Qualitative Analysis of a Modified Leslie-Gower Model with Gestation Delay
DOI:
https://doi.org/10.5890/JAND.2021.09.004Abstract
This paper explores the qualitative analysis of a modified Leslie-Gower prey-predator model where the consumption rate of prey is by per capita predator according to Beddington-DeAngelis functional response. Moreover, time-lag $(\tau)$ is established to exploit a gestation period of predations. The permanence analysis of the proposed system is investigated. We study the local stability of the non-delayed model at all possible equilibrium points. It is demonstrated that the given model experiences Hopf bifurcation about the interior equilibrium point with respect to delay $\tau$. Thereafter the stability and direction of Hopf bifurcation are formulated through normal and center manifold theorems. The derived criteria are justified with the help of numerical simulations.References
[1] Lotka, A.J. (1956), Elements of Mathematical Biology, Cambridge University Press.
[2] Maynard-Smith, J. (1978), Models in Ecology, Cambridge University Press Archive.
[3] Cantrell, R.S. and Cosner, C. (2001), On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, Journal of Mathematical Analysis and Applications, 257(1), 206-222.
[4] Zhu, H., Campbell, S.A., and Wolkowicz, G.S. (2003), Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 63(2), 636-682.
[5] Yuan, S. and Song, Y. (2009), Bifurcation and stability analysis for a delayed Leslie-Gower predator-prey system, IMA Journal of Applied Mathematics, 74(4), 574-603.
[6] Feng, P. (2014), Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting, Journal of Applied Mathematics and Computing, 44(1-2), 251-262.
[7] Celik, C. (2015), Stability and Hopf bifurcation in a delayed ratio dependent Holling-Tanner type model, Applied Mathematics and Computation, 255, 228-237.
[8] Aziz-Alaoui, M.A. and Okiye, M.D. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Applied Mathematics Letters, 16(7), 1069-1075.
[9] Beddington, J.R. (1975), Mutual interference between parasites or predators and its effect on searching efficiency, The Journal of Animal Ecology, 44(1), 331-340.
[10] DeAngelis, D.L., Goldstein, R.A., and Oneill, R.V. (1975), A model for tropic interaction, Ecology, 56(4), 881-892.
[11] Pal, P.J. and Mandal, P.K. (2014), Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-Deangelis functional response and strong Allee effect, Mathematics and Computers in Simulation, 97, 123-146.
[12] Yu, S. (2014), Global stability of a modified Leslie-Gower model with Beddington-Deangelis functional response, Advances in Difference Equations, 2014(1), 84.
[13] Abid, W., Yafia, R., Aziz-Alaoui, M.A., Bouhafa, H., and Abichou, A. (2015), Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type, Evolution Equations & Control Theory, 4(2), 115.
[14] Indrajaya, D., Suryanto, A., and Alghofari, A. R. (2016), Dynamics of modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and additive Allee effect, International Journal of Ecology and Development, 31(3), 60-71.
[15] Lahrouz, A., Settati, A., and Mandal, P.S. (2016), Dynamics of a switching diffusion modified Leslie-Gower predator-prey system with Beddington-Deangelis functional response, Nonlinear Dynamics, 85(2), 853-870.
[16] Roy, B., Roy, S.K., and Gurung, B.D. (2017), Holling-Tanner model with Beddington-Deangelis functional response and time delay introducing harvesting, Mathematics and Computers in Simulation, 142, 1-14.
[17] Song, Y. and Wei, J. (2005), Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Journal of Mathematical Analysis and Applications, 301(1), 1-21.
[18] Saha, T. and Chakrabarti, C. (2009), Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model, Journal of Mathematical Analysis and Applications, 358(2), 389-402.
[19] Karaoglu, E. and Merdan, H. (2014), Hopf bifurcations of a ratio-dependent predator-prey model involving two discrete maturation time delays, Chaos, Solitons $&$ Fractals, 68, 159-168.
[20] El Foutayeni, Y. and Khaladi, M. (2016), Equilibrium Points and Their Stability Properties of a Multiple Delays Model, Differential Equations and Dynamical Systems, 1-18.
[21] Huang, J., Gong, Y., and Chen, J. (2013), Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, International Journal of Bifurcation and Chaos, 23(10), 1350164.
[22] Huang, J., Liu, S., Ruan, S., and Zhang, X. (2016), Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting, Communication in Pure and Applied Analysis, 15(3), 1041-1055.
[23] Chakraborty, K., Jana, S., and Kar, T.K. (2012), Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting, Applied Mathematics and Computation, 218(18), 9271-9290.
[24] Gupta, R.P. and Chandra, P. (2013), Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, Journal of Mathematical Analysis and Applications, 398(1), 278-295.
[25] Yuan, R., Jiang, W., and Wang, Y. (2015), Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, Journal of Mathematical Analysis and Applications, 422(2), 1072-1090.
[26] Hu, D. and Cao, H. (2017), Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Analysis: Real World Applications, 33, 58-82.
[27] Yang, R. and Zhang, C. (2017), Dynamics in a diffusive modified Leslie-Gower predator-prey model with time delay and prey harvesting, Nonlinear Dynamics, 87(2), 863-878.
[28] Hassard, B.D., Kazarinoff, N.D., and Wan, Y.W. (1981), Theory and applications of Hopf bifurcation, CUP Archive.
[29] Saifuddin, M., Biswas, S., Samanta, S., Sarkar, S., and Chattopadhyay, J. (2016), Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator, Chaos, Solitons & Fractals, 91, 270-285.
[30] Singh, M.K., Bhadauria, B.S., and Singh, B.K. (2018), Bifurcation analysis of modified Leslie-Gower predator-prey model with double Allee effect, Ain Shams Engineering Journal, 9(4), 1263-1277.
[31] Peng, Y. and Zhang, T. (2016), Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect, Applied Mathematics and Computation, 275, 1-12.
[32] Slobodkin, L.B. (1980), Growth and regulation of animal populations, Dover Publications.
[33] Dennis, B. (1989), Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3(4), 481-538.
[34] Sivasamy, R., Sathiyanathan, K., and Balachandran, K. (2019), Dynamics of a Modified Leslie-Gower Model with Crowley-Martin Functional Response and Prey Harvesting, Journal of Applied Nonlinear Dynamics, 8(4), 621-636.
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