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<p>In this paper, a reaction-diffusion model is proposed to investigate the growing process of invasive plants. Two free boundaries are introduced to describe the spreading frontiers are caused only by the seeds. The main objective is to realize the variation of the invasive plants and the free boundaries. Similar to the basic reproduction number of epidemic diseases, we define the threshold parameters <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_2695_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(J_0\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_2695_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(J_0^F(t)\)</EquationSource> </InlineEquation> to discuss the dynamic behaviors of equilibrium solutions of invasive plants model with free boundary conditions. Based on them, we give the sufficient conditions for whether plants invade successfully or not. We show that the invasive plants will successfully persist in the new environment if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_2695_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\(J^F_0(0)&lt;1&lt;J_0\)</EquationSource> </InlineEquation> with large initial values or <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_2695_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">\(J_0^F(t_0)\ge 1 ~(\exists t_0\ge 0)\)</EquationSource> </InlineEquation>, while failing and dying out in the long run if <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_2695_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(J^F_0(0)&lt;1\)</EquationSource> </InlineEquation> with small initial values or <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_2695_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(J_0\le 1\)</EquationSource> </InlineEquation>.</p>

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The conditions of failing and successful plant invasions with free boundary

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月23日

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  • Rong Li,
  • Linling Zhu,
  • You Zhou

摘要

In this paper, a reaction-diffusion model is proposed to investigate the growing process of invasive plants. Two free boundaries are introduced to describe the spreading frontiers are caused only by the seeds. The main objective is to realize the variation of the invasive plants and the free boundaries. Similar to the basic reproduction number of epidemic diseases, we define the threshold parameters \(J_0\) and \(J_0^F(t)\) to discuss the dynamic behaviors of equilibrium solutions of invasive plants model with free boundary conditions. Based on them, we give the sufficient conditions for whether plants invade successfully or not. We show that the invasive plants will successfully persist in the new environment if \(J^F_0(0)<1<J_0\) with large initial values or \(J_0^F(t_0)\ge 1 ~(\exists t_0\ge 0)\) , while failing and dying out in the long run if \(J^F_0(0)<1\) with small initial values or \(J_0\le 1\) .