Performing Sensitivity Analysis for Ecological Models Michael Mikucki

Performing Sensitivity Analysis for Ecological Models Michael Mikucki Colorado State University 21 November 2009

Stage Structured Ecological Models n Discrete Time: Iterated Maps n n Previous Talk by Reid Thornton Life cycles grouped in states Transitioning between states: probabilities Continuous Time: Differential Equations

Sensitivity Analysis n What is sensitivity? n n n States x 1, x 2, …, xi Parameters p 1, p 2, …, pj Initial Conditions x 10, x 20, …, xi 0 (treated as pj+1, …, pj+i) Relative change in state with respect to a parameter Interpretation What are the benefits? n n Data Collection Management Strategies

Linear Sensitivity Calculations n Linear maps for ecological systems n xt+1 = Axt n n n Sensitivity of state xi with respect to parameter pj = dxi/dpj (at time t) n n xt is vector of states evaluated at time t A is the transition matrix defining the system dxt+1/dp = (d. A/dp)xt + A(dxt/dp) Easy to calculate

Adding Nonlinearity (Caswell 2009) n 2 states n n xt+1 = Axt where n Suppose σ1 depends on states n A depends on p and x x 1 – Juveniles x 2 – Adults 4 parameters n n f – adult fertility σ1 – juvenile survival σ2 – adult survival γ – maturation probability d. A/dp = (∂A/∂x)(dx/dp) + ∂A/∂p Caswell: Perturbation Analysis of Nonlinear Models (2009)

Adding Nonlinearity (Caswell 2009) n n n Caswell: Perturbation Analysis of Nonlinear Models (2009)

Adding Nonlinearity (Caswell 2009) n n Found ∂A/∂p and ∂A/∂x Given initial values for f, γ, σ, σ2, we can iterate dx/dp in time dxt+1/dp = (d. A/dp)xt + A(dxt/dp) = [(∂A/∂p) + (∂A/∂xt)(dxt/dp)] xt + A(dxt/dp) n Most difficult nonlinear sensitivity analysis to date n Increasingly difficult with more complicated models Caswell: Perturbation Analysis of Nonlinear Models (2009)

Form of Nonlinear Model n Caswell Example form n n Sensitivity of Caswell Example form n n x(p, t+1) = A(x(p, t), p)*x(p, t) dx(t+1)/dp = [(∂A/∂p) + (∂A/∂xt)(dxt/dp)] xt + A(dxt/dp) Treat as x(p, t+1) = g(x(p, t), p)) n dx(t+1)/dp = (∂g/∂x)(dx/dp) + ∂g/∂p

Caswell Example Rewritten n Want form x(t+1, p) = g(x(t, p) dx(t+1)/dp = (∂g/∂x)(dx/dp) + ∂g/∂p n f n n f

The Need for a Computerized System n Pine Model by R. Thornton n n Nonlinear form 12 states, 29 parameters = 348 derivatives Longest derivative required 3, 241 characters (no spaces) Total of 97, 846 characters (no spaces) Automated differentiation

Graphical User Interface (GUI) n Direct Implementation n n Indirect Implementation n Model Equations Parameters Initial conditions Number of Iterations Create own Maple code to input the above information General outline provided Press “Create Matlab files using Maple” button

GUI Technology n Creates MATLAB files using Maple n Automatic differentiation n Maple: ∂g/∂x, ∂g/∂p MATLAB executes files by iterating dx/dp over time n Requires MATLAB 7. 8. 0 (R 2009 a), Maple 13, and “Maple toolbox for MATLAB” n

Sensitivity Manipulation Amount of plots (IPlot) n Want only specific states (IList) n Want only specific parameters (KList) n Want sensitivities at specific iterations n Want linear combination of solutions n n Example: Sum/difference of 2 classes QOI = < ψ, x > d(QOI)/dp = < ψ, dx/dp >

Implementation of the GUI n Input Quantity of Interest Information n Iplot, IList, KList Ψ Vector, QOI Times Press the “Execute Matlab files” button

Continuous-time Sensitivity n ODE extension n n x(t+1) = g(x(t, p) dx/dt = g(x(t, p) Let z = dx/dp dz/dt = d/dt (dx/dp) = d/dp (dx/dt) **sensitivity of dx/dt = d/dp (g(x(t, p)) = (∂g/∂x)(dx/dp) + (∂g/∂p) = (∂g/∂x)z + ∂g/∂p Numerical Error in dz/dt evaluation Can do sensitivity analysis for continuous nonlinear models n SIR, cellular processes, Hodgkin Huxley, electrical circuits

Conclusions n Sensitivity analysis is crucial for ecological models n n Need for automated differentiation (GUI) n n n Discrete or continuous models Complicated models Edits to the model Extensions of the GUI lead to further research n n Numerical error in continuous model analysis Adjoint techniques in solving data

Acknowledgements Simon Tavener, Mike Antolin Colorado State University Funded in part by National Science Foundation Anna Schoettle United States Forest Service