A probabilistic method for food web modeling

A probabilistic method for food web modeling Bayesian Networks methodology, challenges, and possibilities Anna Åkesson, Linköping University, Sweden 2 nd international symposium on Ecological Networks, Bristol, UK

Motivation To efficiently predict extinction risk of species in ecological networks Species function in complex networks a single species extinction can cause a cascade of secondary extinctions There is the danger of simplification, and there is complexity can we find a middle ground approach?

Common methods - Topological approach Advantages Requires only network structure as input Possible to analyse very large networks Drawbacks All nodes have identical characteristics Secondary extinctions only occur when all resources are lost

Common methods - Dynamical modeling Advantages Possible to capture indirect effects such as top-down effects Species can be given various properties depending on, for example, trophy level Drawbacks Requires extensive set of parameters Slightly different initial conditions can produces different outcomes many replicates necessary

- a graphical model Middle-ground approach: topological structure no extensive simulations, but with some of the complexity used in dynamical models included Applications Probability of the presence of various diseases Modeling beliefs in bioinformatics (gene regulatory networks, protein structure, gene expression analysis) Artificial Intelligence

- structure Nodes Bernoulli random variables Links directed arcs, representing conditional dependencies among variables Extinction probabilities - a function of the state of a species parent nodes (resources)

- structure Extinction probability of species i; P(i f) = π + (1-π) f P(D A,C)=0.2 P(D A,C)=0.6 P(D A,C)=0.6 P(D A,C)=1 where f is the fraction of resources lost P(C A,B)=0.2 P(C A,B)=0.6 P(C A,B)=0.6 P(C A,B)=1 P(A)= π (0.2) P(B)= π (0.2)

- structure P(D A,C)=0.2 P(D A,C)=0.2 P(D A,C)=0.2 P(D A,C)=1 Topological Bayesian network P(D A,C)=0.2 P(D A,C)=0.6 P(D A,C)=0.6 P(D A,C)=1 P(C A,B)=0.2 P(C A,B)=0.2 P(C A,B)=0.2 P(C A,B)=1 P(C A,B)=0.2 P(C A,B)=0.6 P(C A,B)=0.6 P(C A,B)=1 P(A)=0.2 P(B)=0.2 P(A)=0.2 P(B)=0.2

- structure P(D A,C)=0.4 P(D A,C)=0.7 P(D A,C)=0.7 P(D A,C)=1 Bayesian network Different baseline probability of extinction P(C A,B)=0.3 P(C A,B)=0.65 P(C A,B)=0.65 P(C A,B)=1 P(A)=0.2 P(B)=0.2

- structure Bayesian network Different baseline probability of extinction Interaction strengths: resources weighted for their relative contribution (e.g. proportion biomass flowing from resource to consumer)

- marginal probabilities Builds a table for each species, specifying its probability of extinction defining the Bayesian network Need to combine all possible states (tables) of all species solving the Bayesian network Receives marginal probabilities for every species P(D A,C)=0.2 P(D A,C)=0.6 P(D A,C)=0.6 P(D A,C)=1 P(A)=0.2 P(C A,B)=0.2 P(C A,B)=0.6 P(C A,B)=0.6 P(C A,B)=1 P(B)=0.2

- testing the method Can we capture the secondary extinctions produced in dynamical simulations? 100 networks built with the niche model Extinction scenarios simulated by the Allometric Trophic Network (ATN) model provide reference extinction scenarios Computation of the likelihood that the Bayesian network algorithms replicate ATN-simulated extinctions

- performance Eklöf et al. (2013): Results close to result of the ATN model, however; secondary extinctions where all of the species resources are extant cannot be predicted Can top-down effects be implemented in a Bayesian network?

- attempts to improve the model Calculate marginal probabilities for bottom-up controlled network and somehow include bi-directional forces, such as pressure from predator to prey?

Dynamic Bayesian Networks - a possible solution? Extension of Bayesian networks variables are related to each other over adjacent time steps. Enables modeling of sequential data, e.g. temporal data Unfold the network in time to enable bi-directional forces t

Conclusions Bayesian networks - combine the simplicity of the topological approach with important features of dynamical models, without an extensive set of parameters - builds a bridge between theoretical biology and conservation biology; includes results from conservation-oriented research into algorithms for the analysis of networks

- practical usage Take the network structure for some ecological system Use the IUCN Red List to assign baseline probabilities Calculate each species probability of going extinct; Pinpoint species particularly threatened; Simulate primary extinctions and consequences for the remaining system

Thank you for listening! anna.akesson@liu.se