×

Dynamics of a stochastic Markovian switching predator-prey model with infinite memory and general Lévy jumps. (English) Zbl 1524.92078

Summary: This paper investigates a stochastic Markovian switching predator-prey model with infinite memory and general Lévy jumps. Firstly, we transfer a classic infinite memory predator-prey model with weak kernel case into an equivalent model through integral transform. Then, for the corresponding stochastic Markovian switching model, we establish the sufficient conditions for permanence in time average and the threshold between stability in time average and extinction. Finally, sufficient criteria for a unique ergodic stationary distribution of the model are derived. Our results show that, firstly, both white noise and infinite memory are unfavorable to the existence of the stationary distribution; secondly, the general Lévy jumps could make the stationary distribution vanish as well as happen; finally, the Markovian switching could make the stationary distribution appear.

MSC:

92D25 Population dynamics (general)
60J27 Continuous-time Markov processes on discrete state spaces
60J76 Jump processes on general state spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bao, J.; Mao, X.; Yin, G.; Yuan, C., Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74, 6601-6616 (2011) · Zbl 1228.93112
[2] Bao, J.; Wang, F.; Yuan, C., Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory, Stochastic Process. Appl., 129, 4576-4596 (2019) · Zbl 1433.60032
[3] Barǎalat, I., Systems d’equations differential d’oscillations nonlineairies, Rev. Roumaine Math. Pures Appl., 4, 267-270 (1959) · Zbl 0090.06601
[4] Chang, Z.; Meng, X.; Zhang, T., A new way of investigating the asymptotic behaviour of a stochastic SIS system with multiplicative noise, Appl. Math. Lett., 87, 80-86 (2019) · Zbl 1426.60069
[5] Deng, Y.; Liu, M., Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations, Appl. Math. Model., 78, 482-504 (2020) · Zbl 1481.92059
[6] Deng, Y.; Liu, M., Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations, Appl. Math. Model., 78, 482-504 (2020) · Zbl 1481.92059
[7] Geng, J.; Liu, M.; Zhang, Y., Stability of a stochastic one-predator-two-prey population model with time delays, Commun. Nonlinear Sci. Numer. Simul., 53, 65-82 (2017) · Zbl 1510.92161
[8] Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0752.34039
[9] Hale, J. K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21, 11-41 (1978) · Zbl 0383.34055
[10] He, S.; Tang, S.; Wang, W., A stochastic SIS model driven by random diffusion of air pollutants, Physica A, 532 (2019), Article 121759 · Zbl 07570905
[11] Higham, D. J., An algorithmic introduction to numerical simulation of stochastic diffrential equations, SIAM Rev., 43, 525-546 (2001) · Zbl 0979.65007
[12] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[13] Kuang, Y.; Smith, H. L., Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103, 221-246 (2003) · Zbl 0786.34077
[14] Li, D.; Liu, M., Invariant measure of a stochastic food-limited population model with regime switching, Math. Comput. Simulation, 178, 16-26 (2020) · Zbl 1510.92168
[15] Li, X.; Wang, R.; Yin, G., Moment bounds and ergodicity of switching diffusion systems involving two-time-scale Markov chains, Systems Control Lett., 132 (2019), Article 104514 · Zbl 1425.93260
[16] Li, X.; Yin, G., Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376, 11-28 (2011) · Zbl 1205.92058
[17] Lipster, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228 (1980) · Zbl 0435.60037
[18] Liu, M.; Bai, C., Optimal harvesting of a stochastic mutualism model with regime-switching, Appl. Math. Comput., 375, Article 125040 pp. (2020) · Zbl 1433.92035
[19] Liu, Q.; Chen, Q., Analysis of a general stochastic non-autonomous logistic model with delays and Lévy jumps, J. Math. Anal. Appl., 433, 95-120 (2016) · Zbl 1326.92060
[20] Liu, M.; Deng, M., Permanence and extinction of a stochastic hybrid model for tumor growth, Appl. Math. Lett., 94, 66-72 (2019) · Zbl 1423.92098
[21] Liu, M.; Fan, M., Stability in distribution of a three-species stochastic casade predator-prey system with time delays, IMA J. Appl. Math., 82, 396-423 (2017) · Zbl 1404.92158
[22] Liu, Q.; Jiang, D., Stationary distribution and extinction of a stochastic predator-prey model with distributed delay, Appl. Math. Lett., 78, 79-87 (2018) · Zbl 1382.92223
[23] Liu, Q.; Jiang, D.; Hayat, T.; Alsaedi, A., Asymptotic behavior of a food-limited Lotka-Volterra mutualism model with Markovian switching and Lévy jumps, Physica A, 505, 94-104 (2018) · Zbl 1514.92092
[24] Liu, Q.; Jiang, D.; Hayat, T.; Alsaedi, A., Dynamical behavior of a stochastic epidemic model for cholera, J. Franklin Inst., 356, 7486-7514 (2019) · Zbl 1418.92179
[25] Liu, M.; Wang, K., On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl., 63, 871-886 (2012) · Zbl 1247.60085
[26] Liu, M.; Wang, K., Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps, Nonlinear Anal. Theory Methods Appl., 85, 204-213 (2013) · Zbl 1285.34047
[27] Liu, M.; Wang, K., Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410, 750-763 (2014) · Zbl 1327.92046
[28] Liu, M.; Zhu, Y., Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps, Nonlinear Anal. Hybrid Syst., 30, 225-239 (2018) · Zbl 1420.60071
[29] Lu, C.; Ding, X., Persistence and extinction of an impulsive stochastic logistic model with infinite delay, Osaka J. Math., 53, 1-29 (2016) · Zbl 1345.90006
[30] Lu, C.; Ding, X., Dynamical behavior of stochastic delay Lotka-Volterra competitive model with general Lévy jumps, Physica A, 531 (2019), Article 121730 · Zbl 07569431
[31] Lu, C.; Ding, X., Periodic solutions and stationary distribution for a stochastic predator-prey system with impulsive perturbations, Appl. Math. Comput., 350, 313-322 (2019) · Zbl 1428.34056
[32] Lv, J.; Zhang, Y.; Zou, X., Recurrence and strong stochastic persistence of a stochastic single-species model, Appl. Math. Lett., 89, 64-69 (2019) · Zbl 1426.60077
[33] Macdonald, N., (Time Lags in Biological Models. Time Lags in Biological Models, Lecture Notes in Biomathematics (1978), Springer-Verlag: Springer-Verlag Heidelberg) · Zbl 0403.92020
[34] Mao, W.; Hu, L.; Mao, X., Neutral stochastic functional differential equations with Lévy jumps under the local Lipschitz condition, Adv. Differential Equations, 57, 1-24 (2017) · Zbl 1422.34238
[35] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97, 96-110 (2002) · Zbl 1058.60046
[36] Qiu, H.; Deng, W., Optimal harvesting of a stochastic delay competitive Lotka-Volterra model with Lévy jumps, Appl. Math. Comput., 317, 210-222 (2019) · Zbl 1426.92062
[37] Rudnicki, R.; Pichór, K., Influence of stochastic perturbation on prey-predator systems, Math. Biosci., 206, 108-119 (2007) · Zbl 1124.92055
[38] Song, Y.; Baker, C. T.H., Qualitative behaviour of numerical approximations to Volterra integro-differential equations, J. Comput. Appl. Math., 172, 101-115 (2004) · Zbl 1059.65129
[39] Wang, Y.; Wu, F.; Mao, X., Stability in distribution of stochastic functional differential equations, Systems Control Lett., 132 (2019), Article 104513 · Zbl 1425.93299
[40] Wang, Y.; Wu, F.; Mao, X., Stability in distribution of stochastic functional differential equations, Syst. Control Lett., 132 (2019), Article 104513 · Zbl 1425.93299
[41] Wu, F.; Xu, Y., Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70, 641-657 (2009) · Zbl 1197.34164
[42] Wu, R.; Zou, X.; Wang, K., Dynamics of Logistic system driven Lévy noise under regime switching, Electron. J. Differ. Equ., 76, 1-16 (2014) · Zbl 1296.60160
[43] Xu, Y.; Wu, F.; Tan, Y., Stochastic Lotka-Volterra system with infinite delay, J. Comput. Appl. Math., 232, 472-480 (2009) · Zbl 1205.34103
[44] Yu, X.; Yuan, S.; Zhang, T., Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simul., 59, 359-374 (2018) · Zbl 1524.37084
[45] Yu, X.; Yuan, S.; Zhang, T., Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling, Nonlinear Anal. Hybrid Syst., 34, 209-225 (2019) · Zbl 1435.34056
[46] Zhang, S.; Meng, X.; Feng, T.; Zhang, T., Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Anal., 26, 19-37 (2017) · Zbl 1376.92057
[47] Zou, X.; Wang, K., Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simul., 19, 1557-1568 (2014) · Zbl 1457.65007
[48] Zou, X.; Wang, K., Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps, Nonlinear Anal.-Hybrid Syst., 13, 32-44 (2014) · Zbl 1325.91040
[49] Zuo, W.; Jiang, D.; Sun, X.; Hayat, T.; Alsaedi, A., Long-time behaviors of a stochastic cooperative Lotka-Volterra system with distributed delay, Physica A, 506, 542-559 (2018) · Zbl 1514.92107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.