The spreading of global solutions of chemotaxis systems with logistic source and consumption on R^N

This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \{ aligned (ii) the chemical substance does not induce infinitely fast spreading of u; (iii) the spreading speed remains unaffected under conditions that v (0, ) decays spatially or 0<- 1 and =1. Additionally, our numerical simulations reveal a noteworthy phase transition in: for v (0, ) uniformly distributed across space, the spreading speed accelerates only when surpasses a critical positive value.