Abstract
We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.
Highlights
We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration
Theoretical studies on cyclic competition among three species are based on the Lotka–Volterra model of ordinary differential equations (ODEs), which ignores the effects of the spatial domain and predicts a solution of unstable periodic dynamics that leads to the extinction of two of the species after a short transient[24,25]
We show that the balanced model admits a unique coexistence equilibrium that is asymptotically stable if the dispersal network is heterogeneous, whereas the same equilibrium is neutrally stable in the case of a homogeneous network
Summary
We highlight some terminologies, methods, and results from literature that will be used in this paper. The properties (i.e., existence and stability) of the coexistence equilibrium of the mean-field ODE system (1) can be deduced from the behavior of the Nash equilibrium (NE) strategy of a corresponding zero-sum game This is because the ODE system (1) is equivalent with the celebrated replicator equation of evolutionary game theory, which describes how a population of pure strategies evolves over time in a symmetric, two-player zero-sum game defined by the payoff matrix T10,39,40. It has already been shown that all twoperson zero-sum games with skew-symmetric payoff matrices of even order are never completely mixed and the coexistence of an even number of species is not p ossible[42]. Finding the stable coexistence equilibrium of (1) is equivalent to finding the optimal strategy NE of the corresponding completely mixed matrix game.
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