Performing Sensitivity Analysis for

Ecological Models Michael Mikucki

Colorado State University

21 November 2009

Stage Structured Ecological Models Discrete Time: Iterated

Maps Previous Talk by Reid

Thornton Life cycles grouped in

states Transitioning between

states: probabilities

Continuous Time: Differential Equations

Sensitivity Analysis What is sensitivity?

States x1, x2, …, xi

Parameters p1, p2, …, pj

Initial Conditions x10, x20

, …, xi0

(treated as pj+1, …, pj+i) Relative change in state

with respect to a parameter Interpretation

What are the benefits? Data Collection Management Strategies

Linear Sensitivity Calculations Linear maps for ecological systems

xt+1 = Axt

xt is vector of states evaluated at time t A is the transition matrix defining the system

Sensitivity of state xi with respect to parameter pj = dxi/dpj (at time t) dxt+1/dp = (dA/dp)xt + A(dxt/dp)

Easy to calculate

Adding Nonlinearity (Caswell 2009) 2 states

x1 – Juveniles x2 – Adults

4 parameters f – adult fertility σ1 – juvenile survival σ2 – adult survival γ – maturation

probability

xt+1 = Axt where

Suppose σ1 depends on states

A depends on p and xdA/dp = (∂A/∂x)(dx/dp) + ∂A/∂p

Caswell: Perturbation Analysis of Nonlinear Models (2009)

Adding Nonlinearity (Caswell 2009)

Caswell: Perturbation Analysis of Nonlinear Models (2009)

Adding Nonlinearity (Caswell 2009) Found ∂A/∂p and ∂A/∂x Given initial values for f, γ, σ, σ2, we can iterate dx/dp

in time

dxt+1/dp = (dA/dp)xt + A(dxt/dp)

= [(∂A/∂p) + (∂A/∂xt)(dxt/dp)] xt + A(dxt/dp)

Most difficult nonlinear sensitivity analysis to date Increasingly difficult with more complicated models

Caswell: Perturbation Analysis of Nonlinear Models (2009)

Form of Nonlinear Model Caswell Example form

x(p,t+1) = A(x(p,t),p)*x(p,t) Sensitivity of Caswell Example form

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))

dx(t+1)/dp = (∂g/∂x)(dx/dp) + ∂g/∂p

Caswell Example Rewritten Want form x(t+1,p) = g(x(t,p),p)

dx(t+1)/dp = (∂g/∂x)(dx/dp) + ∂g/∂p

f f

The Need for a Computerized System

Pine Model by R. Thornton 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) Direct Implementation

Model Equations Parameters Initial conditions Number of Iterations

Indirect Implementation Create own Maple code to

input the above information General outline provided

Press “Create Matlab files using Maple” button

GUI Technology Creates MATLAB files using Maple

Automatic differentiation Maple: ∂g/∂x, ∂g/∂p

MATLAB executes files by iterating dx/dp over time

Requires MATLAB 7.8.0 (R2009a), Maple 13, and “Maple toolbox for MATLAB”

Sensitivity Manipulation Amount of plots (IPlot) Want only specific states (IList) Want only specific parameters (KList) Want sensitivities at specific iterations Want linear combination of solutions

Example: Sum/difference of 2 classes QOI = < ψ, x > d(QOI)/dp = < ψ, dx/dp >

Implementation of the GUI Input Quantity of

Interest Information Iplot, IList, KList Ψ Vector, QOI Times

Press the “Execute Matlab files” button

Continuous-time Sensitivity ODE extension

x(t+1) = g(x(t,p),p) dx/dt = g(x(t,p),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),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 SIR, cellular processes, Hodgkin Huxley, electrical circuits

Conclusions Sensitivity analysis is crucial for ecological

models Discrete or continuous models

Need for automated differentiation (GUI) Complicated models Edits to the model

Extensions of the GUI lead to further research Numerical error in continuous model analysis Adjoint techniques in solving data

AcknowledgementsSimon Tavener, Mike Antolin

Colorado State University

Funded in part by National Science Foundation

Anna Schoettle

United States Forest Service