On the existence of forced extinction wave and spreading speed for delayed population in time-periodic deteriorated environment

In this communication, we are concerned with the propagation phenomena of a delayed equation without quasimonotonicity modeling of the dynamics of a population in a time-periodic environment, in which is assumed that temporally-average shifts from a favorable zone to an unfavorable zone. First, by employing the technique in [J. S. Guo, A. A. L. Poh and C. C. Wu, Forced waves of saturation type for Fisher-KPP equation in a shifting environment, Appl. Math. Lett. 140 (2023) 108573], we prove the existence of periodic forced wave as [Formula: see text] in the spirit of [G. Lin, Spreading speeds and traveling wave solutions for a delayed periodic equation without quasimonotonicity, J. Dyn. Differ. Equations 31(4) (2019) 2275–2292], employing Schauders fixed point theorem and regularity of analytic semigroups by tactfully constructing a pair of generalized super- and sub-solutions without assuming monotonicity on the initial growth rate of population. Our result confirms that the delayed death rate in our model does not prevent the occurrence of forced extinction waves in deteriorated environment. Furthermore, by providing suitable auxiliary equations, we derive the spreading speed for this model. Our work is a counterpart of the interesting results obtained in [J. Fang, R. Peng and X. Q. Zhao, Propagation dynamics of a reaction–diffusion equation in a time-periodic shifting environment, J. Math. Pures Appl. 147 (2021) 1–28; J. S. Guo, A. A. L. Poh and C. C. Wu, Forced waves of saturation type for Fisher-KPP equation in a shifting environment, Appl. Math. Lett. 140 (2023) 108573; G. Lin, Spreading speeds and traveling wave solutions for a delayed periodic equation without quasimonotonicity, J. Dyn. Differ. Equations 31(4) (2019) 2275–2292].