Discrete-time prey-predator model with θ-logistic growth for prey incorporating square root functional response

P.K. Santra

Abstract


This article presents the dynamics of a discrete-time prey-predator system with square root functional response incorporating θ-logistic growth. This type of functional response is used to study the dynamics of the prey--predator system where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The existence and stability of fixed points and Neimark-Sacker Bifurcation (NSB) are analyzed. The phase portraits, bifurcation diagrams and Lyapunov exponents are presented and analyzed for different parameters of the model. Numerical simulations show that the discrete model exhibits rich dynamics as the effect of θ-logistic growth.

Keywords


Discrete prey-predator model; θ-logistic growth; Lyapunov exponents; Stability; Neimark-Sacker Bifurcation.

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References


Abd-Elalim A. Elsadany, Dynamical complexities in a discrete-time food chain, Computational Ecology and Software, 2(2) (2012) 124-139.

C. Çelik and O. Duman, Allee effect in a discrete-time predator-prey system. Chaos Soliton Fractals, 40 (2009) 1956--1962.

J. Huang, Bifurcations and chaos in a discrete predator--prey system with Holling type-IV functional response, Acta Mathematicae Applicatae Sinica English Series, 21 (2005) 157--176.

J. Huang, S. Liu, S. Ruan and D. Xiao, Bifurcations in a discrete predator--prey model with nonmonotonic functional response, Journal of Mathematical Analysis and Applications, 464(1) (2018) 201-230.

M. Gámeza, I. Lópeza, C. Rodrígueza, Z. Vargab and J. Garayc, Ecological monitoring in a discrete-time prey--predator model, Journal of Theoretical Biology, 429(21) (2017) 52-60.

M. Zhao and Y. Du, Stability of a Discrete-Time Predator-Prey System with Allee Effect, Nonlinear Analysis and Differential Equations, 4(5) (2016) 225 - 233.

P.K. Santra, G.S. Mahapatra, Dynamical study of discrete-time prey predator model with constant prey refuge under imprecise biological parameters, Journal of Biological Systems, (2020) DOI: 10.1142/S0218339020500114.

P.K. Santra, G.S. Mahapatra and G.R. Phaijoo, Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley--Martin Functional Response Incorporating Proportional Prey Refuge, Mathematical Problems in Engineering, (2020) ID 5309814.

X. Liu and D. Xiao, Bifurcations in a discrete time Lotka--Volterra predator--prey system, Discrete And Continuous Dynamical Systems-Series B, 6 (2006) 559--572.

X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator--prey system, Chaos Solitons Fractals, 32 (2007) 80--94.

X. Liu, A note on the existence of periodic solution in discrete predator-prey models, Applied Mathematical Modelling, 34 (2010) 2477--2483.

Z. He and X. Lai, Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Analysis: Real World Applications, 12 (2012) 403--417.

Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator--prey model with nonmonotonic functional response, Nonlinear Analysis: Real World Applications, 12 (2011) 2356--2377.

A. Singh and P. Deolia, Dynamical analysis and chaos control in discrete-time prey-predator model, Communications in Nonlinear Science and Numerical Simulation, 90 (2020).

R. Ma, Y. Bai and F. Wang, Dynamical behavior analysis of a two-dimensional discrete predator-prey model with prey refuge and fear factor, Journal of Applied Analysis and Computation, 10(4) (2020) 1683-1697.

A.Q. Khan and T. Khalique, Bifurcations and chaos control in a discrete-Time biological model, International Journal of Biomathematics, 13(4) (2020).

P. Chakraborty, U. Ghosh and S. Sarkar, Stability and bifurcation analysis of a discrete prey-predator model with square-root functional response and optimal harvesting, Journal of Biological Systems, 28(1) (2020) 91-110.

G. Blé, M.A. Dela-Rosa and I. Loreto-Hernández, Neimark--Sacker bifurcation analysis in an intraguild predation model with general functional responses, Journal of Difference Equations and Applications, 26(2) (2020) 223-243.

M.E. Gilpin and F.J. Ayala, Global models of growth and competition. Proceedings of the National Academy of Sciences of the USA, 70 (1973) 3590--3593.

F. Clark, B. W. Brook, S. Delean, H. R. Akcakaya and C. J. A. Bradshaw, The theta-logistic is unreliable for modelling most census data, Methods in Ecology and Evolution,1 (2010) 253--262




DOI: https://doi.org/10.34312/jjbm.v1i2.7660

Copyright (c) 2020 P.K. Santra

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