Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations (Q499540)

The general problem, to which this work is devoted, belongs to the important mathematical question the representation of spreading phenomena governed by non-local interactions via local partial differential equations, which are much more simple for solving and analyzing. In fact, such a consideration dates back to the works by D. G. Kendall, who first has considered PDEs originated from ODE with non-local (spatially averaged) terms and their interplay with the Fisher-Kolmogorov-Petrovskii-Piskunov equation (FKPP), see, e.g., [\textit{D. G. Kendall}, Mathematics and computer science in biology and medicine. London: H. M. Stationary Off. 213--225 (1965)]. The authors of the present work attack this kind of problems (including FKPP and its counterparts) from a more mathematically rigorous point of view. Three cases are considered: an initial-boundary value problem, the spectral approach to a boundary problem, and an asymptotic correspondence. Using the concept of sub- and super-solutions, which bound the exact one from below and above, the convergence to the desired solutions of non-local problems is proven.