Dynamical behavior of a diseased predator-prey model with fear effect and prey harvesting

Abstract

This article consists of a three-species food web model that has been developed by considering the interaction between susceptible prey, infected prey and predator species. It is assumed that susceptible prey species grow logistically in the absence of predators. It is assumed that predators consume susceptible and infected prey . We consider the effect of fear on susceptible prey due to predator species. Again, the harvesting of susceptible and infected prey has been considered. Furthermore, the predator consumes its prey in the form of Holling-type interactions. The positive invariance, positivity, and boundedness of the system are discussed. The conditions of all biologically feasible equilibrium points have been examined. The local stability of the systems around these equilibrium points is investigated. Furthermore, the occurrence of Hopf-bifurcation concerning the harvesting (h) of the system has been investigated. Finally, we demonstrate some numerical simulation results to illustrate our main analytical findings.

Keywords

• Eco-epidemiological model
• Hopf-bifurcation
• fear effect
• stability
• prey harvesting

References

1. Haeckel, E., The History of Creation, Vol. 1. HS King & Company, 1876.
2. Venturino, E., Ecoepidemiology: a more comprehensive view of population interactions, Mathematical Modelling of Natural Phenomena, 11(1) (2016), 49–90. https://doi.org/10.1051/mmnp/201611104
3. Malchow, H., Petrovskii, S. V., Venturino, E., Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation. Chapman and Hall/CRC, 2007.
4. Kermack, W. O., McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(772) (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118
5. Lotka, A. J., Elements of Physical Biology, Williams & Wilkins, 1925.
6. Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi mem. accad. lincei roma 2 31; fluctuations in the abundance of a species considered mathematically, Nature (London), 118 (1926), 558.
7. Freedman, H. I., Deterministic Mathematical Models in Population Ecology, Vol. 57. Marcel Dekker Incorporated, 1980.
8. Murray, J., Mathematical Biology, Springer-Verlag, New York, 1989.

9. Xu, R., Chaplain, M. A., Davidson, F. A., Persistence and global stability of a ratio- dependent predator–prey model with stage structure, Applied Mathematics and Computation, 158(3) (2004), 729–744. https://doi.org/10.1016/j.amc.2003.10.012
10. Arditi, R., Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, Journal of Theoretical Biology, 139(3) (1989), 311–326.
11. Crowley, P. H., Martin, E. K., Functional responses and interference within and between year classes of a dragonfly population, Journal of the North American Benthological Society, 8(3) (1989), 211–221.
12. Arumugam, D., Muthurathinam, S., Anbulinga, A., Impact of fear on a crowley–martin eco-epidemiological model with prey harvesting, Engineering Proceedings, 56(1) (2023),296. https://doi.org/10.3390/ASEC2023-15908
13. Hassell M., Varley, G., New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133–1137.
14. Beddington, J. R., Mutual interference between parasites or predators and its effect on searching efficiency, The Journal of Animal Ecology, (1975), 331–340.
15. Holling, C. S., The components of predation as revealed by a study of small-mammal predation of the european pine sawfly1, The Canadian Entomologist, 91(5) (1959), 293–320.
16. DeAngelis, D. L., Goldstein, R., O’Neill, R. V., A model for tropic interaction, Ecology, 56(4) (1975), 881–892.
17. Holling, C. S., Some characteristics of simple types of predation and parasitism1, The Canadian Entomologist, 91(7) (1959), 385–398.
18. Pradeep, M. S., Gopal, T. N., Magudeeswaran, S., Deepak, N., Muthukumar, S., Stability analysis of diseased preadator-prey model with holling type ii functional response, AIP Conference Proceedings, vol. 2901, AIP Publishing, 2023.
19. Natesan, R., Shanmugam, M., Manickasundaram, S. P., Nallasamy Prabhumani, D., The effect of fear on a diseased prey–predator model with predator harvesting, Engineering Proceedings, 56(1) (2023), 124. https://doi.org/10.3390/ASEC2023-15248
20. Gaber, T., Herdiana, R., et al., Dynamical analysis of an eco-epidemiological model experiencing the crowding effect of infected prey, Commun. Math. Biol. Neurosci., 2024 (2024). https://doi.org/10.28919/cmbn/8353
21. Fakhry, N. H., Naji, R. K., Fear and hunting cooperation’s impact on the ecoepidemiological model’s dynamics, International Journal of Analysis and Applications, 22 (2024), 15–15.
22. Smith, J., Maynard: Models in ecology, 1974.
23. Kar, T. K., Stability analysis of a prey–predator model incorporating a prey refuge, Com- munications in Nonlinear Science and Numerical Simulation, 10(6) (2005), 681–691. https://doi.org/10.1016/j.cnsns.2003.08.006
24. Gutierrez, A., Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm, Ecology, 73(5) (1992), 1552–1563.
25. Akcakaya, H. R., Arditi, R., Ginzburg, L. R., Ratio-dependent predation: an abstraction that works, Ecology, 76(3) (1995), 995–1004.
26. Cosner, C., DeAngelis, D. L., Ault, J. S., Olson, D. B., Effects of spatial grouping on the functional response of predators, Theoretical Population Biology, 56(1) (1999), 65–75. https://doi.org/10.1006/tpbi.1999.1414