Stability and Coexistence in an Improved Three-Species Predator–Prey System with Logistic Growth and Nonlinear Interactions
DOI:
https://doi.org/10.62054/ijdm/0301.07Abstract
Understanding the mechanisms that sustain coexistence among multiple interacting species remains a fundamental problem in mathematical ecology. In this study, we develop an enhanced three-trophic-level predator–prey model representing a food chain consisting of lions (top predators), leopards (intermediate predators/preys), and hares (preys). The model incorporates logistic growth functions to account for environmental carrying capacities and nonlinear predation terms with energetic conversion efficiencies to reflect realistic trophic interactions. Analytical results establish the positivity, boundedness, and biological feasibility of the model solutions. The equilibrium points and their local stability properties are examined through Jacobian linearization and Routh–Hurwitz conditions, while global stability is investigated via a suitable Lyapunov function. Numerical simulations illustrate diverse dynamic behaviors, including coexistence equilibria, extinction scenarios, and oscillatory coexistence depending on the predation and conversion parameters. The proposed framework provides a more biologically consistent and mathematically robust tool for studying multispecies interactions and can be extended to incorporate stochastic perturbations, time delays, or spatial heterogeneity.
References
Arancibia-Ibarra, C. (2018). The basins of attraction in a modified May–Holling–Tanner predator–prey model with Allee effect and alternative food sources. arXiv. https://arxiv.org/abs/1901.01118
Arancibia-Ibarra, C., Bode, M., Flores, J., Pettet, G., & van Heijster, P. (2019). A modified May–Holling–Tanner predator–prey model with multiple Allee effects on the prey and an alternative food source for the predator. arXiv. https://arxiv.org/abs/1904.02886
Arancibia-Ibarra, C., & Flores, J. (2020). Modelling and analysis of a modified May–Holling–Tanner predator–prey model with Allee effect in the prey and an alternative food source for the predator. arXiv. https://arxiv.org/abs/2009.02482
Bacaër, N. (2011). A short history of mathematical population dynamics. Springer.
Borges, I., Oliveira, L., Barbosa, F., Figueiredo, E., Franco, J. C., Durão, A. C., & Soares, A. O. (2024). Prey consumption and conversion efficiency in females of two feral populations of Macrolophus pygmaeus. Phytoparasitica, 52(2), 31.
Cantrell, R. S., & Cosner, C. (2003). Spatial ecology via reaction–diffusion equations. Wiley.
Chakraborty, K., Jana, S., & Kar, T. K. (2012). Global dynamics and bifurcation in a stage-structured prey–predator fishery model with harvesting. Applied Mathematics and Computation, 218, 9271–9290.
Clinchy, M., Sheriff, M. J., & Zanette, L. Y. (2013). Predator-induced stress and the ecology of fear. Functional Ecology, 27(1), 56–65.
Creel, S., Christianson, D., Liley, S., & Winnie, J. A. (2007). Predation risk affects reproductive physiology and demography of elk. Science, 315(5814), 960.
Diz-Pita, É., & Otero-Espinar, M. V. (2021). Predator–prey models: A review of some recent advances. Mathematics, 9(15), 1783.
Hastings, A., & Powell, T. (1991). Chaos in a three-species food chain. Ecology, 72, 896–903.
Holling, C. S. (1959). The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. The Canadian Entomologist, 91(5), 293–320.
Kot, M. (2001). Elements of mathematical ecology. Cambridge University Press.2
Kuang, Y. (1993). Delay differential equations with applications in population dynamics. Academic Press.
Lotka, A. J. (1925). Elements of physical biology. Williams & Wilkins.
Lv, S., & Zhao, M. (2008). The dynamic complexity of a three-species food chain model. Chaos, Solitons & Fractals, 37(5), 1469–1480.
Lubem M. Kwaghkora, Samuel Adamu, Mohammed Fori. and Hassan Bukar (2024) Mathematical Model of the Dynamics of three Different Species in Predator-Prey System.
International Journal of Development Mathematics Vol 1 Issue 2, Page No. 024 – 037.
https://ijdm.org.ng/index.php/Journals
Pal S, Tiwari PK, Misra AK, Wang H. Fear effect in a three-species food chain model with generalist predator. Math Biosci Eng. 2024 Jan;21(1):1-33. doi: 10.3934/mbe.2024001. Epub 2022 Dec 8. PMID: 38303411.
Murray, J. D. (2007). Mathematical biology I: An introduction (3rd ed.). Springer.
Okubo, A., & Levin, S. A. (2001). Diffusion and ecological problems: Modern perspectives (2nd ed.). Springer.
Panja, P., & Mondal, S. K. (2015). Stability analysis of coexistence of three species prey–predator model. Nonlinear Dynamics, 81(1), 373–382.
Santra, N. (2023). Role of multiple time delays on a stage-structured predator–prey system in a toxic environment. Mathematics and Computers in Simulation, 212, 548–583. https://doi.org/10.1016/j.matcom.2023.05.015
Sen, D., Ghorai, S., & Banerjee, M. (2018). Complex dynamics of a three-species prey–predator model with intraguild predation. Ecological Complexity, 34, 9–22.
Terry, A. J. (2015). Predator–prey models with component Allee effect for predator reproduction. Journal of Mathematical Biology, 71, 1325–1352. https://doi.org/10.1007/s00285-015-0856-5
Verhulst, P.-F. (1838). Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique, 10, 113–121.
Zanette, L. Y., White, A. F., Allen, M. C., & Clinchy, M. (2011). Perceived predation risk reduces the number of offspring songbirds produce per year. Science, 334(6061), 1398–1401.
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