A Mathematical Modelling Approach for a Past-Dependent Prey-Predator System

Year 2022, Volume: 5 Issue: 1, 1 - 7, 30.04.2022

Aytül Gökçe

https://doi.org/10.33187/jmsm.1063954

Abstract

A memory dependent prey-predator model incorporating Allee effect in prey is analysed. For a small and high values of memory rate, the dynamical changes in the prey and predator densities are demonstrated. The equilibria of the proposed model and the local stability analysis corresponding to each equilibrium are presented. The variables of prey and predator species with respect to memory rate are investigated and the existence of the Hopf bifurcation is shown. The analytical part of this paper is supported with detailed numerical simulations.

Keywords

Nonlinear dynamics , prey-predator systems , stability

References

• [1] A. J. Lotka, Elements of physical biology, Science Progress in the Twentieth Century, 21(82) (1926), 341-343.
• [2] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119(2983), 12-13.
• [3] E. Renshaw, Modelling Biological Populations in Space and Time, No. 11, Cambridge University Press, 1993.
• [4] C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91(7) (1959), 385-398.
• [5] B. Sahoo, S. Poria, Dynamics of predator–prey system with fading memory, Appl. Math. Comput., 347 (2019), 319-333.
• [6] S. Djilali, B. Ghanbari, Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative, Adv. Differ. Equ., 2021(1) (2021), 1-19.
• [7] B. Sahoo, B, A predator-prey model with general holling interactions in the presence of additional food, Int. J. Plant Res., 2(1) (2012), 47-50. [8] W. C. Allee, Animal aggregations, Q. Rev. Biol., 2(3) (1927), 367-398.
• [9] P. C. Tabares, J. D. Ferreira, V. Rao, Weak Allee effect in a predator-prey system involving distributed delays, Comput. Appl. Math., 30(3) (2011), 675-699.
• [10] Y. Song, S., Wu, H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differ. Equ., 267(11) (2019), 6316-6351.
• [11] Z. Ma, Hopf bifurcation of a generalized delay-induced predator–prey system with habitat complexity, Int. J. Bifurc. Chaos., 30(06) (2020), 2050082.
• [12] P. Shome, A., Maiti, A., S. Poria, Effects of intraspecific competition of prey in the dynamics of a food chain model, Model. Earth Syst. Environ., 2(4) (2016), 1-11.
• [13] M. Cavani, M. Farkas, Bifurcations in a predator-prey model with memory and diffusion. I: Andronov-Hopf bifurcation, Acta Math. Hungarica, 63(3) (1994), 213-229.
• [14] U. Ghosh, S. Pal, M. Banerjee, Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis, Chaos Solit. Fractals, 143 (2021), 110531.
• [15] F. Courchamp, B. Grenfell, T. Clutton-Brock, Impact of natural enemies on obligately cooperative breeders, Oikos 91(2) (2000), 311-322.
• [16] A. W. Stoner, M. Ray-Culp, Evidence for allee effects in an over-harvested marine gastropod: density-dependent mating and egg production, Mar. Ecol. Prog. Ser. 202 (2000), 297-302.
• [17] J. D. Ferreira, C. A. T. Salazar, P. C. Tabares, Weak Allee effect in a predator–prey model involving memory with a hump, Nonlinear Anal. Real World Appl., 14(1) (2013) 536-548.
• [18] A. Gokce, Numerical bifurcation analysis for a prey-predator type interactions with a time lag and habitat complexity, Bitlis Eren U¨ ni. Fen Bilimleri D., 10(1) (2020), 57-66.
• [19] A. Gokce, A mathematical modeling approach to analyse the effect of additional food in a predator-prey interactions with a white Gaussian noise in Prey’s growth rate, Int. J. Appl. Comput., 8(1) (2022), 1-20.
• [20] A. G¨okc¸e, Exploring a simple stochastic mathematical model including fear with a linear functional response, Fundam. J. of Math. and Appl., 4(4) (2021), 280-288.
• [21] B. Gurbuz, A numerical scheme for the solution of neutral integro-differential equations including variable delay, Math. Sci., (2021), 1-9.
• [22] B. Gurbuz, Laguerre matrix-collocation technique to solve systems of functional differential equations with variable delays, AIP Conf. Proc., AIP Publishing LLC, (2019), 090007.
• [23] S. C¸ akan, Dynamic analysis of a mathematical model with health care capacity for COVID-19 pandemic, Chaos Solit. Fractals, 139 (2020), 110033.