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In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for d ≥ 1 , there exist C 0 , C 1 > 0 , depending on ‖ m ‖ L 1 only, such that C 0 d ≤ supremum of total population ≤ C 1 d. However, when replaced by random diffusion, a conjecture, proposed by Wei-Ming Ni and justified in Bai et al. (Proc. Am. Math. Soc. 144:2161–2170, 2016), indicates that in the one-dimensional case, supremum of total population = 3 ‖ m ‖ L 1. This reflects serious discrepancies between models with local and nonlocal dispersal strategies. Furthermore, we provide an equivalent characterization about the combination of resource distribution and diffusion rate such that the corresponding total population could reach the optimal order d as d goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the results of Evans and Tabrizian (SIAM J Math Anal 48:2944–2961, 2016), by deriving asymptotics for the time-rescaled Kramers–Smoluchowski equations, in the case of a general non-symmetric potential function with multiple wells. The asymptotic limit is described by a system of reaction–diffusion equations whose coefficients are determined by the Kramers constants at the saddle points of the potential function and the Hessians of the potential function at global minima. [ABSTRACT FROM AUTHOR]