We investigate the long-time behavior of phenotype-structured models describing evolutionary dynamics of asexual populations, and analyze the joint effects of nonlocal interactions and spatial resource distributions on the global dynamics of the two species. In the first part, we consider an integro-differential system without diffusion terms, where phenotypic changes are absent and the spatial distribution of resources for one species is heterogeneous while that of the other is homogeneous. Using an entropy method to address nonlocal interactions and resource heterogeneity, we prove that the species subject to heterogeneous resources converges to a Dirac mass concentrated at the peak of the phenotypic fitness landscape, which establishes the selection of the best adapted trait. Numerical experiments further provide a sufficient criterion to identify the positions of fitness peaks. In the second part, we extend our study to a nonlocal reaction-diffusion system involving a linear diffusion operator, where heritable phenotypic changes occur and both resources are spatially heterogeneous. Through an appropriate transformation, we overcome the challenges induced by resource heterogeneity and prove that the long-time limits of the two species under different interspecific competitive coefficients are given by distinct steady states of the parabolic system, with concentrations at the maxima of their respective resource functions. Numerical results confirm the predictions and further reveal phenomena beyond the theoretical analysis.
Shen Bian (Sun,) studied this question.