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” problem How 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 1 Find the smallest number of sites that represents all species The data matrix - A

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 Minimise Subject to if the site is in the reserve system

Simulated annealing and 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} = = - 5 Which is not as fit as {A,B} = = - 4 or {A,B,C} = - 3 = - 3 With best solution {C,E} = = - 2

GAs: Breeding a reserve system cost cost 6... babies infeasible 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 the objective 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

Minimise if the site is in the reserve system Boundary Length Problem Non-linear IP problem 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

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

Effect of South Australia’s existing marine reserves

But reserve systems are not 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

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/s e 1/s m

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

Vegetation dynamics: Transition probability from j early to i early

Fire model

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

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, 2 n, where n is the number of patches. The transition matrix is 2 n by 2 n 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

E5 Management trajectories: 1 – only largest patch occupied C5 E5 E7 DN

E5 Management trajectories: 2 – all patches occupied C2 E2 E7DN E5 E2 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 2 N 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 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

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, 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 be state-dependent? Utility  10 6 Number of Lynx family units (N) 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 interesting, What is interesting is not always important

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

1Background 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

2Visualisation 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 Scaled measure 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 / D t ) – harvest rate (plausible but abstract) Interactive model (Caughley – data hungry) (rainfall  pasture  kangaroos) (more plausible but complex)

Ratio Model Northeast Pastoral Zone Year Population size

Interactive model Population 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

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? Mean net harvest per year 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 Period of monitoring (years) InfrequentAnnual Probability of collapse Cost of monitoring  $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