×

Turing instability and Hopf bifurcation in a modified Leslie-Gower predator-prey model with cross-diffusion. (English) Zbl 1392.35316

Summary: This paper is concerned with some mathematical analysis and numerical aspects of a reaction-diffusion system with cross-diffusion. This system models a modified version of Leslie-Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator-prey model to detect the spatial dynamics in the real life.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B36 Pattern formations in context of PDEs
35B32 Bifurcations in context of PDEs
35K57 Reaction-diffusion equations
35B10 Periodic solutions to PDEs
35B35 Stability in context of PDEs
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abid, W.; Yafia, R.; Aziz-Alaoui, M. A.; Bouhafa, H.; Abichou, A., Instability and pattern formation in three-species food chain model via Holling-type II functional response on a circular domain, Int. J. Bifurcation and Chaos, 25, 1550092-1-25, (2015) · Zbl 1317.92059
[2] Abid, W.; Yafia, R.; Aziz-Alaoui, M. A.; Bouhafa, H.; Abichou, A., Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-gower and beddington-deangelis functional type, Evol. Eqs. Contr. Th., 4, 115-129, (2015) · Zbl 1334.35113
[3] Abid, W.; Yafia, R.; Aziz-Alaoui, M. A.; Bouhafa, H.; Abichou, A., Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain, Appl. Math. Comput., 260, 292-313, (2015) · Zbl 1410.92092
[4] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, J. Theoret. Biol., 139, 311-326, (1989)
[5] Aziz-Alaoui, M. A.; Daher Okiye, M., Boundedness and global stability for a predator-prey model with modified Leslie-gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075, (2003) · Zbl 1063.34044
[6] Camara, B. I.; Aziz-Alaoui, M. A., Dynamics of predator-prey model with diffusion, Dyn. Contin. Discr. Impuls. Syst. Ser. A, 15, 897-906, (2008) · Zbl 1170.35052
[7] Camara, B. I.; Aziz-Alaoui, M. A., Turing and Hopf patterns formation in a predator-prey model with Leslie-gower type functional response, Dyn. Contin. Discr. Impuls. Syst. Ser. B, 16, 897-906, (2008) · Zbl 1170.35052
[8] Chattopadhyay, J.; Sarkar, A. K.; Tapaswi, P. K., Effect of cross-diffusion on a diffusive prey-predator system a nonlinear analysis, J. Biol. Syst., 4, 159-169, (1996)
[9] Chung, J.; Peacock-López, E., Bifurcation diagrams and Turing patterns in a chemical self-replicating reaction-diffusion system with cross diffusion, Chem. Phys., 127, 174903, (2007)
[10] Daher Okiye, M.; Aziz-Alaoui, M. A.; Capasso, V., On the dynamics of a predator-prey model with the Holling-tanner functional, Proc. ESMTB Conf., 270-278, (2002)
[11] Dancer, E. N., On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284, 729-743, (1984) · Zbl 0524.35056
[12] DeAngelis, D. L.; Goldstein, R. A.; O’Neill, R. V., A model for trophic interaction, Ecology, 56, 881-892, (1975)
[13] Dubey, B.; Das, B., A predator-prey interaction model with self and cross-diffusion, Ecol. Model., 141, 67-76, (2001)
[14] Freedman, H. I., Deterministic Mathematical Models in Population Ecology, (1987), HIFR Consulting Ltd., Edmonton · Zbl 0448.92023
[15] Henry, D., Geometric Theory of Semilinear Parabolic Equations, 840, (1981), Springer-Verlag · Zbl 0456.35001
[16] Holling, C. S., The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canad. Entomol., 91, 293-320, (1959)
[17] Hsu, S. B.; Hwang, T. W., Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type, Taiwanese J. Math., 3, 35-53, (1999) · Zbl 0935.34035
[18] Hu, G.-P.; Li, X.-L., Turing patterns of a predator-prey model with a modified Leslie-gower term and cross diffusions, Int. J. Biomath., 5, 203-219, (2012)
[19] Jin, Z.; Liu, Q.-X.; Li, L., Pattern formation induced by cross-diffusion in a predator-prey system, Chin. Phys. B, 17, 3936-3941, (2008)
[20] Kerner, E. H., Further considerations on the statistical mechanics of biological associations, Bull. Math. Biophys., 21, 217-255, (1959)
[21] Kuto, K., Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Diff. Eqs., 197, 293-314, (2004) · Zbl 1210.35122
[22] Kuto, K.; Yamada, Y., Coexistence states for a prey-predator model with cross-diffusion, Discrete and Continuous Dynamical Systems, 536-545, (2005) · Zbl 1156.35318
[23] Leslie, P., Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213-245, (1948) · Zbl 0034.23303
[24] Li, Y.; Xiao, D., Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solit. Fract., 34, 606-620, (2007) · Zbl 1156.34029
[25] Li, C., Global existence of solutions to a cross-diffusion predator-prey system with Holling-type-II functional response, Comput. Math. Appl., 65, 1152-1162, (2013) · Zbl 1319.35277
[26] Lian, X.; Huang, L., Simulation of pattern in a cross-diffusive predator-prey system with the allee effect, J. Inform. Comput. Sci., 11, 527-534, (2014)
[27] Liang, Z.; Pan, H., Qualitative analysis of a ratio-dependent Holling-tanner model, J. Math. Anal. Appl., 334, 954-964, (2007) · Zbl 1124.34030
[28] Lotka, A. J., Elements of Physical Biology, (1925), Williams and Wilkins, Baltimore · JFM 51.0416.06
[29] Murray, J. D., Mathematical Biology, (1993), Springer-Verlag, Berlin · Zbl 0779.92001
[30] Nindjin, A. F.; Aziz-Alaoui, M. A.; Cadivel, M., Analysis of a predator-prey model with modified Leslie-gower and Holling-type II schemes with time delay, Nonlin. Anal.: Real World Appl., 7, 1104-1118, (2006) · Zbl 1104.92065
[31] Okubo, A., Diffusion and Ecological Problems: Mathematical Models, 10, (1980), Springer, Berlin · Zbl 0422.92025
[32] Rosen, G., Effects of diffusion on the stability of the equilibrium in multi-species ecological systems, Bull. Math. Biol., 39, 373-383, (1977) · Zbl 0354.92013
[33] Shi, J.; Xie, Z.; Little, K., Cross-diffusion induced instability and stability in reaction-diffusion systems, J. Appl. Anal. Comput., 1, 95-119, (2011) · Zbl 1304.35329
[34] Shigesada, N.; Kawasaki, K.; Teramoto, E., Spatial segregation of interacting species, J. Theor. Biol., 79, 83-99, (1979)
[35] Sun, X.-K.; Huo, H.-F.; Xiang, H., Bifurcation and stability analysis in predator-prey model with a stage-structure for predator, Nonlin. Dyn., 58, 497-513, (2009) · Zbl 1183.92085
[36] Sun, G.-Q.; Jin, Z.; Li, L.; Haque, M.; Li, B.-L., Spatial patterns of a predator-prey model with cross-diffusion, Nonlin. Dyn., 69, 1631-1638, (2012) · Zbl 1263.34062
[37] Turing, A. M., The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. B, 237, 37-72, (1952) · Zbl 1403.92034
[38] Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie d’animali conviventi, Mem. Acad. Lincei, 2, 31-113, (1926) · JFM 52.0450.06
[39] Wang, M., Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion, Math. Biosci., 212, 149-160, (2008) · Zbl 1138.92034
[40] Wang, W.-M.; Wang, W.-J.; Lin, Y.-Z.; Tan, Y.-J., Pattern selection in a predation model with self and cross-diffusion, Chin. Phys. B, 20, 034702, (2011)
[41] Wen, Z., Turing instability and stationary patterns in a predator-prey systems with nonlinear cross-diffusions, Wen Boundary Value Problems, (2013) · Zbl 1295.35256
[42] Xu, S., Existence of global solutions for predator-prey model with cross-diffusion, Electron. J. Diff. Eqs., 06, 1-14, (2008)
[43] Yafia, R.; Aziz-Alaoui, M. A., Existence of periodic travelling waves solutions in predator-prey model with diffusion, Appl. Math. Model., 37, 3635-3644, (2013) · Zbl 1270.35380
[44] Zhou, J.; Kim, C.-G., Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling-type-II functional response, Sci. China — Math., 57, 991-1010, (2013) · Zbl 1315.35089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.