Minimise

1ix if the site is in the reserve system

Boundary Length ProblemNon-linear IP problem

1

n

ij j ij

a x d

Subject to

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

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

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

Legend

Existing Marine Reserves

No Reserves @ 10% Target

Reserves Fixed @ 10% Target

Effect of South Australia’s existing marine reserves

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

Legend

NUMBER

81 - 119

120 - 273

274 - 999

1000

Cstline.shp

Example 3 – Identifying ‘Irreplaceable’ Areas

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

Optimal Fire Management

for biodiversity conservation

Hugh Possingham, Shane Richards, James Tizard and Jemery Day

The University of Queensland/Adelaide

NCEAS - Santa Barbara

What is decision theory?

• Set a clear objective

• Define decision variables - what do you control?

• Define system dynamics including state variables and constraints

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?

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

Habitat

Assume three successional states

early

late

mid

fire, f

1/se

1/sm

Ngarkat Conservation Park

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35 40 45 50 55

% of Ngarkat CP burnt each year by wildfire

Pro

ba

bilit

yNgarkat data, 1976-1992

All wildfires are left toburn unhinderedAll wildfires are fought

# MidSites

201918171615 ZERO OLD SITES

1413121110

98

7 GOOD

6 STATES

54

32

1

0 0 2 4 6 8 10 12 14 16 18 20

# Early Sites

State space and dynamics

Vegetation dynamics:Transition probability from j early to i early

E i j

s s i jj ij j i i

e e if

otherwise.

1

0

Fire model

otherwise,0

and if)(

,ijij

e

lmnNn

ll

mm

elmn

nji

llmmfF

i jjj

Probability system moves from state(ej,mj,lj) to state (ei,mi,li) due to fire

Where fn is the probability of afire that hits n patches

Fire transition matrix and Succession transition matrix are combined to

generate state dynamics

BUT• Succession Markovian

• Fire model naive

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

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(N+1))/2

# MidSites

201918171615 ZERO OLD SITES

1413121110

98

7 GOOD

6 STATES

54

32

1

0 0 2 4 6 8 10 12 14 16 18 20

# Early Sites

State space and dynamics

# MidSites

20191817 FIGHT WILDFIRES

16 DO NOTHING

15 BURN

1413121110

9876543210

0 2 4 6 8 10 12 14 16 18 20

# Early Sites

Optimal strategy with no costs

# MidSites

20191817 FIGHT WILDFIRES

16 DO NOTHING

15 BURN

1413121110

9876543210

0 2 4 6 8 10 12 14 16 18 20

# Early Sites

Optimal strategy with costs

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

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

Eradicate, Exploit, Conserve

Decision TheoryPure Ecological Theory

Applied

Theoretical

Ecology

+

=

How to manage a metapopulation

Michael Westphal (UC Berkeley),

Drew Tyre (U Nebraska), Scott Field (UQ)

Can we make metapopulation theory useful?

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)

The Mount Lofty Ranges, South Australia

Hugh’s birthplace

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)

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?

Stochastic Patch Occupancy Model(Day and Possingham, 1995)

State at time, t, (0,1,0,0,1,0)

Intermediate states

State at time, t+1, (0,1,1,0,1,0)

(0,0,0,0,1,0)

Extinction process

Colonization process

(0,1,0,0,1,0)

(0,1,0,0,0,0)

Plus fire

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

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

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

E5E5E5E5E5

Management trajectories:1 – only largest patch occupied

C5

E5

E7

DN

E5E5E5E5

Management trajectories:2 – all patches occupied

C2E2

E7DNE5

E5E5E2 C5

E2

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.

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?

Other applications of decision theory to population management

and 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)

Optimal translocation strategies

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

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

Oryx problem

Zoo Population

Growth rate R = 1.3 Capacity = 20

Wild Population

??

Growth rate R = 0.85 Capacity = 50

Result – base parameters

Captive Population

Wild

Pop

ulat

ion

C

R

Captive Population

Wild

Pop

ulat

ion

C

R

R = release, mainly when population in zoo is near capacityC = capture, mainly when zoo population small, capture entire wild population when this would roughly fill the zoo

If zoo growth rate changes, results change – but for a “new” species we

won’t know R in the zoo

Enter – active adaptive management,Management with a plan for learning

Active adaptive management

• 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

Cindy Hauser, Petra Kuhnert, Katriona Shea, Tony Pople, Niclas Jonzen (Lund)

Toy fish problem

Secure

Collapsed

Fragile

Secure

Fragile

Collapsed

Unharvested Harvested

????? ?????

Harvest,Harvest,

Yes or no?Yes or no?

• 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 hasn’t

• Use Baye’s 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

Active adaptive monitoring: the problem of the Swedish lynx Lynx lynx

• 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??

Henrik Andren (SLU), Anna Daniel (SLU), Cindy Hauser, Tony Pople

Information for Swedish lynx problem

• “Population size” (N) is number of Lynx family units• Compensation cost per N – 20,000 SK, higher if N > 200• Cost of current monitoring program• Political cost of N falling below 50 = 100,000,000 SK• Fixed harvest strategy – 15% if N > 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

Results – should monitoring bestate-dependent?

Utility 106

Number of Lynx family units (N)

10

100

40

50 200

States of intensive monitoring?

Applied Theoretical Ecologist Dreaming

Optimal Harvesting

Optimal Monitoring

Optimal Learning

New approaches to the “evolution” of complex ecological systems: kangaroo

population dynamics

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

What is important is

not always interestin

g,

What is interestin

g is not always im

portant

Overview1. 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

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

Data collection: Fixed-wing Survey

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

Animation of Kangaroo survey data

Temporal patterns at a whole region scale

0

5

10

15

20

25

1980 1985 1990 1995 2000 2005

Sca

led

mea

sure

Year

Kangaroo Numbers

Vegetation Index

Growth Rate

??

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?

South Australia: main management zones

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)

Ratio Model

0

5

10

15

20

25

1976 1980 1984 1988 1992 1996 2000 2004

Modelled population

Actual population

Northeast Pastoral Zone

Year

Pop

ulat

ion

size

Interactive model

Northeast Pastoral Zone

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

1976 1980 1984 1988 1992 1996 2000 2004

Pastu

re b

iom

ass (

kg

/ha)

Modelled population

Actual population

Pop

ulat

ion

size

Year

A more complex statistical model

The model – with nested effects of

1. density dependence – bN

2. rainfall – R

3. sheep – S

4. harvesting – H, and

5. correlated environmental variability, E

)()1(,)(,

)(,)(,)1(,)1(,)1(, tititiitiitiii EHSdRcNbatiti eNN

We don’t know as much as we thought

• Use Akaike’s 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

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

Optimal Harvest Strategy?

Logistic model, fish

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 10 20 30 40 50 60

Harvest rate (%)

Mea

n s

easo

nal

har

vest

km

-2

Ratio model

Interactive model

Mea

n ne

t har

vest

per

yea

r

Percentage harvest

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

Monitoring and managing

0.0

0.1

0.2

0.4

0.5

0 1 2 3 4 5

250

300

350

400

450

500

Period of monitoring (years)

InfrequentAnnual

Probability of c

ollapse

Pro

babi

lity

of

coll

apse

Cost of m

onitoring $1000

Cost

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?

• 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?

Take home messages