Computational issues surrounding the management of an ecological food web

Transcription

1 Computatonal ssues surroundng the management of an ecologcal food web Wllam J M Probert, Eve McDonald-Madden, Nathale Peyrard, Régs Sabbadn AIGM 12, ECAI2012 Montpeller, France

2 Ratonale Ecology has many complex problems. Conservaton s management wthn ecology. Management s allocaton of resources. Ecologcal systems as food webs. How should we allocate resources to these systems through tme?

3 The problem - managng a food web - a DAG Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great prey predator

4 Managng a food web Graph metrcs provde a sutable set of heurstcs. If we re gong to manage then we need to quantfy the management problem. What s the problem? Management of graph, G = (V, E) over tme. Has been done over one tme step usng a Bayesan network. Compare heurstcs wth optmal soluton.

5 Dagram of modellng technques decsons & rewards Markov chan 1D random varable Stochastc optmsaton tme structure MDP Bayesan network Dynamc Bayesan networks Influence dagram Structured stochastc optmsaton

6 MDP framework, X, A, P, R, G Fnte tme horzon: t = {1,, T }, T < Speces states: extnct or extant, x t {0, 1} Speces-level actons: to protect or not, a t {0, 1} Budget constrant: c a t B t V

7 Neghbourhoods Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great A speces neghbourhood ncludes all prey and tself N() = {j V (j, ) E} {} Allows decomposton on a local scale of transton probabltes (nto a product) and rewards (nto a sum).

8 Transton probabltes Subject to: P t (x t+1 x t, a t ) = n =1 P t t+1 (x xn() t, at ) Probablty of survval s p 0 tmes proporton of alve prey. ( ) P (x t+1 = 1 x t = 1, x N()\, a t = 0) = p 0 f,t For basal speces, survval probablty s just p 0. Extncton (death) s an absorbng state. A speces must have at least one prey speces extant. A speces wll survve f protected and the above hold. f

9 Rewards Fnal tme-step reward functon: number of extant speces, R T (x T ) = n =1 x T and per-tme-step rewards are zero, R t (x t ) = 0, t < T. Total expected reward of a polcy, δ [ T υδ T (x 1 ) = E R t (x t, a t ) t=1 ] x 1, δ Varous reward functons can be nvestgated.

10 Optmal soluton - Backwards Inducton Algorthm 1. Set the current tme-step to t = T and the value n the fnal tme-step to υ T (x T ) = R T (x T ) x T X 2. Set t = t 1 and calculate υ t (x t ) for each state usng where v (x t t ) = max Q t (x t, a t ) a t A Q t (x t, a t ) = R t (x t ) + x t+1 a t = arg max Q t (x t, a t ) a t A P(x t+1 x t, a t ) υ t+1 (x t+1 ) 3. If t = 1 then stop, otherwse return to step 2.

11 Metrcs polces, δ = (d 1,, d T 1 ), d t : x t a t Manage speces n descendng order of metrc untl B t s exhausted Metrc polces Degree centralty Betweenness centralty Closeness centralty Keystone ndex Trophc level Other polces Bottom-up ndex Return on Investment None Random

12 Metrcs polces Isolates have metrc values of zero (managed last) Tes use randomsaton Dsconnected graphs calculate relatve measures on each subgraph. Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great

13 Metrcs - degree centralty D = D + D Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great

14 Metrcs - prey degree D s the sze of the set V = {j V : (j, ) E} of all prey of speces Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great

15 Metrcs - predator degree D s the sze of the set V = {j V : (, j) E}, the set of all predators of speces Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great

16 Metrcs - betweenness centralty Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great j<k g jk () g jk BC = ( V 1)( V 2), g jk = number of shortest paths between speces j and k, g jk () = number of shortest paths between speces j and speces k whch pass through speces.

17 Metrcs - closeness centralty Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great n d(, k) =1 CC k = n 1 1 d(, k) = dstance from speces to k

18 Metrcs - trophc level Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great Related to Bottom-up prortsaton

19 Metrcs - keystone ndex Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great K = K = K c V + K, 1 D c top-down + bottom-up (1 + Kc ), K = e V 1 D e (1 + K e )

20 Metrcs - keystone top-down ndex Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great

21 Metrcs - keystone bottom-up ndex Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great

22 Metrcs - keystone drected ndex Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great K = K dr = K dr c V + K undr 1 D c + (drect + ndrect) e V 1 D e, K ndr = c V Kc Dc + e V K e D e

23 Metrcs - keystone ndrected ndex Kller Star fsh sea lon Mussels/ barnacles urchn Oceanc Great

24 Fnte horzon metrc polcy evaluaton 1. Set the current tme-step to t = T and the termnal rewards n the fnal tme-step to υ T δ (x T ) = R T (x T ) x T X 2. Set t = t 1 and calculate υ t δ (x t ) for each state usng where v t δ (x t ) = Q t (x t, d(x t )) Q t (x t, d(x t )) = R t (x t ) + P(x t+1 x t, d t (x t )) υ t+1 δ (x t+1 ) x t+1 3. If t = 1 then stop, otherwse return to step 2.

25 Experments For 25 speces, B = 8, we ve more than transton probabltes Varous transtons can be set to zero based on the condtons of the transton probabltes. M = SG S : X 2 X, s a 2 n n Boolean matrx that ndcates for each possble state whch speces s extant. G s adjacency matrx. Gj = 1 f speces s a prey of speces j and otherwse 0. No cannbalsm. M,j = Number of extant prey of speces j when the state s S,: Q = M S P,,a = 0 f Q,j = 0 j s.t. S,j > 0. For 10 speces Alaskan web, > 95% of state transtons are nvald.

26 Prelmnary results - 10 speces Kller sea lon urchn Oceanc Start wth soluton to 10 speces, B = 4, T = 10.

27 Prelmnary results - 10 speces Polcy υ δ (x 1 ) Optmal 5.92 K 5.52 BUP 5.52 D 5.51 K 4.99 K ndr 4.97 K dr 4.76 BC 4.00 D 3.84 CC 3.72 Random 3.66 K 3.49 D 3.45 None 1.10 Parameters: 10 speces, B t = 4, T = 10

28 Results 6 Value / Total expected reward D< 1 D > Random None Optmal Food web sze (n) Parameters: B t = [2, 3, 3, 4, 4, 4, 4] respectvely, T = 10, 25 random food webs, connectance = 0.1

29 Future work Fnd exact soluton for webs wth n up to 20. Code transton probabltes n a faster language Use POMDP solver, eg Perseus Extract decson tree from optmal polcy Smulate management usng heurstcs on large webs Investgate alternatve approxmatng solutons