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Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior. (English) Zbl 07166434
Summary: This paper is concerned with a diffusive predator-prey model with herd behavior. The local and global stability of the unique homogeneous positive steady state \(U^*\) is obtained. Treating the conversion or consumption rate \(\gamma\) as the bifurcation parameter, the steady-state bifurcations both from simple and double eigenvalues are studied near \(U^*\). The techniques include the Lyapunov function, the spectrum analysis of operators, the bifurcation theory, space decompositions and the implicit function theorem.

MSC:
35 Partial differential equations
34 Ordinary differential equations
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[1] Volterra, V., Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali discorso letto il 4 novembre 1901 alla inaugurazione dell’anno scolastico nella r. università di roma dal prof. vito volterra, G. Degli Econ., 436-458 (1901) · JFM 32.0080.03
[2] Voherra, V., Variazione e fluttuazini del numero individui in specie animali conviventi mem, Accad. Naz. Lincei, 2, 31-113 (1926)
[3] Yang, W.; Wu, J.; Nie, H., Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14, 3 (2015) · Zbl 1422.35045
[4] Yang, W., Existence and asymptotic behavior of solutions for a predator-prey system with a nonlinear growth rate, Acta Appl. Math. (2017) · Zbl 1387.35365
[5] Morozov, A.; Petrovskii, S.; Li, B.-L., Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect., J. Theor. Biol., 238, 1, 18-35 (2006)
[6] Petrovskii, S. V.; Malchow, H., Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics, Theor. Popul. Biol., 59, 2, 157-174 (2001) · Zbl 1035.92046
[7] Ko, W.; Ryu, K., Qualitative analysis of a predator-prey model with Holling type ii functional response incorporating a prey refuge, J. Differ. Equ., 231, 2, 534-550 (2006) · Zbl 1387.35588
[8] Skalski, G. T.; Gilliam, J. F., Functional responses with predator interference: viable alternatives to the Holling type ii model, Ecology, 82, 11, 3083-3092 (2001)
[9] González-Olivares, E.; Rojas-Palma, A., Multiple limit cycles in a Gause type predator-prey model with Holling type iii functional response and Allee effect on prey, Bull. Math. Biol., 73, 6, 1378-1397 (2011) · Zbl 1215.92061
[10] Huang, Y.; Chen, F.; Zhong, L., Stability analysis of a prey-predator model with Holling type iii response function incorporating a prey refuge, Appl. Math. Comput., 182, 1, 672-683 (2006) · Zbl 1102.92056
[11] Chen, Y., Multiple periodic solutions of delayed predator-prey systems with type iv functional responses, Nonlinear Anal. Real World Appl., 5, 1, 45-53 (2004) · Zbl 1066.92050
[12] Lian, F.; Xu, Y., Hopf bifurcation analysis of a predator-prey system with Holling type iv functional response and time delay, Appl. Math. Comput., 215, 4, 1484-1495 (2009) · Zbl 1187.34116
[13] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 4, 389-406 (1998) · Zbl 0895.92032
[14] Yang, W.; Li, Y.; Wu, J.; Li, H., Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses, Discrete Contin. Dyn. Syst. - Ser. B, 20, 7, 2269-2290 (2015) · Zbl 1334.35140
[15] Liu, J., Cross-diffusion induced stationary patterns in a prey-predator system with parental care for predators, Appl. Math. Comput., 237, 176-189 (2014) · Zbl 1334.92352
[16] Ko, W.; Ahn, I., A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: II stationary pattern formation, J. Math. Anal. Appl., 397, 1, 29-45 (2013) · Zbl 1253.35189
[17] Zhang, L.; Liu, J.; Banerjee, M., Hopf and steady state bifurcation analysis in a ratio-dependent predator-prey model, Commun. Nonlinear Sci. Numer. Simul., 44, 52-73 (2017)
[18] Banerjee, M.; Petrovskii, S., Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4, 1, 37-53 (2011)
[19] Cantrell, R. S.; Cosner, C., On the dynamics of predator-prey models with the Beddington-Deangelis functional response, J. Math. Anal. Appl., 257, 1, 206-222 (2001) · Zbl 0991.34046
[20] Ivlev, V. S., Experimental ecology of the feeding of fishes (1961), Yale University Press: Yale University Press New Haven
[21] Wang, X.; Wei, J., Dynamics in a diffusive predator-prey system with strong Allee effect and Ivlev-type functional response, J. Math. Anal. Appl., 422, 2, 1447-1462 (2015) · Zbl 1310.35032
[22] Upadhyay, R. K.; Naji, R. K., Dynamics of a three species food chain model with Crowley-Martin type functional response, Chaos, Solitons & Fractals, 42, 3, 1337-1346 (2009) · Zbl 1198.37132
[23] Li, H., Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response, Comput. Math. Appl., 68, 7, 693-705 (2014) · Zbl 1362.92063
[24] Hsu, S.-B.; Hwang, T.-W.; Kuang, Y., Global dynamics of a predator-prey model with Hassell-Varley type functional response, Discrete Contin. Dyn. Syst. Ser. B, 10, 4, 857-871 (2008) · Zbl 1160.34046
[25] Chen, X.; Du, Z., Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse, Qual. Theory Dyn. Syst., 1-14 (2017)
[26] Ajraldi, V.; Pittavino, M.; Venturino, E., Modeling herd behavior in population systems, Nonlinear Anal. Real World Appl., 12, 4, 2319-2338 (2011) · Zbl 1225.49037
[27] Yuan, S.; Xu, C.; Zhang, T., Spatial dynamics in a predator-prey model with herd behavior, Chaos Interdiscip. J. Nonlinear Sci., 23, 3, 033102 (2013) · Zbl 1323.92197
[28] Tang, X.; Song, Y., Bifurcation analysis and turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality, Chaos Solitons Fract., 81, 303-314 (2015) · Zbl 1355.92098
[29] Kooi, B. W.; Venturino, E., Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey, Math. Biosci., 274, 58-72 (2016) · Zbl 1336.37029
[30] Tang, X.; Song, Y.; Zhang, T., Turing-Hopf bifurcation analysis of a predator-prey model with herd behavior and cross-diffusion, Nonlinear Dyn., 86, 1, 73-89 (2016) · Zbl 1349.37091
[31] Salman, S.; Yousef, A.; Elsadany, A., Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response, Chaos Solitons Fract., 93, 20-31 (2016) · Zbl 1372.37134
[32] Tang, X.; Song, Y., Cross-diffusion induced spatiotemporal patterns in a predator-prey model with herd behavior, Nonlinear Anal. Real World Appl., 24, 36-49 (2015) · Zbl 1336.92073
[33] Tang, X.; Song, Y., Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator-prey model with herd behavior, Appl. Math. Comput., 254, 375-391 (2015) · Zbl 1410.37092
[34] Ni, W.-M.; Tang, M., Turing patterns in the Lengyel-Epstein system for the Cima reaction, Trans. Am. Math. Soc., 357, 10, 3953-3969 (2005) · Zbl 1074.35051
[35] Wang, M., Nonlinear Elliptic Equations (Chinese) (2010), Science Press: Science Press Beijing
[36] Crandall, M. G.; Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal., 8, 2, 321-340 (1971) · Zbl 0219.46015
[37] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 3, 487-513 (1971) · Zbl 0212.16504
[38] Jang, J.; Ni, W.-M.; Tang, M., Global bifurcation and structure of turing patterns in the 1-D Lengyel-Epstein model, J. Dyn. Differ. Equ., 16, 2, 297-320 (2004) · Zbl 1072.35091
[39] Lou, Y.; Martínez, S.; Poláčik, P., Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Differ. Equ., 230, 2, 720-742 (2006) · Zbl 1154.35011
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