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

Demou Luo; Qiru Wang

doi:10.3934/dcdsb.2020238

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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 June 2021 Early access 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 and Continuous Dynamical Systems - B, 2021, 26 (6) : 3427-3453. doi: 10.3934/dcdsb.2020238

References:

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[2]	D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effect: Stability, Theory and Applications, Halsted Press, New York, 1998.
[3]	I. Barbalat, System dequations differentielles doscillations nonlinears, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.
[4]	S. Bochner, Abstrakte fastperiodische funktionen, Acta Math., 61 (1933), 149–184, in German. doi: 10.1007/BF02547790.
[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.
[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.
[7]	H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅱ, Acta Math. 46 (1925), 101–214, in German. doi: 10.1007/BF02543859.
[8]	H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅲ, Acta Math., 47 (1926), 237–281, in German. doi: 10.1007/BF02543846.
[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.
[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.
[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.
[12]	A. M. Fink, Almost Periodic Differential Equations, Lecture Notes on Mathematics, vol. 377, Springer-Verlag, Berlin-New York, 1974.
[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.
[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.
[15]	C. Y. He, Almost Periodic Differential Equations, Higher Education Press, Beijing, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, World Scientific, Singapore, 1995.
[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.
[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.
[34]	E. R. van Kampen, Almost periodic functions and compact groups, Ann. of Math., 37 (1936), 78-91. doi: 10.2307/1968688.
[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.
[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.
[37]	L. Wang, Dynamic analysis on an almost periodic predator-prey model with impulses effects, Engineering Letters, 26 (2018), 333-339.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effect: Stability, Theory and Applications, Halsted Press, New York, 1998.
[3]	I. Barbalat, System dequations differentielles doscillations nonlinears, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.
[4]	S. Bochner, Abstrakte fastperiodische funktionen, Acta Math., 61 (1933), 149–184, in German. doi: 10.1007/BF02547790.
[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.
[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.
[7]	H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅱ, Acta Math. 46 (1925), 101–214, in German. doi: 10.1007/BF02543859.
[8]	H. Bohr, Zur theorie der fastperiodischen funktionen. Ⅲ, Acta Math., 47 (1926), 237–281, in German. doi: 10.1007/BF02543846.
[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.
[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.
[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.
[12]	A. M. Fink, Almost Periodic Differential Equations, Lecture Notes on Mathematics, vol. 377, Springer-Verlag, Berlin-New York, 1974.
[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.
[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.
[15]	C. Y. He, Almost Periodic Differential Equations, Higher Education Press, Beijing, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, World Scientific, Singapore, 1995.
[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.
[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.
[34]	E. R. van Kampen, Almost periodic functions and compact groups, Ann. of Math., 37 (1936), 78-91. doi: 10.2307/1968688.
[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.
[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.
[37]	L. Wang, Dynamic analysis on an almost periodic predator-prey model with impulses effects, Engineering Letters, 26 (2018), 333-339.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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	$a_{i}^{l}$	$a_{i}^{u}$	$b_{i}^{l}$	$b_{i}^{u}$	$c_{i}^{l}$	$c_{i}^{u}$	$m_{i}^{l}$	$M_{i}^{u}$	$P_{i}^{l}$	$\tau_{i}^{l}$	$\tau_{i}^{u}$
$x$	$0.25$	$0.35$	$0.94$	$0.96$	$0.11$	$0.11$	$0.2495$	$0.3736$	$0.01$	$0.01$	$0.01$
$y$	$0.45$	$0.55$	$1.20$	$3.80$	$3.00$	$3.00$	$0.0189$	$0.5451$	$0.02$	$0.02$	$0.02$

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2020 Impact Factor: 1.327

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2020 Impact Factor: 1.327

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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