×

Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. (English) Zbl 1480.92173

Summary: The present paper considers a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. The existence of the nontrivial positive equilibria is discussed, and some sufficient conditions for locally asymptotically stability of one of the positive equilibria are developed. Meanwhile, the existence of Hopf bifurcation is discussed by choosing time delays as the bifurcation parameters. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out to support the analytical results.

MSC:

92D25 Population dynamics (general)
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Ávila-Vales, Á. Estrella-González and E. Rivero-Esquivel, Bifurcations of a Leslie Gower predator prey model with Holling type Ⅲ functional response and Michaelis-Menten prey harvesting, arXiv: 1711.08081v1.
[2] A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73, 1530-1535 (1992)
[3] Ả. Brännström; D. Sumpter, The role of competition and clustering in population dynamics, Proc. Biol. Sci., 272, 2065-2072 (2005)
[4] J. Z. Cao; H. Y. Sun, Bifurcation analysis for the Kaldor-Kalecki model with two delays, Adv. Differ. Equ., 107, 1-27 (2019) · Zbl 1459.34162
[5] J. Z. Cao; R. Yuan, Bifurcation analysis in a modified Lesile-Gower model with Holling type Ⅱ functional response and delay, Nonlinear Dynamics, 84, 1341-1352 (2016) · Zbl 1354.92059
[6] J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology, 50, 188-192 (1969)
[7] B. S. Chen and J. J. Chen, Complex dynamic behaviors of a discrete predator-prey model with stage structure and harvesting, Int. J. Biomath., 10 (2017), 1750013, 25 pp. · Zbl 1366.92101
[8] C. W. Clark; M. Mangei, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77, 317-337 (1979)
[9] S. Creel; E. Dröge; J. M’soka; D. Smit; M. Becker; D. Christianson; P. Schuette, The relationship between direct predation and antipredator responses: a test with multiple predators and multiple prey, Ecology, 98, 2081-2092 (2017)
[10] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Berlin Heidelberg New York, 1977. · Zbl 0363.92014
[11] V. Doudoumis, U. Alam and E. Aksoy, et al., Tsetse-Wolbachia symbiosis: Comes of age and has great potential for pest and disease control, J. Invertebr. Pathol., 112 (2013), S94-S103.
[12] M. K. A. Gavina, T. Tahara and K. Tainaka, et al., Multi-species coexistence in Lotka-Volterra competitive systems with crowding effects, Sci. Rep., 8 (2018), 1198.
[13] F. Groenewoud; J. G. Frommen; D. Josi; H. Tanaka; A. Jungwirth; M. Taborsky, Predation risk drives social complexity in cooperative breeders, Proc. Natl.Acad. Sci., 113, 4104-4109 (2016)
[14] Y. X. Guo, N. N. Ji and B. Niu, Hopf bifurcation analysis in a predator-prey model with time delay and food subsidies, Adv. Differ. Equ., 2019 (2019), Paper No. 99, 22 pp. · Zbl 1458.37097
[15] R. P. Gupta; M. Banerjee; P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20, 339-366 (2012) · Zbl 1262.34048
[16] R. P. Gupta; P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398, 278-295 (2013) · Zbl 1259.34035
[17] D. P. Hu and H. J. Cao, Stability and Hopf bifurcation analysis in Hindmarsh-Rose neuron model with multiple time delays, J. Math. Anal. Appl., 11 (2016), 1650187, 27pp. · Zbl 1349.34334
[18] D. P. Hu; H. J. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvest, Nonlinear Anal-RWA., 33, 58-82 (2017) · Zbl 1352.92125
[19] S. Khajanchi, Modeling the dynamics of stage-structure predator-prey system with Monod-Haldane type response function, Appl. Math. Comput., 302, 122-143 (2017) · Zbl 1411.34101
[20] L. Kong; C. R. Zhu, Bogdanov-Takens bifurcations of codimensions 2 and 3 in a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting, Math. Method. Appl. Sci., 40, 6715-6731 (2017) · Zbl 1384.34057
[21] P. Lenzini; J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., 4, 791-803 (2010) · Zbl 1189.37098
[22] L. Z. Li; F. W. Meng; P. J. Ju, Some new integral inequalities and their applications in studying the stability of nonlinear integro differential equations with time delay, J. Math. Anal. Appl., 377, 853-862 (2011) · Zbl 1256.45003
[23] Y. N. Li; Y. G. Sun; F. W. Meng, New criteria for exponential stability of switched time varying systems with delays and nonlinear disturbances, Nonlinear Anal-Hybri., 26, 284-291 (2017) · Zbl 1373.93271
[24] Y. Li; M. X. Wang, Dynamics of a diffusive predator-prey model with modified Leslie-Gower term and Michaelis-Menten type prey harvesting, Acta Appl. Math., 140, 147-172 (2015) · Zbl 1336.35339
[25] B. Liu; R. C. Wu; L. P. Chen, Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting, Math. Biosci., 298, 71-79 (2018) · Zbl 1392.92079
[26] Y. Liu, L. Zhao, X. Y. Huang and H. Deng, Stability and bifurcation analysis of two species amensalism model with Michaelis-Menten type harvesting and a cover for the first species, Adv. Differ. Equ., 2018 (2018), Paper No. 295, 19 pp. · Zbl 1448.92225
[27] J. F. Luo and Y. Zhao, Stability and bifurcation analysis in a predator-prey system with constant harvesting and prey group defense, Int. J. Bifurcat. Chaos, 27 (2017), 1750179, 26pp. · Zbl 1379.92051
[28] Z. H. Ma and S. F. Wang, A delay-induced predator-prey model with Holling type functional response and habitat complexity, Nonlinear Dyn., 93 2018), 1519-1544. · Zbl 1398.92212
[29] R. M. May; J. R. Beddington; C. W. Clark; S. J. Holt; R. M. Laws, Management of multispecies fisheries, Science, 205, 267-277 (1979)
[30] M. Peng, Z. D. Zhang and X. D. Wang, Hybrid control of Hopf bifurcation in a Lotka-Volterra predator-prey model with two delays, Adv. Differ. Equ., 2017 (2017), Paper No. 387, 20 pp. · Zbl 1444.37081
[31] S. N. Raw; P. Mishra; R. Kumar; S. Thakur, Complex behavior of prey-predator system exhibiting group defense: A mathematical modeling study, Chaos Soliton Fract., 100, 74-90 (2017) · Zbl 1373.92110
[32] M. Sen; P. D. N. Srinivasu; M. Banerjee, Global dynamics of an additional food provided predator-prey system with constant harvest in predators, Appl. Math. Comput., 250, 193-211 (2015) · Zbl 1328.37059
[33] J. Shao and F. W. Meng, Oscillation theorems for second order forced neutral nonlinear differential equations with delayed argument, Int. J. Differ. Equ., 2010 (2010), article ID 181784, 1-15. · Zbl 1207.34084
[34] F. E. Smith, Population dynamics in Daphnia Magna and a new model for population growth, Ecology, 44, 651-663 (1963)
[35] Q. N. Song, R. Z. Yang, C. R. Zhang and L. Y. Tang, Bifurcation analysis in a diffusive predator-prey system with Michaelis-Menten-type predator harvesting, Adv. Differ. Equ., 2018 (2018), Paper No. 329, 15 pp. · Zbl 1448.92250
[36] Y. G. Sun and F. W. Meng, Reachable set estimatyion for a class of nonlinear time varying systems, Complexity, 2017 (2017), Article ID 5876371, 6pp. · Zbl 1373.93061
[37] J. M. Wang, H. D. Cheng, H. X. Liu and Y. H. Wang, Periodic solution and control optimization of a prey-predator model with two types of harvesting, Adv. Differ. Equ., 2018 (2018), Paper No. 41, 14 pp. · Zbl 1445.37070
[38] Z. Wang; Y. K. Xie; J. W. Lu; Y. X. Li, Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, Appl. Math. Comput., 347, 360-369 (2019) · Zbl 1428.92094
[39] R. C. Wu; M. X. Chen; B. Liu; L. P. Chen, Hopf bifurcation and Turing instability in a predator-prey model with Michaelis-Menten functional response, Nonlinear Dyn., 91, 2033-2047 (2018) · Zbl 1390.37147
[40] D. M. Xiao; W. X. Li; M. A. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324, 14-29 (2006) · Zbl 1122.34035
[41] R. Z. Yang, C. R. Zhang and Y. Z. Zhang, A delayed diffusive predator-prey system with Michaelis-Menten type predator harvesting, Int. J. Bifurcat. Chaos, 28 (2018), 1850099, 14pp. · Zbl 1395.35186
[42] R. Yuan; W. H. Jiang; Y. Wang, Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting, J. Math. Anal. Appl., 422, 1072-1090 (2015) · Zbl 1306.34132
[43] S. L. Yuan; X. H. Ji; H. P. Zhu, Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations, Math. Biosci. Eng., 14, 1477-1498 (2017) · Zbl 1364.65011
[44] C. H. Zhang; X. P. Yan; G. H. Cui, Hopf bifucations in a predator-prey system with a discrete delay and a distributed delay, Nonlinear Anal-RWA., 11, 4141-4153 (2010) · Zbl 1206.34104
[45] C. R. Zhu; K. Q. Lan, Phase portraits, Hopf-bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin, Dyn. Syst. Ser. B, 14, 289-306 (2010) · Zbl 1201.34065
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.