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<p>In ecosystems, fear response and gestation delays in predator-prey systems play a vital role in shaping population dynamics. Fear response delays affect prey reproduction due to the stress caused by the presence of predators. Gestation delays in predators create a time lag in population growth. It is crucial for predicting the impacts of environmental changes, such as habitat loss or prey depletion, on predator populations. The model considers a fearful prey and a predator, incorporating a fear response delay and the predator’s gestation period, with interaction between them following Holling Type-II functional response. From a mathematical perspective, the inclusion of both delays enhances the dynamical complexity of the system, giving rise to stability changes, Hopf bifurcations, and chaotic dynamics. The analysis highlights that the interplay between the two delays is crucial in triggering the onset of complex and irregular oscillations in predator-prey interactions. We analysed the positivity and boundedness of the system. Stability analysis has been conducted for different pairs of delays and is illustrated graphically. By using center manifold theory and bifurcation techniques we show the transitions between stability and instability. To analyze the dynamics near the critical delay value, we apply center manifold theory, which reduces the infinite-dimensional delayed system to a finite-dimensional one. This reduction enables the derivation of the normal form and helps determine the direction and stability of the Hopf bifurcation. As key system parameters such as time delays continue to change, these periodic oscillations may become unstable, eventually leading to chaotic dynamics. This transition indicates that predator-prey interactions may exhibit strong sensitivity to delay values and fear resulting in irregular and unpredictable fluctuations in population sizes. Our simulations also exhibit similar dynamical behavior. Additionally, to gain a clearer understanding of the influence of these parameters, sensitivity analysis has been conducted, supported by numerical verification.</p>

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Impact of fear and anti-predator behavior on stability and chaos in multi-delayed predator-prey system

  • 期刊论文

  • 发表日期：

    2025年7月29日

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  • K. Parida,
  • B. P. Sarangi,
  • S. Chand

摘要

In ecosystems, fear response and gestation delays in predator-prey systems play a vital role in shaping population dynamics. Fear response delays affect prey reproduction due to the stress caused by the presence of predators. Gestation delays in predators create a time lag in population growth. It is crucial for predicting the impacts of environmental changes, such as habitat loss or prey depletion, on predator populations. The model considers a fearful prey and a predator, incorporating a fear response delay and the predator’s gestation period, with interaction between them following Holling Type-II functional response. From a mathematical perspective, the inclusion of both delays enhances the dynamical complexity of the system, giving rise to stability changes, Hopf bifurcations, and chaotic dynamics. The analysis highlights that the interplay between the two delays is crucial in triggering the onset of complex and irregular oscillations in predator-prey interactions. We analysed the positivity and boundedness of the system. Stability analysis has been conducted for different pairs of delays and is illustrated graphically. By using center manifold theory and bifurcation techniques we show the transitions between stability and instability. To analyze the dynamics near the critical delay value, we apply center manifold theory, which reduces the infinite-dimensional delayed system to a finite-dimensional one. This reduction enables the derivation of the normal form and helps determine the direction and stability of the Hopf bifurcation. As key system parameters such as time delays continue to change, these periodic oscillations may become unstable, eventually leading to chaotic dynamics. This transition indicates that predator-prey interactions may exhibit strong sensitivity to delay values and fear resulting in irregular and unpredictable fluctuations in population sizes. Our simulations also exhibit similar dynamical behavior. Additionally, to gain a clearer understanding of the influence of these parameters, sensitivity analysis has been conducted, supported by numerical verification.