September  2021, 14(9): 3197-3222. 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  September 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. 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.

[2]

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

[3]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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