Generalized transition waves in nonlocal dispersal cooperative systems with temporal heterogeneity

This paper investigates the existence, nonexistence, and stability of generalized transition waves in nonlocal dispersal cooperative systems exhibiting temporal heterogeneity in both the kernel functions and reaction terms. Utilizing the theory of the principal Lyapunov exponent and Floquet bundle for the corresponding linearized system, we construct appropriate lower and upper solutions for these cooperative systems. Based on establishing suitable regularity estimates for spatially monotone solutions of the corresponding Cauchy problem, we then prove the existence of generalized transition waves for all sufficiently large wave speed functions in a certain averaged sense. Conversely, such solutions do not exist for the wave speed function with small average. Furthermore, when the initial data is uniformly bounded with respect to a weighted maximum norm, we demonstrate that all solutions converge exponentially over time to the generalized transition wave. Finally, we apply these results to two biological models: (ⅰ) a man-environment-man epidemic model, and (ⅱ) an age-structured juvenile-adult population model.