Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping

Fethi Souna¹ ; Salih Djilali^2, 3 ; Fayssal Charif^4, 5

¹ Biomathematics Laboratory, University of Djillali Liabes, Sidi-Bel-Abbès, Algeria.
² Faculty of Exact Sciences and Informatics, Mathematic Department, University of Hassiba Benbouali, Chlef, Algeria.
³ Laboratoire d’Analyse Non Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algeria.
⁴ Faculty of sciences, Mathematic Department, University of Moulay Tahar, Saida, Algeria.
⁵ Laboratoire de Géométrie, Analyse, Contrôle et Applications, University of Moulay Tahar, Saida, Algeria.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 23

Résumé

In this paper, we consider a new approach of prey escaping from herd in a predator-prey model with the presence of spatial diffusion. First, the sensitivity of the equilibrium state density with respect to the escaping rate has been studied. Then, the analysis of the non diffusive system was investigated where boundedness, local, global stability, Hopf bifurcation are obtained. Besides, for the diffusive system, we proved the occurrence of Hopf bifurcation and the non existence of diffusion driven instability. Furthermore, the direction of Hopf bifurcation has been proved using the normal form on the center manifold. Some numerical simulations have been used to illustrate the obtained results.

Détail

DOI : 10.1051/mmnp/2019044

Bibliographie

[1] V. Ajraldi, M. Pittavino, E. Venturino Modeling herd behavior in population systems Nonlin. Anal. Real World Appl 2011 2319 2338

[2] A. Bahar, X. Mao Stochastic delay Lotka-Volterra model J. Math. Anal. Appl 2004 364 380

[3] P.A. Braza Predator-prey dynamics with square root functional responses Nonlin. Anal. Real World Appl 2012 1837 1843

[4] J. Carr, Applications of Center Manifold Theory. SpringerVerlag, New York (1981).

[5] S.N. Chow and J.K. Hale, Methods of Bifurcation Theory. Springer, New York (1982).

[6] E. Cagliero, E. Venturino Ecoepidemics with infected prey in herd defense: the harmless and toxic cases Int. J. Comput. Math 2016 108 127

[7] S. Djilali Herd behavior in a predator-prey model with spatial diffusion: bifurcation analysis and Turing instability J. Appl. Math. Comp 2017 125 149

[8] S. Djilali Impact of prey herd shape on the predator-prey interaction Chaos Solitons Fractals 2019 139 148

[9] S. Djilali Effect of herd shape in a diffusive predator-prey model with time delay J. Appl. Anal. Comput 2019 638 654

[10] S. Djilali, S. Bentout Spatiotemporal patterns in a diffusive predator-prey model with prey social behavior Acta Appl. Math 2019

[11] S. Djilali, T.M. Touaoula, S.E.H. Miri A Heroin epidemic model: very general non linear incidence, treat-age, and global stability Acta Appl. Math 2017 171 194

[12] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981).

[13] Y. Huang, F. Chen, L. Zhong Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge Appl. Math. Comput 2006 672 683

[14] G. Huang, W. Ma, Y. Takeuchi Global properties for virus dynamics model with Beddington-DeAngelis functional response Appl. Math. Lett 2009 1690 1693

[15] W. Ko, K. Ryu Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge J. Differ. Equ 2006 534 550

[16] Y. Kuang, E. Beretta Global qualitative analysis of a ratio-dependent predator-prey system J. Math. Biol 1998 389 406

[17] C. Li A diffusive Holling-Tanner prey-predator model with free boundary Int. J. Biomath 2018 1850066

[18] Z. Ma, S. Wang A delay-induced predator-prey model with Holling type interaction functional response and habitat complexity Nonlin. Dyn 2018 1519 1544

[19] I. Martina, E. Venturino Shape effects on herd behavior in ecological interacting population models Math. Comput. Simul 2017 40 55

[20] Q. Meng, L. Yang Steady state in a cross-diffusion predator-prey model with the Beddington-DeAngelis functional response Nonlinear Anal. Real World Appl 2019 401 413

[21] A. Mousaoui, S. Bassaid, E.H.A. Dads The impact of water level fluctuations on a delayed prey-predator model Nonlin. Anal. Real World Appl 2015 170 184

[22] K. Ryu, W. Ko, M. Haque Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities Nonlin. Dyn 2018 1639 1656

[23] H.-B. Shi, W.-T. Li, G. Lin Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response Nonlin. Anal. Real World Appl 2010 3711 3721

[24] Y. Song, X. Tang Stability, steady-state bifurcation, and Turing patterns in predator-prey model with herd behavior and prey-taxis Stud. Appl. Math 2017 371 404

[25] Y. Song, X. Zou Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point Comput. Math. Appl 2014 1978 1997

[26] X. Tang, Y. Song Bifurcation analysis and Turing instability in a diffusive predator prey model with herd behavior and hyperbolic mortality Chaos Solit. Frac 2015 303 314

[27] X. Tang, Y. Song Turing-hopf bifurcation analysis of a predator-prey model with herd behavior and cross-diffusion Nonlinear Dyn 2016 73 89

[28] E. Venturino A minimal model for ecoepidemics with group defense J. Biol. Syst 2011 763 85

[29] E. Venturino Modeling herd behavior in population systems Nonlin. Anal. Real World Appl 2013 2319 38

[30] E. Venturino, S. Petrovskii Spatiotemporal behavior of a prey-predator system with a group defense for prey Ecol. Complex 2013 37 47

[31] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York (1991).

[32] Z. Xu, Y. Song Bifurcation analysis of a diffusive predator-prey system with a herd behavior and quadratic mortality Math. Meth. Appl. Sci 2015 2994 3006

[33] C. Xu, S. Yuan, T. Zhang Global dynamics of a predator-prey model with defense mechanism for prey Appl. Math. Lett 2016 42 48

[34] R. Yang, M. Liu, C. Zhang A diffusive predator-prey system with additional food and intra-specific competition among predators Int. J. Biomath 2018 1850060

[35] W. Yang Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior Appl. Math. Model 2018 433 446

[36] S. Yuan, C. Xu, T. Zhang Spatial dynamics in a predator-prey model with herd behavior Chaos 2013 033102

[37] J. Zhang, W. Li, X. Yan Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models Appl. Math. Model 2011 1883 1893

[38] H. Zhu, X. Zhang Dynamics and patterns of a diffusive prey-predator system with a group defence for prey Disc. Dyn. Nat. Soc 2018 6519696