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 the dynamics of a population in a time-periodic environment, which is assumed that temporally-average shifts from a favorable zone to an unfavorable zone. First, by employing the technique in [9], we prove the existence of periodic forced wave as c > c * in the spirit of Lin [17], employing Schauder’s 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 [7, 9, 17].