Applied Mathematical Ecology/ Ecological Modelling

1. Applied Mathematical Ecology/Ecological Modelling Dr Hugh Possingham The University of Queensland (Professor of Mathematics and Professor of Ecology) AMSI Winter School 2004

2. 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

3. 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.

4. Optimal Reserve System Design Hugh Possingham and Ian Ball (Australian Antarctic Division) and others

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

6. 13,000 planning units

7. 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

8. Example Problem 1 Find the smallest number of sites that represents all species The data matrix - A

9. Algorithms to solve the reserve system design problem • Wild guess • Heuristics • Mathematical Programming • Heuristic algorithms • Simulated annealing • Genetic Algorithms

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

11. ILP formulation Minimise Subject to if the site is in the reserve system

13. 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

14. 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 ...

15. 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

16. 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

17. 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

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

21. 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

23. 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

24. 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

25. Effect of South Australia’s existing marine reserves

26. 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

27. Example 3 – Identifying ‘Irreplaceable’ Areas

28. 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

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

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

31. 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?

32. 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

33. Habitat

34. Assume three successional states fire, f late mid 1/sm 1/se early

35. Ngarkat Conservation Park

36. 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

39. Vegetation dynamics:Transition probability from j early to i early

40. Fire model

41. Fire transition matrix and Succession transition matrix are combined to generate state dynamicsBUT • Succession Markovian • Fire model naive

42. 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

43. 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

47. 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

48. 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

49. Applied Theoretical Ecology Eradicate, Exploit, Conserve + Pure Ecological Theory Decision Theory =

50. How to manage a metapopulation Michael Westphal (UC Berkeley), Drew Tyre (U Nebraska), Scott Field (UQ) Can we make metapopulation theory useful?