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Predator–prey system with multiple delays: prey’s countermeasures against juvenile predators in the predator–prey conflict

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Abstract

Usually, juvenile predators are small in size, comparatively feeble and in the earlier phase of learning the tricky manoeuvre skills of predation. Due to small prey–predator size-ratio, juvenile predators and preys become two deadliest opponents, therefore, whenever both face off, the outcome may be utterly unpredictable. In response to the juvenile predators’ attack, preys also come up with a countermeasure and try to defend themselves but a sharp role-reversal in the predator–prey ecology occurs when preys kill the juvenile predators during the counter-attacking strategies. In this paper, we propose and analyze a biologically motivated delayed predator–prey system to explore the joint impact of counterattacking strategies and multiple delays on the predator–prey interactions. We have introduced the transformation of immature to mature predators including delay \(\tau _1\) as an average time to reach maturity. Another time delay \(\tau _2\) owing to the gestation of the adult predators, is taken into account. After the brief discussion of the basic ecosystem properties for the model in absence of delay, we have performed the stability analysis of the delayed system comprehensively. In this context, existence of Hopf bifurcation at the interior equilibrium is established and its stability as well as direction of the Hopf bifurcation is also investigated. Numerical simulations are also performed to corroborate the theoretical findings. Through this work, it has been illustrated that counterattacking of preys suppresses the unstable oscillations in the population, on the other hand, large delays always destabilize the interior steady state. Hence, in the realistic predator–prey ecology, where counterattack and delays are showing opposite behavior in the same ecosystem, the results of this paper are important to disclose the actual crossover and synergies between both the ecological properties.

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Acknowledgements

First author thanks the Council of Scientific and Industrial Research (CSIR), India (Grant No. 9013-11-044) for the financial support. Authors are also grateful to the anonymous referees for their valuable suggestions.

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  1. Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    Rajat Kaushik & Sandip Banerjee

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  1. Rajat Kaushik
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  2. Sandip Banerjee
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Correspondence to Rajat Kaushik.

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Kaushik, R., Banerjee, S. Predator–prey system with multiple delays: prey’s countermeasures against juvenile predators in the predator–prey conflict. J. Appl. Math. Comput. 68, 2235–2265 (2022). https://doi.org/10.1007/s12190-021-01615-9

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  • DOI: https://doi.org/10.1007/s12190-021-01615-9

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