Propagation in a time-periodic environment governed by a reaction–diffusion model of KPP type with a new free boundary condition

In this paper, we investigate the spreading behavior of an invasive species modeled by a KPP type reaction–diffusion equation, in time-periodic medium of one space dimension, with a new free boundary condition that arises from a “preferred population density” assumption at the range boundary. For the model in homogeneous medium, Du [Propagation dynamics of the monostable reaction-diffusion equation with a new free boundary condition, Discrete Contin. Dyn. Syst.44 (2024) 2524–2563] obtained a fairly complete description of the long-time dynamics and showed that the concerned species always spreads successfully, which contrasts sharply to the corresponding free boundary models in Du and Lin [Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal.42 (2010) 377–405] and other related works, where vanishing of the species is possible. In this paper, we are interested in the effect of time-periodic perturbation of the environment on the propagation dynamics of such a new free boundary model. Our results show that the main features of the model are retained under the perturbation, namely the species always spreads successfully and there is an asymptotic spreading speed. Furthermore, as the preferred population density at the range boundary approaches zero, we show that the solution of our free boundary problem converges to the solution of the corresponding Cauchy problem.