ABSTRACT The goal of this work is to explore a two‐species predator‐prey chemotaxis system with two chemicals and logistic sources on where , and are positive constants. For the fully parabolic case with , we first prove the global boundedness of the classical solution, under the assumption that , , are sufficiently small. Subsequently, we also study the asymptotic behavior of solutions to the above system and show that: when and are sufficiently small, the global solution of this system exponentially converges to as ; when , are sufficiently small and is sufficiently large, the global solution of this system exponentially converges to as . For the parabolic‐parabolic‐elliptic‐elliptic case with , we also obtain the global boundedness of classical solution, under the assumption that , are sufficiently small. Moreover, when and are sufficiently small, the global solution of this system exponentially converges to as ; when , , are sufficiently small and is sufficiently large, the global solution of this system exponentially converges to as .
Su et al. (Tue,) studied this question.