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An investigation into the impact of odour: A dynamical study of two predators and one prey model, taking into account both integer order and fractional order derivatives

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  • Das, Dipam
  • Bhattacharjee, Debasish

Abstract

This article presents a prey–predator system that takes into account two important factors: the negative impact of predator odour on its competitors and the positive impact of predator odour on the prey. The system is analysed using two models: one with ODEs and another with FDEs. We have extensively validated the model system biologically, ensuring that the solutions are both nonnegative and bounded. An in-depth investigation has been carried out to thoroughly investigate the stability of all potential equilibrium points of the model systems in a systematic manner. Our observations reveal that our model systems exhibit various types of bifurcations, including transcritical and Hopf bifurcations, around the interior equilibrium point for three distinct parameters. These parameters include the rate of conversion of the first predator r4, the degree of resistance or avoidance exhibited by the prey due to predator odour m, and the level of disruption to competitors in predation due to the presence of predator odour a. A significant finding in this paper is that the resistance shown by prey towards the first predator in predation in reaction to the odour is vital for sustaining the population of the second predator. The survival of the second predator within the biosystem is heavily dependent on the growth rate of the first predator. The possibility of the second predator facing extinction becomes much less likely when the first predator is absent from the system, which is another significant result. Within the context of fractional order derivatives, the system dynamics demonstrate a higher level of stability in comparison to the traditional integer order derivative. It has been noticed that where the parameter values are identical, the fluctuations exhibited by the integer order system are stabilised in the fractional order system. Thus, the significance of predator odour and the effect of memory in the system have been thoroughly established. Ultimately, the study backs up the theoretical findings with convincing numerical simulations.

Suggested Citation

  • Das, Dipam & Bhattacharjee, Debasish, 2025. "An investigation into the impact of odour: A dynamical study of two predators and one prey model, taking into account both integer order and fractional order derivatives," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 233(C), pages 341-368.
    • Handle: RePEc:eee:matcom:v:233:y:2025:i:c:p:341-368
    DOI: 10.1016/j.matcom.2025.01.026
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    References listed on IDEAS

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    1. Roy, Banani & Roy, Sankar Kumar & Gurung, Dil Bahadur, 2017. "Holling–Tanner model with Beddington–DeAngelis functional response and time delay introducing harvesting," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 142(C), pages 1-14.
    2. Barman, Dipesh & Roy, Jyotirmoy & Alrabaiah, Hussam & Panja, Prabir & Mondal, Sankar Prasad & Alam, Shariful, 2021. "Impact of predator incited fear and prey refuge in a fractional order prey predator model," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    3. Narayan Mondal & Dipesh Barman & Shariful Alam, 2021. "Impact of adult predator incited fear in a stage-structured prey–predator model," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 23(6), pages 9280-9307, June.
    4. Sk, Nazmul & Tiwari, Pankaj Kumar & Pal, Samares, 2022. "A delay nonautonomous model for the impacts of fear and refuge in a three species food chain model with hunting cooperation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 136-166.
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