Semi-waves for delayed fisher-KPP equations without quasimonotonicity

Abstract Traveling waves for Fisher-KPP equations with/without time delay have been studied widely in existing literature. These waves are defined in the whole space and play an important role in population dynamics. In this paper, we study the existence of traveling waves defined only on a half-space (called semi-waves) for the delayed Fisher-KPP equation (or diffusive Hutchinson equation) without quasimonotonicity, which may be used to describe the spread of species in a hostile environment. We show that semi-wave solutions exist as long as its wave speed c is less than the minimal speed $$c^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> for the Fisher-KPP equation and the delay is not too large; while there are no semi-wave solutions with the speed $$c \ge c^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>≥</mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> for any delay. The appearance of monotone and non-monotone waves and the effect of the delay on the wave profile is also discussed numerically.