Numerical bifurcation analysis for a prey-predator type interactions with a time lag and habitat complexity

Year 2021, , 57 - 66, 21.03.2021

Aytül Gökçe

https://doi.org/10.17798/bitlisfen.840245

Cited By: 6

Abstract

In this paper, a two-component generic prey-predator system incorporated with habitat complexity in predator functional response, and with constant time delay in predator gestation is considered. Although the role of time delay on the system dynamics is widely studied in the literature, only a few researchers have addressed the effect of habitat complexity in the prey-predator type interactions. In the first part of the paper the equilibria and stability analysis of the mathematical model is mentioned. In the second part, particular attention is paid on the numerical bifurcation analysis of the prey and predator densities based on two system parameters:(i) the strength of homogeneous habitat complexity and (ii) predator attack rate with and without time delay. It is found that dynamics with time delay in predator gestation are found to be much richer compared to that without time delay. The system stability may change from stable to unstable through a Hopf bifurcation and the solution branches emanating from these Hopf points are usually stable and supercritical. However, delay driven system may lead unstable orbits arising from Hopf bifurcations. It is also found that increasing the strength of habitat complexity may lead the stability change from unstable to stable.

Keywords

Habitat Complexity, Delay Differential Equations, Numerical Bifurcation Analysis

Numerical bifurcation analysis for a prey-predator type interactions with a time lag and habitat complexity

Abstract

In this paper, a two-component generic prey-predator system incorporated with habitat complexity in predator functional response, and with constant time delay in predator gestation is considered. Although the role of time delay on the system dynamics is widely studied in the literature, only a few researchers have addressed the effect of habitat complexity in the prey-predator type interactions. In the first part of the paper the equilibria and stability analysis of the mathematical model is mentioned. In the second part, particular attention is paid on the numerical bifurcation analysis of the prey and predator densities based on two system parameters:(i) the strength of homogeneous habitat complexity and (ii) predator attack rate with and without time delay. It is found that dynamics with time delay in predator gestation are found to be much richer compared to that without time delay. The system stability may change from stable to unstable through a Hopf bifurcation and the solution branches emanating from these Hopf points are usually stable and supercritical. However, delay driven system may lead unstable orbits arising from Hopf bifurcations. It is also found that increasing the strength of habitat complexity may lead the stability change from unstable to stable.

Keywords

Habitat complexity, delay differential equations, numerical bifurcation analysis

References

• Bairagi N., Jana D. 2011. On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity. Applied Mathematical Modeling, 35 (7): 3255-3267.
• Dubey B.K., Maiti A.P. 2019. Global stability and Hopf-bifurcation of prey-predator system with two discrete delays including habitat complexity and prey refuge. Communications in Nonlinear Science and Numerical Simulation, 67: 528-554.
• Bairagi N., Jana D. 2012. Age-structured predator-prey model with habitat complexity: oscillations and control. Dynamical Systems, 27 (4): 475-499.
• Ghorai S., Poria S. 2016. Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity. Chaos, Solitons & Fractals, 91: 421-429.
• Din Q. 2017. Complexity and chaos control in a discrete-time prey-predator model. Communications in Nonlinear Science and Numerical Simulation, 49: 113-134.
• Gökçe A., Yazar S., Sekerci Y. 2020. Delay induced nonlinear dynamics of oxygen-plankton interactions. Chaos, Solitons & Fractals, 141: 110327.
• Chakraborty K., Chakraborty M., Kar T.K. 2011. Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay. Nonlinear Analysis Hybrid Systems, 5 (4): 613-625.
• Liao T., Yu H., Zhao M. 2017. Dynamics of a delayed phytoplankton-zooplankton system with Crowley-Martin functional response. Advances in Difference Equations, 2017 (1): 1-30.
• Tang Y., Zhou L. 2007. Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay. Journal of Mathematical Analysis and Applications, 334 (2): 1290-1307.
• Yu H., Zhao M., Agarwal R.P. 2014. Stability and dynamics analysis of time delayed eutrophication ecological model based upon the Zeya reservoir. Mathematics and Computers in Simulations, 97: 53-67.
• Chattopadhyay J., Sarkar R.R., El Abdllaoui A. 2002. A delay differential equation model on harmful algal blooms in the presence of toxic substances. Mathematical Medicine and Biology: A Journal of IMA, 19 ( 2): 137-161.
• Jiang Z., Wang L. 2017. Global Hopf bifurcation for a predator-prey system with three delays. International Journal of Bifurcation and Chaos, 27 (7): 1750108.
• Misra A.K., Chandra P., Raghavendra V. 2011. Modeling the depletion of dissolved oxygen in a lake due to algal bloom: Effect of time delay. Advances in Water Resources, 34 (10): 1232-1238.
• Rehim M., Imran M. 2012. Dynamical analysis of a delay model of phytoplankton--zooplankton interaction. Appllied Mathematical Modeling, 36 (2): 638-647.
• Sharma A., Sharma A.K., Agnihotri K. 2015. Analysis of a toxin producing phytoplankton-zooplankton interaction with Holling IV type scheme and time delay. Nonlinear Dynamics, 81 (1-2): 13-25.
• Jana D., Bairagi N. 2014. Habitat complexity, dispersal and metapopulations: Macroscopic study of a predator--prey system. Ecological Complexity, 17: 131-139.
• Engelborghs K., Luzyanina T., Roose D. 2002. Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Transactions on Mathematical Software, 28 (1): 1-21.
• Engelborghs K., Luzyanina T., Samaey G. 2000. DDE-BIFTOOL: a Matlab package for bifurcation analysis of delay differential equations. TW Report, 305: 1-36.
• Ji C., Jiang D., Li X. 2011. Qualitative analysis of a stochastic ratio-dependent predator-prey system. Journal of Computational and Applied Mathematics, 235 (5): 1326-1341.
• Han B.S., Wang Z.C. 2018. Turing patterns of a Lotka-Volterra competitive system with nonlocal delay. International Journal of Bifurcation and Chaos, 28 (7): 1830021.