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doi: 10.3934/dcdss.2020259

Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

*Corresponding author: Fanwei Meng

Received  May 2019 Revised  September 2019 Published  February 2020

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.

Citation: Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020259
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. Google Scholar

[2]

A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.  doi: 10.2307/1940005.  Google Scholar

[3]

Ả. Brännström and D. Sumpter, The role of competition and clustering in population dynamics, Proc. Biol. Sci., 272 (2005), 2065-2072.   Google Scholar

[4]

J. Z. Cao and H. Y. Sun, Bifurcation analysis for the Kaldor-Kalecki model with two delays, Adv. Differ. Equ., 107 (2019), 1-27.   Google Scholar

[5]

J. Z. Cao and R. Yuan, Bifurcation analysis in a modified Lesile-Gower model with Holling type Ⅱ functional response and delay, Nonlinear Dynamics, 84 (2016), 1341-1352.  doi: 10.1007/s11071-015-2572-5.  Google Scholar

[6]

J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.  doi: 10.2307/1934845.  Google Scholar

[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. doi: 10.1142/S1793524517500139.  Google Scholar

[8]

C. W. Clark and M. Mangei, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979), 317-337.   Google Scholar

[9]

S. CreelE. DrögeJ. M'sokaD. SmitM. BeckerD. Christianson and P. Schuette, The relationship between direct predation and antipredator responses: a test with multiple predators and multiple prey, Ecology, 98 (2017), 2081-2092.  doi: 10.1002/ecy.1885.  Google Scholar

[10]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Berlin Heidelberg New York, 1977. doi: 10.1007/978-3-642-93073-7.  Google Scholar

[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. doi: 10.1016/j.jip.2012.05.010.  Google Scholar

[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. doi: 10.1038/s41598-017-19044-9.  Google Scholar

[13]

F. GroenewoudJ. G. FrommenD. JosiH. TanakaA. Jungwirth and M. Taborsky, Predation risk drives social complexity in cooperative breeders, Proc. Natl.Acad. Sci., 113 (2016), 4104-4109.  doi: 10.1073/pnas.1524178113.  Google Scholar

[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. doi: 10.1186/s13662-019-2050-3.  Google Scholar

[15]

R. P. GuptaM. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.  doi: 10.1007/s12591-012-0142-6.  Google Scholar

[16]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.  doi: 10.1016/j.jmaa.2012.08.057.  Google Scholar

[17] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifucation, Cambridge University Press, Cambridge, 1981.   Google Scholar
[18]

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. doi: 10.1142/S021812741650187X.  Google Scholar

[19]

D. P. Hu and H. J. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvest, Nonlinear Anal-RWA., 33 (2017), 58-82.  doi: 10.1016/j.nonrwa.2016.05.010.  Google Scholar

[20]

S. Khajanchi, Modeling the dynamics of stage-structure predator-prey system with Monod-Haldane type response function, Appl. Math. Comput., 302 (2017), 122-143.  doi: 10.1016/j.amc.2017.01.019.  Google Scholar

[21]

L. Kong and 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 (2017), 6715-6731.  doi: 10.1002/mma.4484.  Google Scholar

[22] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.   Google Scholar
[23]

P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., 4 (2010), 791-803.   Google Scholar

[24]

L. Z. LiF. W. Meng and 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 (2011), 853-862.  doi: 10.1016/j.jmaa.2010.12.002.  Google Scholar

[25]

Y. N. LiY. G. Sun and F. W. Meng, New criteria for exponential stability of switched time varying systems with delays and nonlinear disturbances, Nonlinear Anal-Hybri., 26 (2017), 284-291.  doi: 10.1016/j.nahs.2017.06.007.  Google Scholar

[26]

Y. Li and 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 (2015), 147-172.  doi: 10.1007/s10440-014-9983-z.  Google Scholar

[27]

B. LiuR. C. Wu and L. P. Chen, Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting, Math. Biosci., 298 (2018), 71-79.  doi: 10.1016/j.mbs.2018.02.002.  Google Scholar

[28]

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. doi: 10.1186/s13662-018-1752-2.  Google Scholar

[29]

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. doi: 10.1142/S0218127417501796.  Google Scholar

[30]

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. doi: 10.1007/s11071-018-4274-2.  Google Scholar

[31]

R. M. MayJ. R. BeddingtonC. W. ClarkS. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277.  doi: 10.1126/science.205.4403.267.  Google Scholar

[32]

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. doi: 10.1186/s13662-017-1434-5.  Google Scholar

[33]

S. N. RawP. MishraR. Kumar and S. Thakur, Complex behavior of prey-predator system exhibiting group defense: A mathematical modeling study, Chaos Soliton Fract., 100 (2017), 74-90.  doi: 10.1016/j.chaos.2017.05.010.  Google Scholar

[34]

M. SenP. D. N. Srinivasu and M. Banerjee, Global dynamics of an additional food provided predator-prey system with constant harvest in predators, Appl. Math. Comput., 250 (2015), 193-211.  doi: 10.1016/j.amc.2014.10.085.  Google Scholar

[35]

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. doi: 10.1155/2010/181784.  Google Scholar

[36]

F. E. Smith, Population dynamics in Daphnia Magna and a new model for population growth, Ecology, 44 (1963), 651-663.  doi: 10.2307/1933011.  Google Scholar

[37]

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. doi: 10.1186/s13662-018-1741-5.  Google Scholar

[38]

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. doi: 10.1155/2017/5876371.  Google Scholar

[39]

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. doi: 10.1186/s13662-018-1499-9.  Google Scholar

[40]

Z. WangY. K. XieJ. W. Lu and Y. X. Li, Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, Appl. Math. Comput., 347 (2019), 360-369.  doi: 10.1016/j.amc.2018.11.016.  Google Scholar

[41]

R. C. WuM. X. ChenB. Liu and L. P. Chen, Hopf bifurcation and Turing instability in a predator-prey model with Michaelis-Menten functional response, Nonlinear Dyn., 91 (2018), 2033-2047.  doi: 10.1007/s11071-017-4001-4.  Google Scholar

[42]

D. M. XiaoW. X. Li and M. A. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.  doi: 10.1016/j.jmaa.2005.11.048.  Google Scholar

[43]

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. doi: 10.1142/S0218127418500992.  Google Scholar

[44]

R. YuanW. H. Jiang and 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 (2015), 1072-1090.  doi: 10.1016/j.jmaa.2014.09.037.  Google Scholar

[45]

S. L. YuanX. H. Ji and H. P. Zhu, Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations, Math. Biosci. Eng., 14 (2017), 1477-1498.  doi: 10.3934/mbe.2017077.  Google Scholar

[46]

C. H. ZhangX. P. Yan and G. H. Cui, Hopf bifucations in a predator-prey system with a discrete delay and a distributed delay, Nonlinear Anal-RWA., 11 (2010), 4141-4153.  doi: 10.1016/j.nonrwa.2010.05.001.  Google Scholar

[47]

C. R. Zhu and 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 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.  Google Scholar

show all references

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. Google Scholar

[2]

A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.  doi: 10.2307/1940005.  Google Scholar

[3]

Ả. Brännström and D. Sumpter, The role of competition and clustering in population dynamics, Proc. Biol. Sci., 272 (2005), 2065-2072.   Google Scholar

[4]

J. Z. Cao and H. Y. Sun, Bifurcation analysis for the Kaldor-Kalecki model with two delays, Adv. Differ. Equ., 107 (2019), 1-27.   Google Scholar

[5]

J. Z. Cao and R. Yuan, Bifurcation analysis in a modified Lesile-Gower model with Holling type Ⅱ functional response and delay, Nonlinear Dynamics, 84 (2016), 1341-1352.  doi: 10.1007/s11071-015-2572-5.  Google Scholar

[6]

J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.  doi: 10.2307/1934845.  Google Scholar

[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. doi: 10.1142/S1793524517500139.  Google Scholar

[8]

C. W. Clark and M. Mangei, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979), 317-337.   Google Scholar

[9]

S. CreelE. DrögeJ. M'sokaD. SmitM. BeckerD. Christianson and P. Schuette, The relationship between direct predation and antipredator responses: a test with multiple predators and multiple prey, Ecology, 98 (2017), 2081-2092.  doi: 10.1002/ecy.1885.  Google Scholar

[10]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Berlin Heidelberg New York, 1977. doi: 10.1007/978-3-642-93073-7.  Google Scholar

[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. doi: 10.1016/j.jip.2012.05.010.  Google Scholar

[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. doi: 10.1038/s41598-017-19044-9.  Google Scholar

[13]

F. GroenewoudJ. G. FrommenD. JosiH. TanakaA. Jungwirth and M. Taborsky, Predation risk drives social complexity in cooperative breeders, Proc. Natl.Acad. Sci., 113 (2016), 4104-4109.  doi: 10.1073/pnas.1524178113.  Google Scholar

[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. doi: 10.1186/s13662-019-2050-3.  Google Scholar

[15]

R. P. GuptaM. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.  doi: 10.1007/s12591-012-0142-6.  Google Scholar

[16]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.  doi: 10.1016/j.jmaa.2012.08.057.  Google Scholar

[17] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifucation, Cambridge University Press, Cambridge, 1981.   Google Scholar
[18]

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. doi: 10.1142/S021812741650187X.  Google Scholar

[19]

D. P. Hu and H. J. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvest, Nonlinear Anal-RWA., 33 (2017), 58-82.  doi: 10.1016/j.nonrwa.2016.05.010.  Google Scholar

[20]

S. Khajanchi, Modeling the dynamics of stage-structure predator-prey system with Monod-Haldane type response function, Appl. Math. Comput., 302 (2017), 122-143.  doi: 10.1016/j.amc.2017.01.019.  Google Scholar

[21]

L. Kong and 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 (2017), 6715-6731.  doi: 10.1002/mma.4484.  Google Scholar

[22] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.   Google Scholar
[23]

P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., 4 (2010), 791-803.   Google Scholar

[24]

L. Z. LiF. W. Meng and 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 (2011), 853-862.  doi: 10.1016/j.jmaa.2010.12.002.  Google Scholar

[25]

Y. N. LiY. G. Sun and F. W. Meng, New criteria for exponential stability of switched time varying systems with delays and nonlinear disturbances, Nonlinear Anal-Hybri., 26 (2017), 284-291.  doi: 10.1016/j.nahs.2017.06.007.  Google Scholar

[26]

Y. Li and 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 (2015), 147-172.  doi: 10.1007/s10440-014-9983-z.  Google Scholar

[27]

B. LiuR. C. Wu and L. P. Chen, Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting, Math. Biosci., 298 (2018), 71-79.  doi: 10.1016/j.mbs.2018.02.002.  Google Scholar

[28]

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. doi: 10.1186/s13662-018-1752-2.  Google Scholar

[29]

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. doi: 10.1142/S0218127417501796.  Google Scholar

[30]

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. doi: 10.1007/s11071-018-4274-2.  Google Scholar

[31]

R. M. MayJ. R. BeddingtonC. W. ClarkS. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277.  doi: 10.1126/science.205.4403.267.  Google Scholar

[32]

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. doi: 10.1186/s13662-017-1434-5.  Google Scholar

[33]

S. N. RawP. MishraR. Kumar and S. Thakur, Complex behavior of prey-predator system exhibiting group defense: A mathematical modeling study, Chaos Soliton Fract., 100 (2017), 74-90.  doi: 10.1016/j.chaos.2017.05.010.  Google Scholar

[34]

M. SenP. D. N. Srinivasu and M. Banerjee, Global dynamics of an additional food provided predator-prey system with constant harvest in predators, Appl. Math. Comput., 250 (2015), 193-211.  doi: 10.1016/j.amc.2014.10.085.  Google Scholar

[35]

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. doi: 10.1155/2010/181784.  Google Scholar

[36]

F. E. Smith, Population dynamics in Daphnia Magna and a new model for population growth, Ecology, 44 (1963), 651-663.  doi: 10.2307/1933011.  Google Scholar

[37]

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. doi: 10.1186/s13662-018-1741-5.  Google Scholar

[38]

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. doi: 10.1155/2017/5876371.  Google Scholar

[39]

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. doi: 10.1186/s13662-018-1499-9.  Google Scholar

[40]

Z. WangY. K. XieJ. W. Lu and Y. X. Li, Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, Appl. Math. Comput., 347 (2019), 360-369.  doi: 10.1016/j.amc.2018.11.016.  Google Scholar

[41]

R. C. WuM. X. ChenB. Liu and L. P. Chen, Hopf bifurcation and Turing instability in a predator-prey model with Michaelis-Menten functional response, Nonlinear Dyn., 91 (2018), 2033-2047.  doi: 10.1007/s11071-017-4001-4.  Google Scholar

[42]

D. M. XiaoW. X. Li and M. A. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.  doi: 10.1016/j.jmaa.2005.11.048.  Google Scholar

[43]

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. doi: 10.1142/S0218127418500992.  Google Scholar

[44]

R. YuanW. H. Jiang and 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 (2015), 1072-1090.  doi: 10.1016/j.jmaa.2014.09.037.  Google Scholar

[45]

S. L. YuanX. H. Ji and H. P. Zhu, Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations, Math. Biosci. Eng., 14 (2017), 1477-1498.  doi: 10.3934/mbe.2017077.  Google Scholar

[46]

C. H. ZhangX. P. Yan and G. H. Cui, Hopf bifucations in a predator-prey system with a discrete delay and a distributed delay, Nonlinear Anal-RWA., 11 (2010), 4141-4153.  doi: 10.1016/j.nonrwa.2010.05.001.  Google Scholar

[47]

C. R. Zhu and 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 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.  Google Scholar

Figure 1.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = \tau_2 = 0 $. The positive equilibrium point $ E_2(0.68, 0.32) $ is locally asymptotically stable. Here the initial value is $ (0.8, 0.6)$
Figure 2.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = 0 $, $ \tau_2 = 2.8 < \tau_{20} = 2.91 $. The positive equilibrium point $ E_2(0.48, 0.52) $ is locally asymptotically stable. Here the initial value is $ (0.5,0.5)$
Figure 3.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = 0 $, $ \tau_2 = 2.92 > \tau_{20} = 2.91 $. The positive equilibrium point $ E_2(0.48, 0.52) $ is unstable. Here the initial value is $ (0.5,0.5) $
Figure 4.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = 2.0 < \tau_{30} = 2.11 $, $ \tau_2 = 0 $. The positive equilibrium point $ E_2(0.53, 0.47) $ is locally asymptotically stable. Here the initial value is $ (0.5,0.5)$
Figure 5.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = 2.1185 > \tau_{30} = 2.11 $, $ \tau_2 = 0 $. The positive equilibrium point $ E_2(0.53, 0.47) $ is unstable. Here the initial value is $ (0.5,0.5)$
Figure 6.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = \tau_2 = 1.85 < \tau_{40} = 1.92 $. The positive equilibrium point $ E_2(0.53, 0.47) $ is locally asymptotically stable. Here the initial value is $ (0.55,0.6)$
Figure 7.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = \tau_2 = 1.926 > \tau_{40} = 1.92 $. The positive equilibrium point $ E_2(0.53, 0.47) $ is unstable. Here the initial value is $ (0.55,0.6)$
Figure 8.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = 3 < \tau_{50} = 4.90 $, $ \tau_2 = 1.8 $. The positive equilibrium point $ E_2(0.48, 0.52) $ is locally asymptotically stable. Here the initial value is $ (0.55,0.6)$
Figure 9.  The diagram (a) shows the time series of $ x(t) $, $ y(t) $ and the diagram (b) shows the phase portrait of model (3) with $ \tau_1 = 6 > \tau_{50} = 4.90 $, $ \tau_2 = 1.8 $. The positive equilibrium point $ E_2(0.48, 0.52) $ is unstable. Here the initial value is $ (0.55,0.6)$
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