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Dynamical complexities with effect of additional food and harvesting in the time delay plankton-fish model

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Abstract

In the present work, we have developed a plankton-fish model consisting time delay using functional response Holling type-II and linear harvesting. Two different additional foods are supplied to both the species zooplankton and fish and study which one additional food more influences positively to the system than other. We derive some stipulations for existence of equilibria and its stability. Stipulations for local and global stability, positivity and boundedness of the system are investigated. Stability and instability analysis have been performed through the time series, phase space and bifurcation diagrams with respect to various ecological factors. In both cases, presence of delay and absence of delay, the main observation of the study indicates that the low additional food supply to zooplankton has an extinction risk for both the species fish and themselves, although the certain additional food is provided to the fish. It is beneficial not only for the zooplankton but also for the fish in both cases. It is also observed that enhancing the additional food supply of the zooplankton in presence of delay is more advantageable than in absence of delay, because of, delay enhances that quantity of additional food supply to a certain level without changing the stable nature of the system for which the system exhibits a stable dynamic in absence of delay. The complex behaviour (chaos) which is appeared due to conversion rate and small harvesting of zooplankton that can be completely controlled by the delay. Double Hopf bifurcation behaviour has been inspected due to delay. A small gestation delay has destabilizing effect, whereas a relatively large gestation delay has stabilizing effect.

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Acknowledgements

The work is supported by MATRICS project (File No.: MTR/2021/000472) SERB-DST New Delhi, India to the corresponding author S. N. Raw and is also supported by UGC New Delhi, India to the co-author S. R. Sahu under Junior Research Fellowship (JRF) Award. The authors are thankful to Dr. Pau Martin, Associate Editor and the reviewers for their valuable suggestions and remarks in sha** the manuscript into its current form.

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  1. Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh, 492010, India

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Raw, S.N., Sahu, S.R. Dynamical complexities with effect of additional food and harvesting in the time delay plankton-fish model. SeMA (2023). https://doi.org/10.1007/s40324-023-00339-z

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