Spatiotemporal dynamics of a diffusive predator-prey system incorporating social behavior

Fethi Souna; Salih Djilali; Sultan Alyobi; Anwar Zeb; Nadia Gul; Suliman Alsaeed; Kottakkaran Sooppy Nisar; Fethi Souna; Salih Djilali; Sultan Alyobi; Anwar Zeb; Nadia Gul; Suliman Alsaeed; Kottakkaran Sooppy Nisar

doi:10.3934/math.2023803

AIMS Mathematics

2023, Volume 8, Issue 7: 15723-15748. doi: 10.3934/math.2023803

Previous Article Next Article

Research article Special Issues

Spatiotemporal dynamics of a diffusive predator-prey system incorporating social behavior

1.
Laboratory of biomathematics, Department of Mathematics, Djillali Liabés University, BP. 89 Sidi Bel Abbes, 22000, Algeria
2.
Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algeria
3.
Faculty of Exact and Computer Sciences, Mathematics Department, Hassiba Benbouali university, Chlef, Algeria
4.
King Abdulaziz University, College of Science & Arts, Department of Mathematics, Rabigh, Saudi Arabia
5.
Department of Mathematics, COMSATS University Islamabad, Abbottabad, 22060, Pakistan
6.
Department of Mathematics, Shaheed Benazir Bhutto Women University, Peshawar, 25000, Khyber Pakhtunkhwa, Pakistan
7.
Applied Sciences Collage, Department of Mathematical Sciences, Umm Al-Qura Univer-sity P.O. Box 715, 21955 Makkah, Saudi Arabia
8.
Mathematics Department, College of Sciences and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia
9.
School of Technology, Woxsen University- Hyderabad-502345, Telangana State, India

Received: 08 February 2023 Revised: 30 March 2023 Accepted: 05 April 2023 Published: 28 April 2023
MSC : 93A30, 34C23

This research concerned with a new formulation of a spatial predator-prey model with Leslie-Gower and Holling type II schemes in the presence of prey social behavior. The aim interest here is to distinguish the influence of Leslie-Gower term on the spatiotemporal behavior of the model. Interesting results are obtained as Hopf bifurcation, Turing bifurcation and Turing-Hopf bifurcation. A rigorous mathematical analysis shows that the presence of Leslie-Gower can induce Turing pattern, which shows that this kind of interaction is very important in modeling different natural phenomena. The direction of Turing-Hopf bifurcation is studied with the help of the normal form. The obtained results are tested numerically.
- Leslie-Gower predator-prey model,
- social behavior,
- Hopf bifurcation,
- turing pattern,
- T-H bifurcation
Citation: Fethi Souna, Salih Djilali, Sultan Alyobi, Anwar Zeb, Nadia Gul, Suliman Alsaeed, Kottakkaran Sooppy Nisar. Spatiotemporal dynamics of a diffusive predator-prey system incorporating social behavior[J]. AIMS Mathematics, 2023, 8(7): 15723-15748. doi: 10.3934/math.2023803

Related Papers:

Abstract

This research concerned with a new formulation of a spatial predator-prey model with Leslie-Gower and Holling type II schemes in the presence of prey social behavior. The aim interest here is to distinguish the influence of Leslie-Gower term on the spatiotemporal behavior of the model. Interesting results are obtained as Hopf bifurcation, Turing bifurcation and Turing-Hopf bifurcation. A rigorous mathematical analysis shows that the presence of Leslie-Gower can induce Turing pattern, which shows that this kind of interaction is very important in modeling different natural phenomena. The direction of Turing-Hopf bifurcation is studied with the help of the normal form. The obtained results are tested numerically.

References

[1]	V. Ajraldi, M. Pittavino, E. Venturino, Modelling herd behaviour in population systems. Nonlinear Anal. Real World Appl., 12 (2011), 2319–2338. https://doi.org/10.1016/j.nonrwa.2011.02.002
[2]	P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal. Real World Appl., 13 (2012), 1837–1843. https://doi.org/10.1016/j.nonrwa.2011.12.014 doi: 10.1016/j.nonrwa.2011.12.014
[3]	I. M. Bulai, E. Venturino, Shape effects on herd behavior in ecological interacting population models, Math. Comput. Simulat., 141 (2017), 40–55. https://doi.org/10.1016/j.matcom.2017.04.009 doi: 10.1016/j.matcom.2017.04.009
[4]	I. Boudjema, S. Djilali, Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application, Nonlinear Stud., 25 (2018), 665–687.
[5]	S. Djilali, Herd behavior in a predator-prey model with spatial diffusion: bifurcation analysis and Turing instability, J. Appl. Math. Comput., 58 (2018), 125–149. https://doi.org/10.1007/s12190-017-1137-9 doi: 10.1007/s12190-017-1137-9
[6]	S. Djilali, Impact of prey herd shape on the predator-prey interaction, Chaos, Solitons Fract., 120 (2019), 139–148. https://doi.org/10.1016/j.chaos.2019.01.022 doi: 10.1016/j.chaos.2019.01.022
[7]	S. Djilali, Effect of herd shape in a diffusive predator-prey model with time delay, J. Appl. Anal. Comput., 9 (2019), 638–654.
[8]	S. Djilali, S. Bentout, Spatiotemporal patterns in a diffusive predator-prey model with prey social behavior, Acta. Appl. Math., 169 (2020), 125–143. https://doi.org/10.1007/s10440-019-00291-z doi: 10.1007/s10440-019-00291-z
[9]	S. Djilali, Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition, Math. Meth. Appl. Sci., 43 (2020), 2233–2250. https://doi.org/10.1002/mma.6036 doi: 10.1002/mma.6036
[10]	S. Djilali, Spatiotemporal patterns induced by cross-diffusion in predator-prey model with prey herd shape effect, Int. J. Biomath., 13 (2020), 2050030. https://doi.org/10.1142/S1793524520500308 doi: 10.1142/S1793524520500308
[11]	S. Djilali, B. Ghanbari, S. Bentout, A. Mezouaghi, Turing-Hopf bifurcation in a diffusive Mussel-Algae model with time-fractional-order derivative, Chaos, Solitons Fract., 138 (2020) 109954. https://doi.org/10.1016/j.chaos.2020.109954
[12]	B. Ghanabri, S. Djilali, Mathematical and numerical analysis of a three-species predator-prey model with herd behavior and time fractional-order derivative, Math. Meth. Appl. Sci., 43 (2020), 1736–1752. https://doi.org/10.1002/mma.5999 doi: 10.1002/mma.5999
[13]	B. Ghanabri, S. Djilali, Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population, Chaos, Solitons Fract., 138 (2020), 109960. https://doi.org/10.1016/j.chaos.2020.109960 doi: 10.1016/j.chaos.2020.109960
[14]	B. Ghanbari, S. Kumar, R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos, Solitons Fract., 133 (2020), 109619. https://doi.org/10.1016/j.chaos.2020.109619 doi: 10.1016/j.chaos.2020.109619
[15]	J. Gine, C. Valls, Nonlinear oscillations in the modified Leslie-Gower model, Nonlinear Anal. Real World Appl., 51 (2020), 103010. https://doi.org/10.1016/j.nonrwa.2019.103010 doi: 10.1016/j.nonrwa.2019.103010
[16]	E. F. D. Goufo, S. Kumar, S. B. Mugisha, Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos, Solitons Fract., 130 (2020), 109467. https://doi.org/10.1016/j.chaos.2019.109467 doi: 10.1016/j.chaos.2019.109467
[17]	C. S. Holling, The functional response of invertebrate predator to prey density, Mem. Entomol. Soc. Canada, 98 (1966), 5–86. https://doi.org/10.4039/entm9848fv doi: 10.4039/entm9848fv
[18]	C. A. Ibarra, J. Flores, Dynamics of a Leslie-Gower predator-prey model with Holling type II functional response, Allee effect and a generalist predator, Math. Comput. Simul., 188 (2021), 1–22. https://doi.org/10.1016/j.matcom.2021.03.035 doi: 10.1016/j.matcom.2021.03.035
[19]	W. Jiang, Q. An, J. Shi, Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay, J. Differ. Equ., 268 (2020), 6067–6102. https://doi.org/10.1016/j.jde.2019.11.039 doi: 10.1016/j.jde.2019.11.039
[20]	W. Jiang, H. Wang, X. Cao, Turing instability and turing-hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay, J. Dyn. Differ. Equ., 31 (2019), 2223–2247. https://doi.org/10.1007/s10884-018-9702-y doi: 10.1007/s10884-018-9702-y
[21]	S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model., 38 (2014), 3154–3163. https://doi.org/10.1016/j.apm.2013.11.035 doi: 10.1016/j.apm.2013.11.035
[22]	S. Kumar, A. Kumar, D. Baleanu, Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger's equations arise in propagation of shallow water waves, Nonlinear Dyn., 85 (2016), 699–715. https://doi.org/10.1007/s11071-016-2716-2 doi: 10.1007/s11071-016-2716-2
[23]	S. Kumar, M. M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun., 185 (2014), 1947–1954. https://doi.org/10.1016/j.cpc.2014.03.025 doi: 10.1016/j.cpc.2014.03.025
[24]	S. Kumar, D. Kumar, S. Abbasbandy, M. M. Rashidide, Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method, Ain Shams Eng. J., 5 (2014), 569–574. https://doi.org/10.1016/j.asej.2013.11.004 doi: 10.1016/j.asej.2013.11.004
[25]	S. Kumar, S. Ghosh, B. Samet, An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator, Math. Meth. Appl. Sci., 43 (2020), 6062–6080. https://doi.org/10.1002/mma.6347 doi: 10.1002/mma.6347
[26]	S. Kumar, R. Kumar, R. P. Agarwal, B. Samet, A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods, Math. Meth. Appl. Sci., 43 (2020), 5564–5578. https://doi.org/10.1002/mma.6297 doi: 10.1002/mma.6297
[27]	Y. Liu, J. Wei, Spatiotemporal dynamics of a modified Leslie-Gower model with weak allee effect, Int. J. Bifurcat. Chaos, 30 (2020), 2050169. https://doi.org/10.1142/S0218127420501692 doi: 10.1142/S0218127420501692
[28]	Y. Li, F. Zhang, X. Zhuo, Flip bifurcation of a discrete predator-prey model with modified Leslie-Gower and Holling-type $III$ schemes, Math. Biosci. Eng., 17 (2019), 2003–2015. https://doi.org/10.3934/mbe.2020106 doi: 10.3934/mbe.2020106
[29]	C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal, Appl., 198 (1996), 751–779. https://doi.org/10.1006/jmaa.1996.0111 doi: 10.1006/jmaa.1996.0111
[30]	M. M. Rashidi, A. Hosseini, I. Pop, S. Kumar, N. Freidoonimehr, Comparative numerical study of single and two-phase models of nano-fluid heat transfer in wavy channel, Appl. Math. Mech.-Engl. Ed., 35 (2014), 831–848. https://doi.org/10.1007/s10483-014-1839-9 doi: 10.1007/s10483-014-1839-9
[31]	F. Souna, A. Lakmeche, S. Djilali, The effect of the defensive strategy taken by the prey on predator-prey interaction, J. Appl. Math. Comput., 64 (2020), 665–690. https://doi.org/10.1007/s12190-020-01373-0 doi: 10.1007/s12190-020-01373-0
[32]	F. Souna, A. Lakmeche, S. Djilali, Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting, Chaos, Solitons Fract., 140 (2020), 110180. https://doi.org/10.1016/j.chaos.2020.110180 doi: 10.1016/j.chaos.2020.110180
[33]	F. Souna, S. Djilali, A. Lakmeche, Spatiotemporal behavior in a predator-prey model with herd behavior and cross-diffusion and fear effect, Eur. Phys. J. Plus, 136 (2021), 474. https://doi.org/10.1140/epjp/s13360-021-01489-7 doi: 10.1140/epjp/s13360-021-01489-7
[34]	M. E. Taylor, Partial differential equations $III$–Nonlinear equations, Applied Mathematical Science, Springer-Verlag, 1996.
[35]	S. Wang, Z. Xie, R. Zhong, Y. Wu, Stochastic analysis of a predator-prey model with modified Leslie-Gower and Holling type $II$ schemes, Nonlinear Dyn., 101 (2020), 1245–1262. https://doi.org/10.1007/s11071-020-05803-3 doi: 10.1007/s11071-020-05803-3
[36]	D. Xu, M. Liu, X. Xu, Analysis of a stochastic predator-prey system with modified Leslie-Gower and Holling-type $IV$ schemes, Phys. A, 537 (2020), 122761. https://doi.org/10.1016/j.physa.2019.122761 doi: 10.1016/j.physa.2019.122761
[37]	C. Xu, S. Yuan, T. Zhang, Global dynamics of a predator-prey model with defence mechanism for prey, Appl. Math. Lett., 62 (2016), 42–48. https://doi.org/10.1016/j.aml.2016.06.013 doi: 10.1016/j.aml.2016.06.013

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)