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Performing Sensitivity Analysis for Ecological Models

This presentation discusses sensitivity analysis in ecological models, exploring both discrete time iterated maps and continuous time differential equations. Key concepts include the benefits of sensitivity analysis, management strategies, and linear sensitivity calculations. The talk highlights the necessity of computerized systems for complex models, illustrated through a case study involving nonlinear sensitivity analysis. It also emphasizes the importance of numerical methods and automation in analyzing data from ecological systems, aiming to provide insights for effective ecological management.

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Performing Sensitivity Analysis for Ecological Models

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

  2. 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

  3. 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

  4. 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

  5. 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 x dA/dp = (∂A/∂x)(dx/dp) + ∂A/∂p Caswell: Perturbation Analysis of Nonlinear Models (2009)

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

  7. 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)

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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”

  13. 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 >

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

  15. 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

  16. 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

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

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