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<p>In this paper, we analyze the stability of two population models of a species, consisting of an equation and a system. The first model represents the growth of a species that depends on its previous and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1848_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\((k+1){th}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mi mathvariant="italic">th</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> previous generations without the influence of other species. The second model, a system, represents the growth of a species, which depends on its <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1848_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\((k+1){th}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mi mathvariant="italic">th</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> previous generation and another species’ previous generation in which the influencing species mutually depend on its <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1848_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\((k+1){th}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mi mathvariant="italic">th</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> previous generation and the previous generation of the aforesaid species. Moreover, the theorems are verified numerically by animating the plots and bifurcation diagrams to illustrate the boundedness and stability results.</p>

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On the Discrete Population Models Without and With the Influence of \((k+1)^{th}\) Previous Generation of Another Species

  • 期刊论文

  • 发表日期：

    2025年3月8日

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  • Sibi C. Babu,
  • D. S. Dilip

摘要

In this paper, we analyze the stability of two population models of a species, consisting of an equation and a system. The first model represents the growth of a species that depends on its previous and \((k+1){th}\) ( k + 1 ) th previous generations without the influence of other species. The second model, a system, represents the growth of a species, which depends on its \((k+1){th}\) ( k + 1 ) th previous generation and another species’ previous generation in which the influencing species mutually depend on its \((k+1){th}\) ( k + 1 ) th previous generation and the previous generation of the aforesaid species. Moreover, the theorems are verified numerically by animating the plots and bifurcation diagrams to illustrate the boundedness and stability results.