Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting

In this article, we consider the influence of seasonal succession and impulsive harvesting on the dynamical behavior of solutions to a free boundary model. First, the generalized principal eigenvalue is defined and its properties are studied. Next, some sufficient conditions for spreading and for vanishing are given. Then, by introducing a one-parameter family of initial data <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>σ</m:mi> <m:mi>ϕ</m:mi> </m:math> σ φ with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>σ</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:math> σ ≥ 0 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> </m:math> φ being a compactly supported function, we obtain a threshold value <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>σ</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:math> {σ }* such that spreading happens when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>σ</m:mi> <m:mo>></m:mo> <m:msup> <m:mrow> <m:mi>σ</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:math> σ > {σ }* , vanishing happens when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>σ</m:mi> <m:mo>≤</m:mo> <m:msup> <m:mrow> <m:mi>σ</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:math> σ ≤ {σ }* . Finally, we prove the existence and uniqueness of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> </m:math> T -periodic traveling semi-wave. Moreover, when spreading persists, we use the proved <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> </m:math> T -periodic traveling semi-wave to estimate the asymptotic speed of the free boundaries.