In this paper we consider the Cauchy problem for the parabolic system arising in biology. By the method of the analytic semigroup developed in Osaki and Yagi, Yagi we show existence, uniqueness and non-negativity of global solutions.
"Global existence of the Cauchy problem for a chemotactic system with predator-prey dynamics." Hiroshima Math. J. 36 (1) 77 - 111, March 2006. https://doi.org/10.32917/hmj/1147883398