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July  2020, 25(7): 2639-2664. doi: 10.3934/dcdsb.2020026

Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy

 1 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China 2 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China 3 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha, Hunan 410114, China 4 School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Broadway, NSW 2007, Australia

* Corresponding author: Lihong Huang

Received  March 2019 Revised  June 2019 Published  April 2020

Fund Project: This work is supported in part by the National Natural Science Foundation of China (11771059) and the China Scholarship Council (20180613100)

This paper studies the solution behaviour of a general delayed predator-prey model with discontinuous prey control strategy. The positiveness and boundeness of the solution of the system is firstly investigated using the comparison theorem. Then the sufficient conditions are derived for the existence of positive periodic solutions using the differential inclusion theory and the topological degree theory. Furthermore, the positive periodic solution is proved to be globally exponentially stable by employing the generalized Lyapunov approach. The global finite-time convergence is also discussed for the system state. Finally, the numerical simulations of four examples are given to validate the correctness of the theoretical results.

Citation: Wenjie Li, Lihong Huang, Jinchen Ji. Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2639-2664. doi: 10.3934/dcdsb.2020026
References:
 [1] A. A.Berryman and B. A.Hawkins, The refuge as an integrating concept in ecology and evolution, Oikos, 115 (2006), 192-196.   Google Scholar [2] Z. W. Cai and L. H. Huang, Periodic dynamics of delayed Lotka-Volterra competition systems with discontinuous harvesting policies via differential inclusions, Chaos, Solitons Fractals, 54 (2013), 39-56.  doi: 10.1016/j.chaos.2013.05.005.  Google Scholar [3] Z. W. Cai, L. H. Huang, L. L. Zhang and X. L. Hu, Dynamical behavior for a class of predator-prey system with general functional response and discontinuous harvesting policy, Math. Meth. Appl. Sci., 38 (2015), 4679-4701.  doi: 10.1002/mma.3379.  Google Scholar [4] K. Chakraborty, S. Das and T. K. Kar, On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations, Applied Mathematics and Computation, 221 (2013), 581-597.  doi: 10.1016/j.amc.2013.06.065.  Google Scholar [5] C. Chen, Y. Kang and Smith. R, Sliding motion and global dynamics of a Filippov fire-blight model with economic thresholds, Nonlinear Anal. Real World Appl, 39 (2018), 492-519.  doi: 10.1016/j.nonrwa.2017.08.002.  Google Scholar [6] M. I.S. Costa and M. E. M. Meza, Dynamical stabilization of grazing systems: An interplay among plant-water interaction, overgrazing and a thresholdmanagement policy, Mathematical Biosciences, 204 (2006), 250-259.  doi: 10.1016/j.mbs.2006.05.010.  Google Scholar [7] L. Duan, X. Fang and C. Huang, Global exponential convergence in a delayed almost periodic Nicholson's blowflies model with discontinuous harvesting, Mathematical Methods in the Applied Sciences, 41 (2018), 1954-1965.  doi: 10.1002/mma.4722.  Google Scholar [8] L. Duan and L. H. Huang, Global dissipativity of mixed time-varying delayed neural networks with discontinuous activations, Commun Nonlinear Sci Numer Simulat, 19 (2014), 4122-4134.  doi: 10.1016/j.cnsns.2014.03.024.  Google Scholar [9] L. Duan, L. Huang and Y. Chen, Global exponential stability of periodic solutions to a delay Lasota-Wazewska model with discontinuous harvesting, Proceedings of the American Mathematical Society, 144 (2016), 561-573.  doi: 10.1090/proc12714.  Google Scholar [10] Y. H. Fan, W. T. Li and L. L. Wang, Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses, Nonlinear Analysis: Real World Applications, 5 (2004), 247-263.  doi: 10.1016/S1468-1218(03)00036-1.  Google Scholar [11] D. Fang, P. Yu, Y. Lv and L. Chen, Periodicity induced by state feedback controls and driven by disparate dynamics of a herbivore-plankton model with cannibalism, Nonlinear Dyn, 90 (2017), 2657-2672.  doi: 10.1007/s11071-017-3829-y.  Google Scholar [12] M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Transactions on Circuit Theory I: Fund. 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Upadhyay and G. P. Samanta, Ecological dynamics of age selective harvesting of fish population: Maximum sustainable yield and its control strategy, Chaos, Solitons & Fractals, 93 (2016), 111-122.  doi: 10.1016/j.chaos.2016.09.021.  Google Scholar [18] G. R. Jiang and Q. S. Lu, Impulsivee state feedback control of a predator–prey model, J. Comput. Appl.Math., 200 (2007), 193-207.  doi: 10.1016/j.cam.2005.12.013.  Google Scholar [19] D. Q. Jiang, Q. M. Zhang, T. Hayat and A. Alsaedi, Periodic solution for a stochastic non–autonomous competitive Lotka–Volterra model in a polluted environment, Physica A, 471 (2017), 276-287.  doi: 10.1016/j.physa.2016.12.008.  Google Scholar [20] S. Khajanchi, Modeling the dynamics of stage–structure predator-prey system with Monod–Haldane type response function, Applied Mathematics and Computation, 302 (2017), 122-143.  doi: 10.1016/j.amc.2017.01.019.  Google Scholar [21] V. K$\breve{r}$rivan, On the Gause predator-prey model with a refuge: A fresh look at the history, Journal of Theoretical Biology, 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.  Google Scholar [22] B. Leard and J. Rebaza, Analysis of predator-prey models with continuous threshold harvesting, Applied Mathematics and Computation, 217 (2011), 5265-5278.  doi: 10.1016/j.amc.2010.11.050.  Google Scholar [23] W. J. Li, L. H. Huang and J. C. Ji, Periodic solution and its stability of a delayed Beddington–DeAngelis type predator–prey system with discontinuous control strategy, Mathematical Methods in the Applied Sciences, 42 (2019), 4498-4515.  doi: 10.1002/mma.5673.  Google Scholar [24] W. J. Li, J. C. Ji and L. H. Huang, Global dynamic behavior of a predator–prey model under ratio–dependent state impulsive control, Applied Mathematical Modelling, 77 (2020), part 2, 1842–1859. doi: 10.1016/j.apm.2019.09.033.  Google Scholar [25] Y. Li and Z. H. Lin, Periodic solutions of differential inclusions, Nonlinear Anal Theory Methods Appl, 24 (1995), 631-641.  doi: 10.1016/0362-546X(94)00111-T.  Google Scholar [26] H. Y. Li and Z. K. She, Dynamics of a non-autonomous density-dependent predator-prey model with Beddington-DeAngelis type, International Journal of Biomathematics, 9 (2016), 1650050, 25pp. doi: 10.1142/S1793524516500509.  Google Scholar [27] M. Liu and C. Z. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 1493-1508.  doi: 10.3934/dcdsb.2017071.  Google Scholar [28] M. Liu, X. He and J. Y. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Analysis: Hybrid Systems, 28 (2018), 87-104.  doi: 10.1016/j.nahs.2017.10.004.  Google Scholar [29] W. Liu and Y. L. Jiang, Nonlinear dynamical behaviour in a predator-prey model with harvesting, East Asian Journal on Applied Mathematics, 2 (2017), 376-395.  doi: 10.4208/eajam.020916.250217a.  Google Scholar [30] Y. Lu, X. Wang and S. Q. Liu, A non-autonomous predator-prey model with infected prey, Discrete and Continuous Dynamical Systems Series B, 23 (2018), 3817-3836.   Google Scholar [31] D. Luo, Global boundedness of solutions in a reaction-diffusion system of Beddington DeAngelis type predator-prey model with nonlinear prey taxis and random diffusion, Boundary Value Problems, 2018 (2018), Paper No. 33, 11 pp. doi: 10.1186/s13661-018-0952-8.  Google Scholar [32] D. Z. Luo and D. S. Wang, On almost periodicity of delayed predator-preymodel with mutual interference and discontinuous harvesting policies, Math. Meth. Appl. Sci., 39 (2016), 4311-4333.  doi: 10.1002/mma.3861.  Google Scholar [33] A. Martin and S. G. Ruan, Predator-prey models with delay and prey harvesting, Mathematical Biology, 43 (2001), 247-267.  doi: 10.1007/s002850100095.  Google Scholar [34] S. G. Ruan and D. M. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 61 (2000), 1445-1472.  doi: 10.1137/S0036139999361896.  Google Scholar [35] J. Song, M. Hu, Y. Z. Bai and Y. H. Xia, Dynamic analysis of a non-autonomous ratio-dependent predator-prey model with additional food, Journal of Applied Analysis and Computation, 8 (2018), 1893-1909.   Google Scholar [36] S. Y. Tang, J. H. Liang, Y. N. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl Math., 72 (2012), 1061-1080.  doi: 10.1137/110847020.  Google Scholar [37] D. S. Wang, On a non-selective harvesting prey-predator model with Hassell-Varley type functional response, Applied Mathematics and Computation, 246 (2014), 678-695.  doi: 10.1016/j.amc.2014.08.081.  Google Scholar [38] J. M. Wang, H. D. Cheng, Y. Li and X. N. Zhang, The geometrical analysis of a predator-prey model with multi-state dependent impulses, Journal of Applied Analysis and Computation, 8 (2018), 427-442.   Google Scholar [39] P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.   Google Scholar [40] Q. Xiao and B. Dai, Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy, Mathematical Biosciences and Engineering, 5 (2015), 1065-1081.  doi: 10.3934/mbe.2015.12.1065.  Google Scholar [41] S. Q. Zhang, X. Z. Meng, T. Feng and T. H. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, NonlinearAnalysis: Hybrid Systems, 26 (2017), 19-37.  doi: 10.1016/j.nahs.2017.04.003.  Google Scholar [42] K. H. Zhao and Y. P. Ren, Existence of positive periodic solutions for a class of Gilpin-Ayala ecological models with discrete and distributed time delays, Advances in Difference Equations, 2017 (2017), Paper No. 331, 13 pp. doi: 10.1186/s13662-017-1386-9.  Google Scholar [43] R. Zou and S. J. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Computers and Mathematics with Applications, 75 (2018), 1237-1258.  doi: 10.1016/j.camwa.2017.11.002.  Google Scholar [44] W. J. Zuo and D. Q. Jiang, Periodic solutions for a stochastic non-autonomous Holling-Tanner predator-prey system with impulses, NonlinearAnalysis: Hybrid Systems, 22 (2016), 191-201.  doi: 10.1016/j.nahs.2016.03.004.  Google Scholar

show all references

References:
 [1] A. A.Berryman and B. A.Hawkins, The refuge as an integrating concept in ecology and evolution, Oikos, 115 (2006), 192-196.   Google Scholar [2] Z. W. Cai and L. H. Huang, Periodic dynamics of delayed Lotka-Volterra competition systems with discontinuous harvesting policies via differential inclusions, Chaos, Solitons Fractals, 54 (2013), 39-56.  doi: 10.1016/j.chaos.2013.05.005.  Google Scholar [3] Z. W. Cai, L. H. Huang, L. L. Zhang and X. L. Hu, Dynamical behavior for a class of predator-prey system with general functional response and discontinuous harvesting policy, Math. Meth. Appl. Sci., 38 (2015), 4679-4701.  doi: 10.1002/mma.3379.  Google Scholar [4] K. Chakraborty, S. Das and T. K. Kar, On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations, Applied Mathematics and Computation, 221 (2013), 581-597.  doi: 10.1016/j.amc.2013.06.065.  Google Scholar [5] C. Chen, Y. Kang and Smith. R, Sliding motion and global dynamics of a Filippov fire-blight model with economic thresholds, Nonlinear Anal. Real World Appl, 39 (2018), 492-519.  doi: 10.1016/j.nonrwa.2017.08.002.  Google Scholar [6] M. I.S. Costa and M. E. M. Meza, Dynamical stabilization of grazing systems: An interplay among plant-water interaction, overgrazing and a thresholdmanagement policy, Mathematical Biosciences, 204 (2006), 250-259.  doi: 10.1016/j.mbs.2006.05.010.  Google Scholar [7] L. Duan, X. Fang and C. Huang, Global exponential convergence in a delayed almost periodic Nicholson's blowflies model with discontinuous harvesting, Mathematical Methods in the Applied Sciences, 41 (2018), 1954-1965.  doi: 10.1002/mma.4722.  Google Scholar [8] L. Duan and L. H. Huang, Global dissipativity of mixed time-varying delayed neural networks with discontinuous activations, Commun Nonlinear Sci Numer Simulat, 19 (2014), 4122-4134.  doi: 10.1016/j.cnsns.2014.03.024.  Google Scholar [9] L. Duan, L. Huang and Y. Chen, Global exponential stability of periodic solutions to a delay Lasota-Wazewska model with discontinuous harvesting, Proceedings of the American Mathematical Society, 144 (2016), 561-573.  doi: 10.1090/proc12714.  Google Scholar [10] Y. H. Fan, W. T. Li and L. L. Wang, Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses, Nonlinear Analysis: Real World Applications, 5 (2004), 247-263.  doi: 10.1016/S1468-1218(03)00036-1.  Google Scholar [11] D. Fang, P. Yu, Y. Lv and L. Chen, Periodicity induced by state feedback controls and driven by disparate dynamics of a herbivore-plankton model with cannibalism, Nonlinear Dyn, 90 (2017), 2657-2672.  doi: 10.1007/s11071-017-3829-y.  Google Scholar [12] M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Transactions on Circuit Theory I: Fund. Theory Appl., 50 (2003), 1421-1435.  doi: 10.1109/TCSI.2003.818614.  Google Scholar [13] S. J. Gao, L. S. Chen and Z. D. Teng, Impulsivee vaccination of an SEIRS model with time delay and varying total population size, Bull. Math. Biol, 69 (2007), 731-745.   Google Scholar [14] L. N. Guin and S. Acharya, Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting, Nonlinear Dyn, 88 (2017), 1501-1533.  doi: 10.1007/s11071-016-3326-8.  Google Scholar [15] H. J. Guo and L. S. Chen, Periodic solution of a chemostat model with Monod growth rate and impulsivee state feedback control, J. Theor. Biol, 260 (2009), 502-509.  doi: 10.1016/j.jtbi.2009.07.007.  Google Scholar [16] Z. Y. Guo and X. F. Zou, Impact of discontinuous harvesting on fishery dynamics in a stock–effort fishing model, Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 594-603.  doi: 10.1016/j.cnsns.2014.06.014.  Google Scholar [17] D. Jana, R. Agrawal, R. K. Upadhyay and G. P. Samanta, Ecological dynamics of age selective harvesting of fish population: Maximum sustainable yield and its control strategy, Chaos, Solitons & Fractals, 93 (2016), 111-122.  doi: 10.1016/j.chaos.2016.09.021.  Google Scholar [18] G. R. Jiang and Q. S. Lu, Impulsivee state feedback control of a predator–prey model, J. Comput. Appl.Math., 200 (2007), 193-207.  doi: 10.1016/j.cam.2005.12.013.  Google Scholar [19] D. Q. Jiang, Q. M. Zhang, T. Hayat and A. Alsaedi, Periodic solution for a stochastic non–autonomous competitive Lotka–Volterra model in a polluted environment, Physica A, 471 (2017), 276-287.  doi: 10.1016/j.physa.2016.12.008.  Google Scholar [20] S. Khajanchi, Modeling the dynamics of stage–structure predator-prey system with Monod–Haldane type response function, Applied Mathematics and Computation, 302 (2017), 122-143.  doi: 10.1016/j.amc.2017.01.019.  Google Scholar [21] V. K$\breve{r}$rivan, On the Gause predator-prey model with a refuge: A fresh look at the history, Journal of Theoretical Biology, 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.  Google Scholar [22] B. Leard and J. Rebaza, Analysis of predator-prey models with continuous threshold harvesting, Applied Mathematics and Computation, 217 (2011), 5265-5278.  doi: 10.1016/j.amc.2010.11.050.  Google Scholar [23] W. J. Li, L. H. Huang and J. C. Ji, Periodic solution and its stability of a delayed Beddington–DeAngelis type predator–prey system with discontinuous control strategy, Mathematical Methods in the Applied Sciences, 42 (2019), 4498-4515.  doi: 10.1002/mma.5673.  Google Scholar [24] W. J. Li, J. C. Ji and L. H. Huang, Global dynamic behavior of a predator–prey model under ratio–dependent state impulsive control, Applied Mathematical Modelling, 77 (2020), part 2, 1842–1859. doi: 10.1016/j.apm.2019.09.033.  Google Scholar [25] Y. Li and Z. H. Lin, Periodic solutions of differential inclusions, Nonlinear Anal Theory Methods Appl, 24 (1995), 631-641.  doi: 10.1016/0362-546X(94)00111-T.  Google Scholar [26] H. Y. Li and Z. K. She, Dynamics of a non-autonomous density-dependent predator-prey model with Beddington-DeAngelis type, International Journal of Biomathematics, 9 (2016), 1650050, 25pp. doi: 10.1142/S1793524516500509.  Google Scholar [27] M. Liu and C. Z. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 1493-1508.  doi: 10.3934/dcdsb.2017071.  Google Scholar [28] M. Liu, X. He and J. Y. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Analysis: Hybrid Systems, 28 (2018), 87-104.  doi: 10.1016/j.nahs.2017.10.004.  Google Scholar [29] W. Liu and Y. L. Jiang, Nonlinear dynamical behaviour in a predator-prey model with harvesting, East Asian Journal on Applied Mathematics, 2 (2017), 376-395.  doi: 10.4208/eajam.020916.250217a.  Google Scholar [30] Y. Lu, X. Wang and S. Q. Liu, A non-autonomous predator-prey model with infected prey, Discrete and Continuous Dynamical Systems Series B, 23 (2018), 3817-3836.   Google Scholar [31] D. Luo, Global boundedness of solutions in a reaction-diffusion system of Beddington DeAngelis type predator-prey model with nonlinear prey taxis and random diffusion, Boundary Value Problems, 2018 (2018), Paper No. 33, 11 pp. doi: 10.1186/s13661-018-0952-8.  Google Scholar [32] D. Z. Luo and D. S. Wang, On almost periodicity of delayed predator-preymodel with mutual interference and discontinuous harvesting policies, Math. Meth. Appl. Sci., 39 (2016), 4311-4333.  doi: 10.1002/mma.3861.  Google Scholar [33] A. Martin and S. G. Ruan, Predator-prey models with delay and prey harvesting, Mathematical Biology, 43 (2001), 247-267.  doi: 10.1007/s002850100095.  Google Scholar [34] S. G. Ruan and D. M. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 61 (2000), 1445-1472.  doi: 10.1137/S0036139999361896.  Google Scholar [35] J. Song, M. Hu, Y. Z. Bai and Y. H. Xia, Dynamic analysis of a non-autonomous ratio-dependent predator-prey model with additional food, Journal of Applied Analysis and Computation, 8 (2018), 1893-1909.   Google Scholar [36] S. Y. Tang, J. H. Liang, Y. N. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl Math., 72 (2012), 1061-1080.  doi: 10.1137/110847020.  Google Scholar [37] D. S. Wang, On a non-selective harvesting prey-predator model with Hassell-Varley type functional response, Applied Mathematics and Computation, 246 (2014), 678-695.  doi: 10.1016/j.amc.2014.08.081.  Google Scholar [38] J. M. Wang, H. D. Cheng, Y. Li and X. N. Zhang, The geometrical analysis of a predator-prey model with multi-state dependent impulses, Journal of Applied Analysis and Computation, 8 (2018), 427-442.   Google Scholar [39] P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.   Google Scholar [40] Q. Xiao and B. Dai, Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy, Mathematical Biosciences and Engineering, 5 (2015), 1065-1081.  doi: 10.3934/mbe.2015.12.1065.  Google Scholar [41] S. Q. Zhang, X. Z. Meng, T. Feng and T. H. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, NonlinearAnalysis: Hybrid Systems, 26 (2017), 19-37.  doi: 10.1016/j.nahs.2017.04.003.  Google Scholar [42] K. H. Zhao and Y. P. Ren, Existence of positive periodic solutions for a class of Gilpin-Ayala ecological models with discrete and distributed time delays, Advances in Difference Equations, 2017 (2017), Paper No. 331, 13 pp. doi: 10.1186/s13662-017-1386-9.  Google Scholar [43] R. Zou and S. J. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Computers and Mathematics with Applications, 75 (2018), 1237-1258.  doi: 10.1016/j.camwa.2017.11.002.  Google Scholar [44] W. J. Zuo and D. Q. Jiang, Periodic solutions for a stochastic non-autonomous Holling-Tanner predator-prey system with impulses, NonlinearAnalysis: Hybrid Systems, 22 (2016), 191-201.  doi: 10.1016/j.nahs.2016.03.004.  Google Scholar
Periodic solution of Holling Ⅰ type non-autonomous system; (a) phase portraits of the state variables $x(t)$ and $y(t)$, (b) trajectory in three-dimensional space, and (c) trajectories of the state variables $x(t)$ and $y(t)$ with time
The trajectory converging to an equilibrium of the corresponding Holling Ⅰ type autonomous system; (a) phase portrait of the state variables $x(t)$ and $y(t)$, (b) trajectory of the state in three-dimensional space, and (c) trajectories of the variables $x(t)$ and $y(t)$ with time
Periodic solution of Holling Ⅱ type non-autonomous system; (a) phase portraits of the state variables $x(t)$ and $y(t)$, (b) trajectory in three-dimensional space, and (c) trajectories of the state variables $x(t)$ and $y(t)$ with time
The trajectory converging to an equilibrium of the corresponding Holling Ⅱ type autonomous system; (a) phase portrait of the state variables $x(t)$ and $y(t)$, (b) trajectory of the state in three-dimensional space, (c) trajectories of the variables $x(t)$ and $y(t)$ with time
Periodic solution of Holling Ⅲ type non-autonomous system; (a) phase portraits of the state variables $x(t)$ and $y(t)$, (b) trajectory of the state in three-dimensional space, and (c) trajectories of the state variables $x(t)$ and $y(t)$ with time
The trajectory converging to an equilibrium of the corresponding Holling Ⅲ type autonomous system; (a) phase portrait of the state variables $x(t)$ and $y(t)$. (b) trajectory of the state in three-dimensional space, and (c) trajectories of the variables $x(t)$ and $y(t)$ with time
Periodic solution of Ivlev type non-autonomous system; (a) phase portraits of the state variables $x(t)$ and $y(t)$, (b)trajectory in three-dimensional space, and (c) trajectories of the state variables $x(t)$ and $y(t)$ with time
The trajectory converging to an equilibrium of the corresponding Ivlev type autonomous system; (a) phase portrait of the state variables $x(t)$ and $y(t)$, (b) trajectory of the state in three-dimensional space, and (c) trajectories of the variables $x(t)$ and $y(t)$ with time
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