Geometric Approach to Predator-Prey Model with Carrying Capacity on Prey Population

Marshellino; Hengki Tasman; Rahmi Rusin

doi:10.5614/cbms.2024.7.2.1

Authors

Marshellino University of Indonesia
Hengki Tasman University of Indonesia
Rahmi Rusin University of Indonesia

DOI:

https://doi.org/10.5614/cbms.2024.7.2.1

Keywords:

Predator-prey, geometric singular perturbation theory, fast-slow system, entry-exit function

Abstract

In this paper, we explore a classical predator-prey model where the birth rate of the prey is significantly lower than the mortality rate of the predators, while also considering a limited prey population. We incorporate an environmental carrying capacity factor for the prey to account for this. Given the different timescales of the predator and prey populations, some system solutions may exhibit a fast-slow structure. We analyze this fastslow behavior using geometric singular perturbation theory (GSPT), which allows us to separate the system into fast and slow subsystems. Our research investigates the existence and stability of equilibrium solutions and the behavior of solutions near the critical manifold. Additionally, we use an entry-exit function to analytically establish the connection between the solutions of the slow subsystem and those of the fast subsystem.

References

Andrea, C., Prey and predators-a model for the dynamics of biological systems, Medium, 2021. https://towardsdatascience.com/prey-and-predators-a-model-for-the-dynamics-of-biological-systems-747b82d2ea9e, Accessed on March 19, 2023.

Boyce, W. E., Diprima, R. C., and Meade, D. B., Elementary differential equations and boundary value problems, United States, Wiley, 2017.

Brauer, F., A singular perturbation approach to epidemics of vector transmitted diseases, Infectious Disease Modelling, 4, pp. 115-123, 2019.

Brown, J, How do the populations of predator and prey affect each other?, 2021. https://knowledgeburrow.com/how-do-the-populations-of-predator-and-prey-affect-each-other/, Accessed on March 19, 2023.

Chowdhury, P.R., Petrovskii, S. and Banerjee, M., Effect of slow?fast time scale on transient dynamics in a realistic prey-predator system, Mathematics, 10, pp. 1-12, 2022.

Colley, S.J., Vector Calculus, United States, Pearson Education, Inc., 2012.

Hek, G., Geometric singular perturbation theory in biological practice, Journal of Mathematical Biology, 60(3), pp. 347-386, 2020.

Jardon-Kojakhmetov, H., Kuehn, C., Pugliese, A. and Sensi, A., A geometric analysis of the SIR, SIRS, and SIRWS epidemiological models, Nonlinear Analysis: Real World Applications, 58, pp. 1-27, 2021.

Kuehn, C., Multiple time scale dynamics, Springer International Publishing, 191, 2015.

Maesschalck, P.D. and Schecter, S., The entry?exit function and geometric singular perturbation theory, Journal of Differential Equations, 260(8), pp. 6697-6715, 2016.

Martcheva, M., An Introduction to Mathematical Epidemiology, Springer, 2015.

Owen, L. and Tuwankotta, J.M., On slow?fast dynamics in a classical predator?prey system, Mathematics and Computers in Simulation, 177, pp. 306-315, 2020.

Strogatz, S.H., Nonlinear Dynamic and Chaos With Applications to Physics, Biology, Chemistry, and Engineering, New York, CRC Press, 2018.

Tu, L.W., An Introduction to Manifolds, New York, Springer, 2015.