×

A delayed diffusive predator-prey system with predator cannibalism. (English) Zbl 1409.92201

Summary: A diffusive predator-prey system with cannibalism and maturation delay in predator subject to Neumann boundary condition is studied in this paper. Instability and Hopf bifurcation induced by time delay are investigated. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived. Some numerical simulations are given to support our results.

MSC:

92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Yi, F.; Wei, J.; Shi, J., Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246, 5, 1944-1977 (2009) · Zbl 1203.35030
[2] Min, N.; Wang, M., Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72, 6, 1670-1689 (2016) · Zbl 1359.92098
[3] Yuan, R.; Jiang, W.; Wang, Y., Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, J. Math. Anal. Appl., 422, 2, 1072-1090 (2015) · Zbl 1306.34132
[4] Wang, J.; Shi, J.; Wei, J., Dynamics and pattern formation in a difusive predator-prey system with strong Allee efect in prey, J. Differential Equations, 251, 1276-1304 (2011) · Zbl 1228.35037
[5] Ma, X.; Shao, Y.; Wang, Z., An impulsive two-stage predator-prey model with stage-structure and square root functional responses, Math. Comput. Simul., 119, C, 91-107 (2016) · Zbl 07313593
[6] Kang, Y.; Feng, P., Dynamics of a predator-prey model with modified Leslie-Gower model with double Allee effects, Nonlinear Dynam., 80, 1, 1051-1062 (2015) · Zbl 1345.92115
[7] Real, L. A., The kinetics of functional response, Am. Nat., 111, 978, 289-300 (1977)
[8] Boukal, D. S.; Sabelis, M. W.; Berec, L., How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses, Theor. Popul. Biol., 72, 1, 136-147 (2007) · Zbl 1123.92034
[9] Peng, R.; Shi, J., Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations, 247, 3, 866-886 (2009) · Zbl 1169.35328
[10] Ko, W.; Ryu, K., Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differential Equations, 231, 2, 534-550 (2006) · Zbl 1387.35588
[11] Chen, S.; Shi, J.; Wei, J., The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response, Commun. Pure Appl. Anal., 12, 1, 481-501 (2013) · Zbl 1264.35120
[12] Meffe, G. K., Density-dependent cannibalism in the endangered sonoran topminnow (Poeciliopsis occidentalis), Southwest. Nat., 29, 4, 500 (1984)
[13] (Elgar, M. A.; Crespi, B. J., Cannibalism: Ecology and Evolution Among Diverse Taxa (1992), Oxford University Press: Oxford University Press Oxford)
[14] Sun, G. Q.; Zhang, G.; Jin, Z., Predator cannibalism can give rise to regular spatial pattern in a predator-prey system, Nonlinear Dynam., 58, 1, 75-84 (2009) · Zbl 1183.92084
[15] Yuan, R.; Jiang, W.; Wang, Y., Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, J. Math. Anal. Appl., 422, 2, 1072-1090 (2015) · Zbl 1306.34132
[16] Ma, Y., Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays, Nonlinear Anal. RWA, 13, 1, 370-375 (2012) · Zbl 1238.34130
[17] Jiang, J.; Song, Y., Delay-induced Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with nonmonotonic functional response, Commun. Nonlinear Sci. Numer. Simul., 19, 7, 2454-2465 (2014) · Zbl 1457.92140
[18] Xiao, A.; Zhang, G.; Zhou, J., Implicit-explicit time discretization coupled with finite element methods for delayed predator-prey competition reaction-diffusion system, Comput. Math. Appl., 71, 10, 2106-2123 (2016) · Zbl 1443.92171
[19] Ma, Z. P.; Yue, J. L., Competitive exclusion and coexistence of a delayed reaction-diffusion system modeling two predators competing for one prey, Comput. Math. Appl., 71, 9, 1799-1817 (2016) · Zbl 1443.92164
[20] Wu, J., Theory and Applications of Partial Functional Differential Equations (1996), Springer: Springer Berlin · Zbl 0870.35116
[21] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge-New York · Zbl 0474.34002
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.