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Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, Guangdong, China |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |

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