  This paper concerns with the global dynamics of classical solutions to an
important alarm-taxis ecosystem, which demonstrates the behaviors of prey that
attract secondary predator when threatened by primary predator. And the
secondary predator pursues the signal generated by the interaction of the prey
and primary predator. However, it seems that the necessary gradient estimates
for global existence cannot be obtained in critical case due to strong coupled
structure. Thereby, we develop a new approach to estimate the gradient of prey
and primary predator which takes advantage of slightly higher damping power.
Then the boundedness of classical solutions in two dimension with Neumann
boundary conditions can be established by energy estimates and semigroup
theory. Moreover, by constructing Lyapunov functional, it is proved that the
coexistence homogeneous steady states is asymptotically stability and the
convergence rate is exponential under certain assumptions on the system
coefficients.
