<p>Prey vigilance plays a significant role in ecological modeling, as it alters the dynamics of ecosystems by making it more difficult for predators to locate their prey. This study proposes a predator-prey system with a Holling type-II functional response, incorporating prey vigilance behavior to avoid predation, as well as hunting cooperation among predators. The dynamics of the system are analyzed through the examination of the existence and stability of equilibrium points, bifurcation analysis, and the impact of time delays, including maturation delay, gestation delay, and incubation delay. The study investigates the effects of hunting cooperation and prey vigilance on the model. The results show that while hunting cooperation can destabilize the system, prey vigilance has a stabilizing effect. The novelty of this work lies in its exploration of the combined impact of multiple time delays, prey vigilance, and hunting cooperation. The analysis begins by studying the effects of each time delay individually. Maturation delay, denoted as <InlineEquation ID="IEq1585984"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2500_Article_IEq1585984.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau_1\)</EquationSource> </InlineEquation>, exhibits switching behavior, causing the system to alternate between stable and unstable states. In the absence of hunting cooperation, the time delay <InlineEquation ID="IEq1580085005984"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2500_Article_IEq1585984.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau_1\)</EquationSource> </InlineEquation> destabilizes the model through Hopf bifurcation, leading to chaotic behavior. But in the presence of both hunting behavior and prey-vigilance, the model shows the switching behavior through Hopf bifurcation about the time delay parameters <InlineEquation ID="IEq1580105984"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2500_Article_IEq1585984.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau_1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq15118005984"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2500_Article_IEq15118005984.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau_2\)</EquationSource> </InlineEquation>. Also obtained the periodic doubling and chaos about the incubation delay <InlineEquation ID="IEq158005558984"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2500_Article_IEq158005558984.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau_3\)</EquationSource> </InlineEquation> for the model.</p>

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The role of time delay in shaping the dynamics of hunting cooperation and prey vigilance in a predator-prey eco-epidemiology system

  • Bipin Kumar,
  • Rajesh Kumar Sinha

摘要

Prey vigilance plays a significant role in ecological modeling, as it alters the dynamics of ecosystems by making it more difficult for predators to locate their prey. This study proposes a predator-prey system with a Holling type-II functional response, incorporating prey vigilance behavior to avoid predation, as well as hunting cooperation among predators. The dynamics of the system are analyzed through the examination of the existence and stability of equilibrium points, bifurcation analysis, and the impact of time delays, including maturation delay, gestation delay, and incubation delay. The study investigates the effects of hunting cooperation and prey vigilance on the model. The results show that while hunting cooperation can destabilize the system, prey vigilance has a stabilizing effect. The novelty of this work lies in its exploration of the combined impact of multiple time delays, prey vigilance, and hunting cooperation. The analysis begins by studying the effects of each time delay individually. Maturation delay, denoted as \(\tau_1\) , exhibits switching behavior, causing the system to alternate between stable and unstable states. In the absence of hunting cooperation, the time delay \(\tau_1\) destabilizes the model through Hopf bifurcation, leading to chaotic behavior. But in the presence of both hunting behavior and prey-vigilance, the model shows the switching behavior through Hopf bifurcation about the time delay parameters \(\tau_1\) and \(\tau_2\) . Also obtained the periodic doubling and chaos about the incubation delay \(\tau_3\) for the model.