Abstract
Cooperative dynamics are common in ecology and population dynamics. However, their commonly high degree of complexity with a large number of coupled degrees of freedom renders them difficult to analyse. Here, we present a graph-theoretical criterion, via a diakoptic approach (divide-and-conquer) to determine a cooperative system’s stability by decomposing the system’s dependence graph into its strongly connected components (SCCs). In particular, we show that a linear cooperative system is Lyapunov stable if the SCCs of the associated dependence graph all have non-positive dominant eigenvalues, and if no SCCs which have dominant eigenvalue zero are connected by a path.
Highlights
Cooperative systems are a wide class of dynamical systems characterized by a non-negative dependence between components [1]
The conditions stated in theorem 2.10 prescribe a way to simplify the analysis of a high-dimensional linear cooperative system by decomposition into lower dimensional subsystems, the strongly connected components (SCCs) of the dynamical system’s dependence graph
Marginal stability is of importance for linear systems, since marginally stable states represent the only possible non-vanishing stable states—i.e. not identical to the zero-vector—called steady states
Summary
Cooperative systems are a wide class of dynamical systems characterized by a non-negative dependence between components [1]. Since all off-diagonal elements of A (and of each Bk) are non-negative, and each Bk is the adjacency matrix of a strongly connected graph, the matrices Bk are irreducible Metzler matrices, for which the Perron–Frobenius theorem applies, to the shifted eigenvalues [20].
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