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The master stability function is the core element for the description of a multilayer network of the proposed form. While we can define the stability of a system with a particular spatial network using the master stability function, it would be interesting to see if we could define properties of the master stability function that would represent a tendency for stability of networks. Such a definition of stability for meta-communities would be independent of the spatial component and only depend on the matrix of local contributions, P, and the coupling matrix, C. Comparing stable and “forbidden” interval sizes

of the master stability function comes to mind as a intuitive measure of stability but is only possible if we find a suitable range in which we evaluate the master stability function as we can compute the master stability function up to an arbitrary point. A proposal for such a range would be, for example, up to the last stability change. We (usually) cannot compute the stability changes analytically but using eigenvalue approximations like the Gershgorin circle theorem or the Brauer-Cassini ovals allows us to make estimations of the limits of the eigenvalues of the Jacobian. Under certain conditions it is possible to say that the sign of the master stability function cannot possibly change outside a certain range and thus the system cannot change its stability anymore.

We can get the eigenvalues of the Jacobian J of the meta-community with the combination matrix

P− κC =   P_ii− κCii · · · PiN− κCiN .. . ... ... P_{N i− κCN i · · · PN N− κCN N}   . (213)

We can use the Gershgorin theorem to compute boundaries of the eigenvalues of the combination matrix. To define the stability of the system we only need to know the real parts of the eigenvalues. All eigenvalues are within the Gershgorin disk D(X , R) which lies in the complex plane and has a center at X _{∈ C and a range of R ∈ R. For a real combination matrix we know that the center X has no imaginary} part and thus sits on the real axis of the complex plane at the value X . The largest real part of an eigenvalue is thus

Re(λ_max) = Re(X ) + R = X + R (214)

The center X and range R can be written in terms of the matrix elements. The disk D(X , R) then be- comes D€P_ii− κCii, PN i6= j|Pi j− κCi j| Š or D€P_ii− κCii, PN i6= j|Pji− κCji| Š

depending on whether we take the row or column sum for the approximation. For the sake of brevity we will omit the discs computed by using row sums as it is completely analogous to the computation with column sums. The maximum real part of the eigenvalue for column i that exists within that disk is

λS,i= Pii− κCii+ N X

i6= j

|Pi j− κCi j| (215)

under the condition that no discs overlap. If discs overlap the eigenvalues can be situated anywhere within the connected discs and thus the largest λ_S,i would be the limit for all eigenvalues associated

with the overlapping discs. If the largest real part of all eigenvalues is below zero, we have a stable system. We also know that a system cannot be stable if there is a eigenvalue that lies within a Gershgorin disk that does not allow for eigenvalues with a below zero real part. The lower limit of the disk can be described by λI,i= Pii− κCii− N X i6= j |Pi j− κCi j| . (216)

The eigenvalues depend on the Laplacian eigenvalue κ. The master stability function is usually de- picted in the form of real part of leading eigenvalue overκ and therefore solve the largest real part of all eigenvalues forκ: κ = Pii− PN i6= j±Pi j C_ii−PN i6= j±Ci j . (217)

The signs of theP and C elements depend on the exact values. This formula does not necessarily have

a feasible solution (say negativeκ) and extraneous solutions have to be discarded. The value of κ for a disk that just reachesλ_I= 0 can be computed analogously to the equation (217).

A more graphical interpretation of the gersghorin disk may express the eigenvalue problem as such: If the radius of the gersghorin disk ( PN_{i6= j}|Pi j, ji− κCi j, ji| ) increases faster than the center of the disk

(P_ii− κCii) moves in a direction then we cannot establish a point at which we can guarantee the stability or instability of the system. This is the case when

C_ii < N X

i6= j

|Ci j, ji| (218)

as the local parameters covered byP are negligible for sufficiently large coupling strengths C_{i j} and/or κ.

The sign of the diagonal element C_ii gives the direction of the movement of the center of the disk. In most cases of migration we expect a positive value for the diagonal elements of the coupling matrix as we expect migration to increase with higher population densities.

An alternative to the Gershgorin discs as a way to approximate eigenvalue boundaries are the Brauer- Cassini ovals. While more ovals than discs need to be considered to get the limits of the eigenvalues, the boundaries are more precise.

The Cassini ovals for a complex n× n matrix A = [ai, j] are defined as

|z − ai,i| · |z − aj, j| ≤ Ri· Rj; z∈ C. (219) with R_i defined as the i-th column sum. We are only interested in the boundaries of the inequality and thus can write

|z − ai,i| · |z − aj, j| = Ri· Rj. (220) We can now solve the equation for z. There are four distinct solutions:

z_±,±=1 2

€

±Ça2_{i,i− 2ai,i}a_{j, j}+ a2_{j, j± 4Ri· Rj}+ a_i,i+ a_{j, j}Š (221) The largest real value is obtained for z_+,+, where both plus-minus signs are plus. If z_+,+= 0 we know that all eigenvalues are at or below zero and thus the system should be stable for non-pathological cases. In our caseA is the combination matrix P− κC.

The limit value for the Laplace eigenvalue is always smaller or equal to the one computed with the Gershgorin discs. The trade-off is the slightly more costly computation of the boundaries.