Appendix S1: Numerical procedure for finding the ... - PLOS
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Appendix S1: Numerical procedure for finding the ... - PLOS
Advice for step 1: The equation in step 1 can be solved by using a procedure offered in many numerical software to find the eigenvector corresponding to the ...
1
Appendix S1: Numerical procedure for finding the metapopulation equilibrium ˆ requires solving Eqs. 13. We use the following scheme of iterative substiFinding the equilibrium (ˆ p, d) tution, beginning with an initial guess d0 for the disperser-pool state: 0. Let d = d0 . 1. Construct B(d) and solve B(d)p = 0 for p. 2. If f (d, p) 6= 0 up to numerical precision, solve f (d, p) = 0 for d and return to step 1; ˆ = d, and stop. otherwise, let p ˆ = p and d Advice for step 1: The equation in step 1 can be solved by using a procedure offered in many numerical software to find the eigenvector corresponding to the zero eigenvalue of B (i.e., its null space). Normalizing this eigenvector gives the estimate for p. Alternatively, one can exploit the sparseness of B with the following procedure: a) Find bmax = maxj {−bjj }, where bjj are the diagonal elements of B. b) Construct C = B + bmax I. c) From an initial guess q0 , iterate qn+1 = Cqn / kqn k to convergence; then let p = qfinal / kqfinal k. Advice for step 2: We find that for dim(d) = m > 1 the equation in step 2 is best solved by considering it in 2 terms of the minimization problem p = argmind0 kf (d0 , p)k and using a multi-dimensional optimization procedure to obtain the estimate for d. Since this scheme depends on multi-dimensional optimization, it may take some experimentation with initial values to obtain satisfactory convergence to a non-trivial equilibrium.
