Asymptotic Behavior of Solutions to Delayed Nonlinear Integral Equations Influenced by Growth Rates and Time Delays

This study examines partial differential equation models with delay on both unbounded and bounded domains. The mathematical model is reformulated as a system of nonlinear integral equations. The primary objective is to investigate the long-time behavior of solutions to nonlinear integral equations and to compare these findings with those from the partial differential equation model under varying growth rates. The theory of asymptotic spreading speed is employed to achieve this objective. The existence and uniqueness of solutions to the nonlinear integral equations are demonstrated. Minimal wave speeds are calculated for the model with bounded and unbounded domains, considering different growth rates and time delays. Numerical experiments are conducted to validate the theoretical results.