Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary
The paper analyzes the solution of a certain boundary-value problem for an elliptic partial differential equation. The work studies the following three cases: the trivial solution, the semi-trivial solutions, and the solutions with both components positive, i.e. the coexistence states. The trivial solution always exists. The paper derives the necessary and sufficient conditions for existence of the semi-trivial solutions. The sufficient conditions for the stability and instability of these solutions are also obtained. The case ``no chemotaxis'' and the properties of the coexistence state are studied in separate sections, Section 5 and 6, respectively. Section 7 proposes an interpretation of the main results for certain application. The paper can be recommended to the readers who are interested in qualitative theory of elliptic problems. The work also claims that the models under consideration are associated with angiogenesis in tumor development and chemotaxis. However, no informal picture in terms of specific biological phenomena is included. The role of the so-called endothelial cells (ECs) is solely outlined by formal mathematical representations. The abbreviation TAF remains to be unexplained to the readers-mathematicians. The reading of chemotaxis in terms of the related physical forces is also omitted. Reviewer's remark: On the whole, the convincing connection of the obtained, purely mathematical results to the corresponding biological processes in tumor development is unclear to the reviewer. Consequently, the paper is hardly expected to be of interest to biologists or oncologists who are interested in biological mathematics.
scientific article; zbMATH DE number 5285689
scientific article