Propagation dynamics of cline and gap states for spatially-periodic Lotka–Volterra competition systems in shifting media

Abstract We study the propagation dynamics of a Lotka–Volterra competition system in which one growth rate behaves like a monotonically decreasing wave profile that shifts with a given speed and is also periodic in the first spatial variable, while the other growth rate behaves similarly, except that its profile is monotonically increasing with respect to the shifting variable. Furthermore, both growth functions are assumed to be sign-changed, which implies that the environments in which the species live switch spatially from ‘good’ regions (suitable for survival) to ‘bad’ regions (not suitable for survival) and vice versa. We reveal that the model admits a forced pulsating wave only when the forced speed lies within a finite interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> that contains zero. Biologically, this corresponds to the formation of a shifting cline. Moreover, we find that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> can be calculated in terms of the Fisher-KPP speeds related to the linearized equations of each species. By applying a sliding technique, we show that the forced pulsating wave is unique. We also prove that the forced pulsating wave is Lyapunov-stable. Finally, the spreading dynamics of spatial gap formation in the two species are also investigated when the forced wave speed is either less than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> </mml:mrow> </mml:math> or greater that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> . We employ a novel approach to demonstrate how the species invade in response to a shifting environment.