On the influence of spatial network in multi-patch systems with species' diffusion and selection

Demonstrating precise influence of spatial topology on population dynamics of meta-community is a central challenge in ecology. Recent experiments to solve the challenge reported that different topologies result in significantly different population abundance of species, where the species is manually made to move between adjacent patches in the experiments. In this paper, we study an extended situation where the species can move between all patches on patch-networks. Using graph-theoretic method and dynamical systems theory, we show global equilibrium stability in the system. Further analysis on the equilibrium leads to three formulas, which quantitatively characterize the influence of spatial (network) topology: (i) When the diffusive rate is small, influence of the network topology is a weighted sum of the influence of diffusion on each path in isolation, where the weight is just the diffusive asymmetry on the path. (ii) When the diffusive rate is large, the influence is characterized by the vector of total asymmetry to each patch, where the diffusion to sink patches could increase the total population abundance. (iii) When there is no diffusion, the influence is characterized by resource-distribution on network, where the best and worst distributions are given. Moreover, species' monotone selection on diffusive asymmetry increases the total abundance and could make the network evolve to the best topology. Our results are important for constructing efficient and concise networks in conservation and management of species.