<p>We investigate the positive steady states of an age-structured population model with nonlinear density-dependent diffusion, density-dependent fertility, and direct intra-age competition. Treating the mortality intensity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> as the bifurcation parameter, we study the emergence of positive equilibria from the trivial branch. The linearized problem yields a compact strongly positive next-generation operator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(Q_\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation>, and the critical threshold <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\lambda _0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is characterized by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(r(Q_{\lambda _0})=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <msub> <mi>λ</mi> <mn>0</mn> </msub> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Using the mortality-driven local bifurcation framework, we obtain a local branch of positive steady states bifurcating from <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((\lambda _0,0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. A second-order expansion gives an explicit computable quantity <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\Theta _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Θ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, whose sign determines both the bifurcation direction and the local asymptotic stability of the bifurcating positive equilibria. A Rabinowitz-type global continuation argument yields a global alternative for the real-parameter continuum, while the biologically relevant positive component cannot return to the nonnegative trivial branch except at <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\((\lambda _0,0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>0</mn> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Finally, concrete examples are used to compute <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\Theta _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Θ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\lambda '(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>λ</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and to visualize the corresponding bifurcation profiles.</p>

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Bifurcation of positive steady states in an age-structured population model with nonlinear diffusion

  • Mengmeng Hou,
  • Shangjiang Guo

摘要

We investigate the positive steady states of an age-structured population model with nonlinear density-dependent diffusion, density-dependent fertility, and direct intra-age competition. Treating the mortality intensity \(\lambda \) λ as the bifurcation parameter, we study the emergence of positive equilibria from the trivial branch. The linearized problem yields a compact strongly positive next-generation operator \(Q_\lambda \) Q λ , and the critical threshold \(\lambda _0\) λ 0 is characterized by \(r(Q_{\lambda _0})=1\) r ( Q λ 0 ) = 1 . Using the mortality-driven local bifurcation framework, we obtain a local branch of positive steady states bifurcating from \((\lambda _0,0)\) ( λ 0 , 0 ) . A second-order expansion gives an explicit computable quantity \(\Theta _2\) Θ 2 , whose sign determines both the bifurcation direction and the local asymptotic stability of the bifurcating positive equilibria. A Rabinowitz-type global continuation argument yields a global alternative for the real-parameter continuum, while the biologically relevant positive component cannot return to the nonnegative trivial branch except at \((\lambda _0,0)\) ( λ 0 , 0 ) . Finally, concrete examples are used to compute \(\Theta _2\) Θ 2 and \(\lambda '(0)\) λ ( 0 ) and to visualize the corresponding bifurcation profiles.