Strong maximum principle for nonlocal diffusion in measure spaces: new characterization and population dynamics applications

We study the existence of nontrivial solutions to nonlocal semilinear diffusion equations of the form $$a_0(x)u(x)-{\int }_{\mathrm{\Omega }}k(x,y)(u(y)-u(x))d\mu (y)=F(x,u),u\ge 0,$$ on a general measure space $(\mathrm{\Omega },M,\mu )$, where the kernel $k$ is nonnegative and symmetric. In this framework, a major challenge is establishing the strict positivity of solutions. To this end, we provide a novel complete characterization of the strong maximum principle in terms of a connectivity structure induced by the kernel, which leads to a decomposition of the domain into independent components. We also develop the method of sub- and supersolutions in this setting, obtaining existence, uniqueness, and nonexistence results, as well as a regularity result that ensures the continuity of solutions under suitable assumptions. The main interest of working within an abstract measure space lies in its rich applicability, as it unifies and extends a wide variety of frameworks, including discrete, continuous, and hybrid models, or systems interacting across different dimensions. Finally, numerical experiments are presented to illustrate the potential of our theoretical results.