<p>This research proposes a novel predator-prey model that simultaneously incorporates fear of predation among prey, Allee effect in predator and Holling type II functional response, with a particular focus on the role of memory in ecological interactions. By incorporating Caputo fractional derivatives, the model captures identical as well as rarely studied distinct memory intensities for prey and predator populations, thereby providing a more realistic description of ecosystems in which current dynamics depend on past experiences. Fundamental analytical properties of the model, including existence, uniqueness, positivity and boundedness of solutions are rigorously established to ensure ecological feasibility. The existence of biologically meaningful equilibria, their local stability and system persistence are investigated. Hopf bifurcation is theoretically analyzed with respect to both the fractional order and the fear parameter. Numerical simulations demonstrate that increased memory and fear generally enhance system stability, low and high level of Allee in predator stabilizes the system whereas moderate level of Allee results oscillation. The model further reveals bi-stability and the “Paradox of enrichment". A key finding is that species memory can significantly modulate both prey’s fear intensity and predator Allee strength, potentially destabilizing the system even when these effects are strong. Moreover, stability depends on different memory of both species, where stronger prey memory has greater impact than predator memory in stable coexistence. These results highlight previously unexplored interactions between memory-driven fear and Allee mechanisms and offer new insights into the complex dynamics of predator-prey systems.</p>

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A Fractional-Order Commensurate and Incommensurate Study on Predator-Prey System with Allee in Predator and Fear Effect

  • Arnabi Saha,
  • Debjit Pal,
  • Dipak Kesh,
  • Debasis Mukherjee

摘要

This research proposes a novel predator-prey model that simultaneously incorporates fear of predation among prey, Allee effect in predator and Holling type II functional response, with a particular focus on the role of memory in ecological interactions. By incorporating Caputo fractional derivatives, the model captures identical as well as rarely studied distinct memory intensities for prey and predator populations, thereby providing a more realistic description of ecosystems in which current dynamics depend on past experiences. Fundamental analytical properties of the model, including existence, uniqueness, positivity and boundedness of solutions are rigorously established to ensure ecological feasibility. The existence of biologically meaningful equilibria, their local stability and system persistence are investigated. Hopf bifurcation is theoretically analyzed with respect to both the fractional order and the fear parameter. Numerical simulations demonstrate that increased memory and fear generally enhance system stability, low and high level of Allee in predator stabilizes the system whereas moderate level of Allee results oscillation. The model further reveals bi-stability and the “Paradox of enrichment". A key finding is that species memory can significantly modulate both prey’s fear intensity and predator Allee strength, potentially destabilizing the system even when these effects are strong. Moreover, stability depends on different memory of both species, where stronger prey memory has greater impact than predator memory in stable coexistence. These results highlight previously unexplored interactions between memory-driven fear and Allee mechanisms and offer new insights into the complex dynamics of predator-prey systems.