<p>We explore the dynamical study of a three-dimensional food chain with diverse food supply predator species using a model system in this article. We aim to emulate the dynamic behavior of the model system using alternative food sources for temporal and spatial systems. Dynamical techniques such as bifurcation, permanence, persistence, boundedness, and stability are applied. Hopf bifurcation is also demonstrated about parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1905_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega _{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1905_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega _{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is the growth rate of intermediate predator. Using normal form theory, we analyze the Hopf bifurcation direction. The effect of the normalizing coefficient on the environment <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1905_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> is also discussed. In a spatial system, we investigated Turing instability conditions and patterns, focusing on the effect of diffusion variation. In addition, we obtained the time evaluation pattern generation of the spatial system. Predator population density is also a major factor in the formation of Turing patterns. The results of the analytical analysis are validated by numerical simulation, which also shows that the dynamics of the system stabilize as the growth rate of the intermediate predator increases.</p>

Modeling Spatiotemporal Predator–Prey Interactions in the Presence of Alternative Food Sources for Predators

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发表日期：

2025年5月2日

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Surabhi Pareek,
Randhir Singh Baghel

摘要

We explore the dynamical study of a three-dimensional food chain with diverse food supply predator species using a model system in this article. We aim to emulate the dynamic behavior of the model system using alternative food sources for temporal and spatial systems. Dynamical techniques such as bifurcation, permanence, persistence, boundedness, and stability are applied. Hopf bifurcation is also demonstrated about parameter $\omega _{1}$ $ω_{1}$ , the $\omega _{1}$ $ω_{1}$ is the growth rate of intermediate predator. Using normal form theory, we analyze the Hopf bifurcation direction. The effect of the normalizing coefficient on the environment $\gamma $ $γ$ is also discussed. In a spatial system, we investigated Turing instability conditions and patterns, focusing on the effect of diffusion variation. In addition, we obtained the time evaluation pattern generation of the spatial system. Predator population density is also a major factor in the formation of Turing patterns. The results of the analytical analysis are validated by numerical simulation, which also shows that the dynamics of the system stabilize as the growth rate of the intermediate predator increases.