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In this paper, we are concerned with a model of the indirect transmission of an epidemic disease between two spatially distributed host populations having non-coincident spatial domains with nonlocal and cross-diffusion, the epidemic disease transmission occurring through a contaminated environment. The mobility of each class is assumed to be influenced by the gradient of the other classes. We address the questions of existence of weak solutions and existence and uniqueness of classical solution by using, respectively, a regularization method and an interpolation result between Banach spaces. Moreover, we propose a finite volume scheme and proved the well-posedness, nonnegativity and convergence of the discrete solution. The convergence proof is based on deriving a series of a priori estimates and by using a general L p compactness criterion. Finally, the numerical scheme is illustrated by some examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with a system of nonlinear partial differential equations modeling a predator–prey system with cross-diffusion in heterogeneous habitats. Predators are assumed to feed on preys with a Holling type II functional response to prey density and preys are assumed to follow a logistic growth in the absence of predation. The mobility of each classes is assumed to be influenced by the gradient of other classes. The existence result is proved by means of an approximation system, the Faedo–Galerkin method, and the compactness method. The global existence of classical solutions is proved under certain restrictions on the coefficients. [ABSTRACT FROM AUTHOR]

Abstract: We analyze a finite volume scheme for the bidomain model, which is a degenerate reaction–diffusion system modeling electrophysiological waves in cardiac tissue. We establish existence and uniqueness of solutions to the finite volume scheme, and show that it converges to a weak solution of the bidomain model. The convergence proof is based on deriving series of a priori estimates and using a general compactness criterion. [Copyright &y& Elsevier]