  Many populations of animals or plants, exhibit a metapopulation structure
with close, spatially-separated subpopulations. The field of metapopulation
theory has made significant advancements since the influential Levins model.
Various modeling approaches have provided valuable insights to theoretical
Ecology. Despite extensive research on metapopulation models, there are still
challenging questions that are difficult to answer from ecological
metapopulational data or multi-patch models. Low-dimension mathematical models
offer a promising avenue to address these questions, especially for global
dynamics which have been scarcely investigated. In this study, we investigate a
two-patch metapopulation model with logistic growth and diffusion between
patches. By using analytical and numerical methods, we thoroughly analyze the
impact of diffusion on the dynamics of the metapopulation. We identify the
equilibrium points and assess their local and global stability. Furthermore, we
analytically derive the optimal diffusion rate that leads to the highest
metapopulation values. Our findings demonstrate that increased diffusion plays
a crucial role in the preservation of both subpopulations and the full
metapopulation, especially under the presence of stochastic perturbations.
Specifically, at low diffusion values, the origin is a repeller, causing orbits
starting around it to travel closely parallel to the axes. This configuration
makes the metapopulation less resilient and thus more susceptible to local and
global extinctions. However, as diffusion increases, the repeller transitions
to a saddle point, and orbits starting near the origin rapidly converge to the
unstable manifold of the saddle. This phenomenon reduces the likelihood of
stochastic extinctions and the metapopulation becomes more resilient due to
these changes in the vector field of the phase space.
