| Abstract: |
| Generalized Lotka-Volterra (GLV) systems, as models of ecological communities, can display diverse dynamics, ranging from global stability, to periodic orbits, and even chaos. We propose a framework of studying GLV systems by borrowing ideas from reaction network theory, specifically the notions of detailed-balanced and complex-balanced. We associate any GLV system to a directed graph embedded in $\mathbb{R}^n$, and prove theorems of the form: If the graph has property $P$, then the associated GLV system has dynamical property $X$. For example, if the embedded graph is strongly connected, then the GLV system has a globally stable coexistence equilibrium (within each invariance manifold). The stability is guaranteed by a different Lyapunov function than the Volterra Lypaunov function. Other dynamical properties we can infer include persistence, and ruling out limit cycles. |
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