Supplementary Material
Appendix A: Sensitivitycalculationin forest landscape model
In the analyses for matrix population models, researchers mainly focused on the perturbation (sensitivity and elasticity) analysis of long-term equilibrium dynamics to model parameters (Caswell 1978). Since we are concerned with both equilibrium and non-equilibrium forest dynamics, for the sensitivity analysis for a forest landscape model, we need to use sensitivityanalysis of transient dynamics in matrix population modeling(Koons et al. 2005; Yearsley 2004; Fox and Gurevitch 2000; Caswell 2007; Mertens et al. 2006), which can account for both short-term non-equilibrium and long-term equilibrium dynamics. Two approaches are available to calculate the sensitivity and elasticityfor transient dynamics: 1) based on the derivatives of eigenvalues (Yearsley 2004; Fox and Gurevitch 2000); and 2) based on an iterative derivative approach (Caswell 2007). The iterative approach is not as computationally-intensive as the eigenvalue approach, and it does not require separate calculations of eigenvalues or eigenvectors.It also provides the potential to incorporate non-constant transition matrix. Thus,for this study, we adopted the Caswell’s iterative derivative approach for calculation of sensitivity for transient population dynamics. For a time period from t1 to t2, we define
(A1)
as a random matrix which can take values in {, ,…, }. Following the iterative derivative approach proposed by Caswell(2007), we can calculate the sensitivity of model output to the transition matrix entries by
, (A2)
where is a m2 by m sensitivity matrix,
.
is a transformation of the matrix to column vectors; represents the Kronecker product and is a m by m identity matrix.We apply eq. (A2) until time t2, with the initialization at time t1
(A3)
where is a m2 by mzero matrix.
The model formused in eq.(1) of the main text is subject to one important condition that the transition probability in each column should be sum to one. The sensitivity normally refers to the change of a specific matrix entry (i.e., transition probability) while holding other entries constant, which contradicts the condition that the column entries sum to 1. Alternative compensation methods have been proposed to address this issue (Caswell 2001). Specifically, the sensitivity can be adjusted as follows,
(A4)
where y can be a linear combination of . There are different ways to select based on different compensation methods (e.g., uniform compensation, proportional compensation and random compensation) (Caswell 2001). However, these methods are inherently subjective and our experience shows that for any of the compensation methods, some entries not important in themselves may become important after the compensation method is applied. Thus, in this study, we did not incorporate the compensation, which is equivalent to assuming that the compensation coefficient,, be equal to zero.
Appendix B: Elasticity calculationfor combination of forest types
If we are concerned with the elasticity of the sum of different forest types (e.g, , where ) to transition from forest type j to forest type i, in view of the additive properties for derivatives, the elasticity (, where and is a m by 1 vector of 1 or 0’s) can be calculated based on the sum of sensitivity for individual forest types. Namely,
. (B1)
For notational convenience, we use to represent the elasticity matrix for kth forest type in the model output,
. (B2)
Similarly, we use to represent elasticity matrix for the sum of different forest types in model outputs.
References
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