<p>We investigate the rich dynamics of the discrete Moran-Ricker model with delay, highlighting various bifurcation structures. One-parameter bifurcation diagrams reveal discontinuity-like periodic transitions arising from the coexistence of periodic attractors and saddle-node bifurcations. Additionally, we examine a two-sided crisis, where periodic orbits of different periods exist on either side. We also demonstrate the coexistence of nested mode-locked periodic orbits with distinct periods. Furthermore, two-parameter bifurcation diagrams in the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1713_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha -\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>-</mo> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation> plane reveal web-like structures across a broad parameter space. A distinct pattern emerges in determining the periodicity of primary and secondary shrimps at the center of parallelogram-like formations. Lastly, we identify period-adding structures, which manifest in two cases: (a) with increasing <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1713_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> parameter and (b) at higher <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1713_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> values, where striped regions exhibit these structures.</p>

Discontinuous-like periodic transitions and coexistence of mode-locked orbits in delayed Moran-Ricker map

期刊论文
发表日期：

2025年5月10日

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S. S. Muni

摘要

We investigate the rich dynamics of the discrete Moran-Ricker model with delay, highlighting various bifurcation structures. One-parameter bifurcation diagrams reveal discontinuity-like periodic transitions arising from the coexistence of periodic attractors and saddle-node bifurcations. Additionally, we examine a two-sided crisis, where periodic orbits of different periods exist on either side. We also demonstrate the coexistence of nested mode-locked periodic orbits with distinct periods. Furthermore, two-parameter bifurcation diagrams in the $\alpha -\sigma $ $α - σ$ plane reveal web-like structures across a broad parameter space. A distinct pattern emerges in determining the periodicity of primary and secondary shrimps at the center of parallelogram-like formations. Lastly, we identify period-adding structures, which manifest in two cases: (a) with increasing $\alpha $ $α$ parameter and (b) at higher $\sigma $ $σ$ values, where striped regions exhibit these structures.