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June  2021, 26(6): 3427-3453. doi: 10.3934/dcdsb.2020238

Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays

Demou Luo and  Qiru Wang ,

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, Guangdong, China

* Corresponding author: mcswqr@mail.sysu.edu.cn

Received  May 2020 Revised  June 2020 Published  August 2020

Fund Project: This research was supported by the National Natural Science Foundation of China (No. 11671406)

This article is concerned with a generalized almost periodic predator-prey model with impulsive effects and time delays. By utilizing comparison theorem and constructing a feasible Lyapunov functional, we obtain sufficient conditions to guarantee the permanence and global asymptotic stability of the system. By applying Arzelà-Ascoli theorem, we establish the existence and uniqueness of almost-periodic positive solutions. A feasible numerical simulation is provided to explain the suitability of our main criteria.

Keywords: Almost periodic predator-prey model, impulsive effects and time delays, permanence and global stability, existence and uniqueness of almost periodic positive solutions.
Mathematics Subject Classification: Primary: 34K14, 34K20, 34K45; Secondary: 92D25.
Citation: Demou Luo, Qiru Wang. Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3427-3453. doi: 10.3934/dcdsb.2020238
References:
[1]

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, 28. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812831804.  Google Scholar

[2] D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effect: Stability, Theory and Applications, Halsted Press, New York, 1998.   Google Scholar
[3]

I. Barbalat, System dequations differentielles doscillations nonlinears, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

[4]

S. Bochner, Abstrakte fastperiodische funktionen, Acta Math., 61 (1933), 149–184, in German. doi: 10.1007/BF02547790.  Google Scholar

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S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, Ⅰ. Funktionen einer Variablen, Math. Ann., 96 (1927), 119–147, in German. doi: 10.1007/BF01209156.  Google Scholar

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H. Bohr, Zur theorie der fastperiodischen funktionen: Ⅰ. Eine verallgemeinerung der theorie der fourierreihen, Acta Math. 45 (1925), 29–127, in German. doi: 10.1007/BF02395468.  Google Scholar

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H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅱ, Acta Math. 46 (1925), 101–214, in German. doi: 10.1007/BF02543859.  Google Scholar

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H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅲ, Acta Math., 47 (1926), 237–281, in German. doi: 10.1007/BF02543846.  Google Scholar

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T. Diagana and H. Zhou, Existence of positive almost periodic solutions to the hematopoiesis model, Appl. Math. Comput., 274 (2016), 644-648.  doi: 10.1016/j.amc.2015.10.029.  Google Scholar

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H.-S. Ding, Q.-L. Liu and J. J. Nieto, Existence of positive almost periodic solutions to a class of hematopoiesis model, Appl. Math. Model., 40 (2016), 3289-3297.  doi: 10.1016/j.apm.2015.10.020.  Google Scholar

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H.-S. Ding, G. M. N'Guérékata and J. J. Nieto, Weighted pseudo almost periodic solutions to a class of discrete hematopoiesis model, Rev. Mat. Complut., 26 (2013), 427-443.  doi: 10.1007/s13163-012-0114-y.  Google Scholar

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J. Gao, Q. Wang and Y. Lin, Existence and exponential stability of almost-periodic solutions for neutral BAM neural networks with time-varying delays in leakage terms on time scales, Math. Methods Appl. Sci., 39 (2016), 1361-1375.  doi: 10.1002/mma.3574.  Google Scholar

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J. Gao, Q.-R. Wang and L.-W. Zhang, Existence and stability of almost-periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales, Appl. Math. Comput., 237 (2014), 639-649.  doi: 10.1016/j.amc.2014.03.051.  Google Scholar

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K. Hong and P. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-Ⅱ functional response and harvesting, Nonlinear Anal. Real World Appl., 14 (2013), 83-103.  doi: 10.1016/j.nonrwa.2012.05.004.  Google Scholar

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F. Kong, Q. Zhu, K. Wang and J. J. Nieto, Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator, J. Franklin Inst., 356 (2019), 11605-11637. doi: 10.1016/j.jfranklin.2019.09.030.  Google Scholar

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X. Lin and F. Chen, Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response, Appl. Math. Comput., 214 (2009), 548-556.  doi: 10.1016/j.amc.2009.04.028.  Google Scholar

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X. Lin, Z. Du and Y. Lv, Global asymptotic stability of almost periodic solution for a multispecies competition-predator system with time delays, Appl. Math. Comput., 219 (2013), 4908-4923.  doi: 10.1016/j.amc.2012.10.083.  Google Scholar

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B. Lisena, Global stability of a periodic Holling-Tanner predator-prey model, Math. Methods Appl. Sci., 41 (2018), 3270-3281.  doi: 10.1002/mma.4814.  Google Scholar

[25]

B. Liu, New results on the positive almost periodic solutions for a model of hematopoiesis, Nonlinear Anal. Real World Appl., 17 (2014), 252-264.  doi: 10.1016/j.nonrwa.2013.12.003.  Google Scholar

[26]

Z. Ma, F. Chen, C. Wu and W. Chen, Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference, Appl. Math. Comput., 219 (2013), 7945-7953.  doi: 10.1016/j.amc.2013.02.033.  Google Scholar

[27]

X. Meng, W. Xu and L. Chen, Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion, Appl. Math. Comput., 188 (2007), 365-378.  doi: 10.1016/j.amc.2006.09.133.  Google Scholar

[28]

V. D. Mil'man and A. D. Myshkis, On the stability of motion in the presence of impulses, Siberian Math. Ž., 1 (1960), 233-237.   Google Scholar

[29]

L. Nie, Z. Teng, L. Hu and J. Peng, Qualitative analysis of a modified Leslie-Gowerand Holling-type Ⅱ predator-prey model with state dependent impulsive effects, Nonlinear Anal. Real World Appl., 11 (2010), 1364-1373.  doi: 10.1016/j.nonrwa.2009.02.026.  Google Scholar

[30]

J. Qiu and J. Cao, Exponential stability of a competitive Lotka-Volterra system with delays, Appl. Math. Comput., 201 (2008), 819-829.  doi: 10.1016/j.amc.2007.11.046.  Google Scholar

[31]

A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, World Scientific, Singapore, 1995. Google Scholar

[32]

Y. Shan, K. She, S. Zhong, Q. Zhong, K. Shi and C. Zhao, Exponential stability and extended dissipativity criteria for generalized discrete-time neural networks with additive time-varying delays, Appl. Math. Comput., 333 (2018), 145-168.  doi: 10.1016/j.amc.2018.03.101.  Google Scholar

[33]

C. Shen, Permanence and global attractivity of the food-chain system with Holling Ⅳ type functional response, Appl. Math. Comput., 194 (2007), 179-185.  doi: 10.1016/j.amc.2007.04.019.  Google Scholar

[34]

E. R. van Kampen, Almost periodic functions and compact groups, Ann. of Math., 37 (1936), 78-91.  doi: 10.2307/1968688.  Google Scholar

[35]

J. von Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc., 36 (1934), 445-492.  doi: 10.1090/S0002-9947-1934-1501752-3.  Google Scholar

[36]

K. Wang and Y. Zhu, Global attractivity of positive periodic solution for a Volterra model, Appl. Math. Comput., 203 (2008), 493-501.  doi: 10.1016/j.amc.2008.04.005.  Google Scholar

[37]

L. Wang, Dynamic analysis on an almost periodic predator-prey model with impulses effects, Engineering Letters, 26 (2018), 333-339.   Google Scholar

[38]

X. Yu and Q. Wang, Weighted pseudo-almost periodic solutions for Shunting inhibitory cellular neural networks on time scales, Bull. Malays. Math. Sci. Soc., 42 (2019), 2055-2074.  doi: 10.1007/s40840-017-0595-4.  Google Scholar

[39]

X. Yu, Q. Wang and Y. Bai, Permanence and almost periodic solutions for $N$-species nonautonomous Lotka-Volterra competitive systems with delays and impulsive perturbations on time scales, Complexity, 2018 (2018), Article ID 2658745, 12 pp. doi: 10.1155/2018/2658745.  Google Scholar

[40]

H. Zhang, Y. Li, B. Jing and W. Zhao, Global stability of almost periodic solution of multispecies mutualism system with time delays and impulsive effects, Appl. Math. Comput., 232 (2014), 1138-1150.  doi: 10.1016/j.amc.2014.01.131.  Google Scholar

[41]

H. Zhang and J. Shao, Almost periodic solutions for cellular neural networks with time-varying delays in leakage terms, Appl. Math. Comput., 219 (2013), 11471-11482.  doi: 10.1016/j.amc.2013.05.046.  Google Scholar

[42]

H. Zhang, M. Yang and L. Wang, Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis, Appl. Math. Lett., 26 (2013), 38-42.  doi: 10.1016/j.aml.2012.02.034.  Google Scholar

[43]

H. Zhou, W. Wang and Z. Zhou, Positive almost periodic solution for a model of hematopoiesis with infinite time delays and a nonlinear harvesting term, Abstr. Appl. Anal., 2013 (2013), Article ID 146729, 6 pp. doi: 10.1155/2013/146729.  Google Scholar

[44]

H. Zhou and L. Yang, A new result on the existence of positive almost periodic solution for generalized hematopoiesis model, J. Math. Anal. Appl., 462 (2018), 370-379.  doi: 10.1016/j.jmaa.2018.01.075.  Google Scholar

[45]

X. Zhou, X. Shi and X. Song, Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay, Appl. Math. Comput., 196 (2008), 129-136.  doi: 10.1016/j.amc.2007.05.041.  Google Scholar

[46]

Z.-Q. Zhu and Q.-R. Wang, Existence of nonoscillatory solutions to neutral dynamic equations on time scales, J. Math. Anal. Appl., 335 (2007), 751-762.  doi: 10.1016/j.jmaa.2007.02.008.  Google Scholar

[47]

L. Zu, D. Jiang, D. O'Regan, T. Hayat and B. Ahmad, Ergodic property of a Lotka-Volterra predator-prey model with white noise higher order perturbation under regime switching, Appl. Math. Comput., 330 (2018), 93-102.  doi: 10.1016/j.amc.2018.02.035.  Google Scholar

show all references

References:
[1]

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, 28. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812831804.  Google Scholar

[2] D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effect: Stability, Theory and Applications, Halsted Press, New York, 1998.   Google Scholar
[3]

I. Barbalat, System dequations differentielles doscillations nonlinears, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

[4]

S. Bochner, Abstrakte fastperiodische funktionen, Acta Math., 61 (1933), 149–184, in German. doi: 10.1007/BF02547790.  Google Scholar

[5]

S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, Ⅰ. Funktionen einer Variablen, Math. Ann., 96 (1927), 119–147, in German. doi: 10.1007/BF01209156.  Google Scholar

[6]

H. Bohr, Zur theorie der fastperiodischen funktionen: Ⅰ. Eine verallgemeinerung der theorie der fourierreihen, Acta Math. 45 (1925), 29–127, in German. doi: 10.1007/BF02395468.  Google Scholar

[7]

H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅱ, Acta Math. 46 (1925), 101–214, in German. doi: 10.1007/BF02543859.  Google Scholar

[8]

H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅲ, Acta Math., 47 (1926), 237–281, in German. doi: 10.1007/BF02543846.  Google Scholar

[9]

T. Diagana and H. Zhou, Existence of positive almost periodic solutions to the hematopoiesis model, Appl. Math. Comput., 274 (2016), 644-648.  doi: 10.1016/j.amc.2015.10.029.  Google Scholar

[10]

H.-S. Ding, Q.-L. Liu and J. J. Nieto, Existence of positive almost periodic solutions to a class of hematopoiesis model, Appl. Math. Model., 40 (2016), 3289-3297.  doi: 10.1016/j.apm.2015.10.020.  Google Scholar

[11]

H.-S. Ding, G. M. N'Guérékata and J. J. Nieto, Weighted pseudo almost periodic solutions to a class of discrete hematopoiesis model, Rev. Mat. Complut., 26 (2013), 427-443.  doi: 10.1007/s13163-012-0114-y.  Google Scholar

[12]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes on Mathematics, vol. 377, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[13]

J. Gao, Q. Wang and Y. Lin, Existence and exponential stability of almost-periodic solutions for neutral BAM neural networks with time-varying delays in leakage terms on time scales, Math. Methods Appl. Sci., 39 (2016), 1361-1375.  doi: 10.1002/mma.3574.  Google Scholar

[14]

J. Gao, Q.-R. Wang and L.-W. Zhang, Existence and stability of almost-periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales, Appl. Math. Comput., 237 (2014), 639-649.  doi: 10.1016/j.amc.2014.03.051.  Google Scholar

[15] C. Y. He, Almost Periodic Differential Equations, Higher Education Press, Beijing, 1992.   Google Scholar
[16]

K. Hong and P. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-Ⅱ functional response and harvesting, Nonlinear Anal. Real World Appl., 14 (2013), 83-103.  doi: 10.1016/j.nonrwa.2012.05.004.  Google Scholar

[17]

T. Hu, Z. He, X. Zhang and S. Zhong, Finite-time stability for fractional-order complex-valued neural networks with time delay, Appl. Math. Comput., 365 (2020), 548-556.  doi: 10.1016/j.amc.2019.124715.  Google Scholar

[18]

T. Hu, X. Zhang and S. Zhong, Global asymptotic synchronization of nonidentical fractional-order neural networks, Neurocomputing, 313 (2018), 39-46.  doi: 10.1016/j.neucom.2018.05.098.  Google Scholar

[19]

F. Kong and J. J. Nieto, Almost periodic dynamical behaviors of the hematopoiesis model with mixed discontinuous harvesting terms, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5803-5830.   Google Scholar

[20]

F. Kong, Q. Zhu, K. Wang and J. J. Nieto, Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator, J. Franklin Inst., 356 (2019), 11605-11637. doi: 10.1016/j.jfranklin.2019.09.030.  Google Scholar

[21]

N. A. Kudryashov and A. S. Zakharchenko, Analytical properties and exact solutions of the Lotka-Volterra competition system, Appl. Math. Comput., 254 (2015), 219-228.  doi: 10.1016/j.amc.2014.12.113.  Google Scholar

[22]

X. Lin and F. Chen, Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response, Appl. Math. Comput., 214 (2009), 548-556.  doi: 10.1016/j.amc.2009.04.028.  Google Scholar

[23]

X. Lin, Z. Du and Y. Lv, Global asymptotic stability of almost periodic solution for a multispecies competition-predator system with time delays, Appl. Math. Comput., 219 (2013), 4908-4923.  doi: 10.1016/j.amc.2012.10.083.  Google Scholar

[24]

B. Lisena, Global stability of a periodic Holling-Tanner predator-prey model, Math. Methods Appl. Sci., 41 (2018), 3270-3281.  doi: 10.1002/mma.4814.  Google Scholar

[25]

B. Liu, New results on the positive almost periodic solutions for a model of hematopoiesis, Nonlinear Anal. Real World Appl., 17 (2014), 252-264.  doi: 10.1016/j.nonrwa.2013.12.003.  Google Scholar

[26]

Z. Ma, F. Chen, C. Wu and W. Chen, Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference, Appl. Math. Comput., 219 (2013), 7945-7953.  doi: 10.1016/j.amc.2013.02.033.  Google Scholar

[27]

X. Meng, W. Xu and L. Chen, Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion, Appl. Math. Comput., 188 (2007), 365-378.  doi: 10.1016/j.amc.2006.09.133.  Google Scholar

[28]

V. D. Mil'man and A. D. Myshkis, On the stability of motion in the presence of impulses, Siberian Math. Ž., 1 (1960), 233-237.   Google Scholar

[29]

L. Nie, Z. Teng, L. Hu and J. Peng, Qualitative analysis of a modified Leslie-Gowerand Holling-type Ⅱ predator-prey model with state dependent impulsive effects, Nonlinear Anal. Real World Appl., 11 (2010), 1364-1373.  doi: 10.1016/j.nonrwa.2009.02.026.  Google Scholar

[30]

J. Qiu and J. Cao, Exponential stability of a competitive Lotka-Volterra system with delays, Appl. Math. Comput., 201 (2008), 819-829.  doi: 10.1016/j.amc.2007.11.046.  Google Scholar

[31]

A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, World Scientific, Singapore, 1995. Google Scholar

[32]

Y. Shan, K. She, S. Zhong, Q. Zhong, K. Shi and C. Zhao, Exponential stability and extended dissipativity criteria for generalized discrete-time neural networks with additive time-varying delays, Appl. Math. Comput., 333 (2018), 145-168.  doi: 10.1016/j.amc.2018.03.101.  Google Scholar

[33]

C. Shen, Permanence and global attractivity of the food-chain system with Holling Ⅳ type functional response, Appl. Math. Comput., 194 (2007), 179-185.  doi: 10.1016/j.amc.2007.04.019.  Google Scholar

[34]

E. R. van Kampen, Almost periodic functions and compact groups, Ann. of Math., 37 (1936), 78-91.  doi: 10.2307/1968688.  Google Scholar

[35]

J. von Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc., 36 (1934), 445-492.  doi: 10.1090/S0002-9947-1934-1501752-3.  Google Scholar

[36]

K. Wang and Y. Zhu, Global attractivity of positive periodic solution for a Volterra model, Appl. Math. Comput., 203 (2008), 493-501.  doi: 10.1016/j.amc.2008.04.005.  Google Scholar

[37]

L. Wang, Dynamic analysis on an almost periodic predator-prey model with impulses effects, Engineering Letters, 26 (2018), 333-339.   Google Scholar

[38]

X. Yu and Q. Wang, Weighted pseudo-almost periodic solutions for Shunting inhibitory cellular neural networks on time scales, Bull. Malays. Math. Sci. Soc., 42 (2019), 2055-2074.  doi: 10.1007/s40840-017-0595-4.  Google Scholar

[39]

X. Yu, Q. Wang and Y. Bai, Permanence and almost periodic solutions for $N$-species nonautonomous Lotka-Volterra competitive systems with delays and impulsive perturbations on time scales, Complexity, 2018 (2018), Article ID 2658745, 12 pp. doi: 10.1155/2018/2658745.  Google Scholar

[40]

H. Zhang, Y. Li, B. Jing and W. Zhao, Global stability of almost periodic solution of multispecies mutualism system with time delays and impulsive effects, Appl. Math. Comput., 232 (2014), 1138-1150.  doi: 10.1016/j.amc.2014.01.131.  Google Scholar

[41]

H. Zhang and J. Shao, Almost periodic solutions for cellular neural networks with time-varying delays in leakage terms, Appl. Math. Comput., 219 (2013), 11471-11482.  doi: 10.1016/j.amc.2013.05.046.  Google Scholar

[42]

H. Zhang, M. Yang and L. Wang, Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis, Appl. Math. Lett., 26 (2013), 38-42.  doi: 10.1016/j.aml.2012.02.034.  Google Scholar

[43]

H. Zhou, W. Wang and Z. Zhou, Positive almost periodic solution for a model of hematopoiesis with infinite time delays and a nonlinear harvesting term, Abstr. Appl. Anal., 2013 (2013), Article ID 146729, 6 pp. doi: 10.1155/2013/146729.  Google Scholar

[44]

H. Zhou and L. Yang, A new result on the existence of positive almost periodic solution for generalized hematopoiesis model, J. Math. Anal. Appl., 462 (2018), 370-379.  doi: 10.1016/j.jmaa.2018.01.075.  Google Scholar

[45]

X. Zhou, X. Shi and X. Song, Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay, Appl. Math. Comput., 196 (2008), 129-136.  doi: 10.1016/j.amc.2007.05.041.  Google Scholar

[46]

Z.-Q. Zhu and Q.-R. Wang, Existence of nonoscillatory solutions to neutral dynamic equations on time scales, J. Math. Anal. Appl., 335 (2007), 751-762.  doi: 10.1016/j.jmaa.2007.02.008.  Google Scholar

[47]

L. Zu, D. Jiang, D. O'Regan, T. Hayat and B. Ahmad, Ergodic property of a Lotka-Volterra predator-prey model with white noise higher order perturbation under regime switching, Appl. Math. Comput., 330 (2018), 93-102.  doi: 10.1016/j.amc.2018.02.035.  Google Scholar

Figure 1.  Numeric simulation of the prey $ x(t) $ and the predator $ y(t) $ of (42) with the initial conditions $ (x(0),y(0))^{T} = (0.6,0.3)^{T} $, $ (x(0),y(0))^{T} = (0.2,0.1)^{T} $ and $ (x(0),y(0))^{T} = (0.4,0.2)^{T} $
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Table 1.  The biological parameters of $ x $ and $ y $
$a_{i}^{l}$ $a_{i}^{u}$ $b_{i}^{l}$ $b_{i}^{u}$ $c_{i}^{l}$ $c_{i}^{u}$ $m_{i}^{l}$ $M_{i}^{u}$ $P_{i}^{l}$ $\tau_{i}^{l}$ $\tau_{i}^{u}$
$x$ $0.25$ $0.35$ $0.94$ $0.96$ $0.11$ $0.11$ $0.2495$ $0.3736$ $0.01$ $0.01$ $0.01$
$y$ $0.45$ $0.55$ $1.20$ $3.80$ $3.00$ $3.00$ $0.0189$ $0.5451$ $0.02$ $0.02$ $0.02$
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$a_{i}^{l}$ $a_{i}^{u}$ $b_{i}^{l}$ $b_{i}^{u}$ $c_{i}^{l}$ $c_{i}^{u}$ $m_{i}^{l}$ $M_{i}^{u}$ $P_{i}^{l}$ $\tau_{i}^{l}$ $\tau_{i}^{u}$
$x$ $0.25$ $0.35$ $0.94$ $0.96$ $0.11$ $0.11$ $0.2495$ $0.3736$ $0.01$ $0.01$ $0.01$
$y$ $0.45$ $0.55$ $1.20$ $3.80$ $3.00$ $3.00$ $0.0189$ $0.5451$ $0.02$ $0.02$ $0.02$
[1]

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[2]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51
[3]

Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure & Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423
[4]

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