ABSTRACT To investigate the joint effects of individual movement and prey fear response on predator–prey interaction dynamics, we develop a mathematical model that incorporates age structure and delayed nonlocal interactions. We establish a comparison principle for a linear parabolic operator with delayed nonlocal interaction and apply it to prove the well‐posedness of the system. By introducing the basic reproduction number as a threshold index, we fully classify the system dynamics and demonstrate the existence and uniqueness of a positive constant steady state. We further analyze the stability of this equilibrium, showing that neither Hopf bifurcation nor Turing instability can occur, and excluding the possibility of spatially heterogeneous steady states bifurcating from this equilibrium. These results indicate that the linear nonlocal delay structure does not destabilize the positive constant steady state or generate periodic dynamics. In the absence of advection and with symmetric kernel functions, we construct a Lyapunov functional to establish the global asymptotic stability of the positive constant steady state, and further exclude heterogeneous steady states for specific kernel forms. Ecologically, our findings reveal that the fear effect suppresses both predator and prey densities, while maturation delay increases prey density but inhibits predator growth. Moreover, asymmetric kernel functions can readily generate spatially heterogeneous steady states, highlighting the critical role of kernel structure in spatial pattern formation.
Xin et al. (Wed,) studied this question.