Dynamical behavior of a stochastic delayed one-predator and two-mutualistic-prey model with Markovian switching and different functional responses. (English) Zbl 1417.92132
Summary: We propose a stochastic delayed one-predator and two-mutualistic-prey model perturbed by white noise and telegraph noise. By the \(M\)-matrix analysis and Lyapunov functions, sufficient conditions of stochastic permanence and extinction are established, respectively. These conditions are all dependent on the subsystems’ parameters and the stationary probability distribution of the Markov chain. We also investigate another asymptotic property and finally give two examples and numerical simulations to illustrate main results.
MSC:
| 92D25 | Population dynamics (general) |
References:
| [1] | May, R., Theoretical Ecology: Principles and Applications (1976), Philadelphia, Pa, USA: WB Saunders Company, Philadelphia, Pa, USA |
| [2] | Dean, A. M., A simple model of mutualism, The American Naturalist, 121, 3, 409-417 (1983) · doi:10.1086/284069 |
| [3] | Wolin, C. L.; Lawlor, L. R., Models of facultative mutualism: density effects, The American Naturalist, 124, 6, 843-862 (1984) · doi:10.1086/284320 |
| [4] | Boucher, D. H., Lotka-volterra models of mutualism and positive density-dependence, Ecological Modelling, 27, 3-4, 251-270 (1985) · doi:10.1016/0304-3800(85)90006-7 |
| [5] | Wright, D. H., A simple, stable model of mutualism incorporating handling time, American Naturalist, 134, 4, 664-667 (1989) · doi:10.1086/285003 |
| [6] | García-Algarra, J.; Galeano, J.; Pastor, J. M.; Iriondo, J. M.; Ramasco, J. J., Rethinking the logistic approach for population dynamics of mutualistic interactions, Journal of Theoretical Biology, 363, 332-343 (2014) · Zbl 1309.92067 · doi:10.1016/j.jtbi.2014.08.039 |
| [7] | Holling, C. S., Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91, 7, 385-398 (1959) · doi:10.4039/ent91385-7 |
| [8] | Jang, S. R.-J., Dynamics of herbivore-plant-pollinator models, Journal of Mathematical Biology, 44, 2, 129-149 (2002) · Zbl 0991.92031 · doi:10.1007/s002850100117 |
| [9] | Kar, T. K.; Ghorai, A., Dynamic behaviour of a delayed predator-prey model with harvesting, Applied Mathematics and Computation, 217, 22, 9085-9104 (2011) · Zbl 1215.92065 · doi:10.1016/j.amc.2011.03.126 |
| [10] | Wang, Y.; Zhao, M., Dynamic analysis of an impulsively controlled predator-prey model with Holling type IV functional response, Discrete Dynamics in Nature and Society, 2012 (2012) · Zbl 1233.92079 · doi:10.1155/2012/141272 |
| [11] | Pal, P. J.; Mandal, P. K.; Lahiri, K. K., A delayed ratio-dependent predator-prey model of interacting populations with Holling type III functional response, Nonlinear Dynamics, 76, 1, 201-220 (2014) · Zbl 1319.92047 · doi:10.1007/s11071-013-1121-3 |
| [12] | Zu, L.; Jiang, D.; O’Regan, D., Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regime switching, Communications in Nonlinear Science and Numerical Simulation, 29, 1-3, 1-11 (2015) · Zbl 1510.92195 · doi:10.1016/j.cnsns.2015.04.008 |
| [13] | Liu, M.; Bai, C., Global asymptotic stability of a stochastic delayed predator-prey model with Beddington-DeAngelis functional response, Applied Mathematics and Computation, 226, 581-588 (2014) · Zbl 1354.92067 · doi:10.1016/j.amc.2013.10.052 |
| [14] | Tripathi, J. P.; Tyagi, S.; Abbas, S., Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response, Communications in Nonlinear Science and Numerical Simulation, 30, 1-3, 45-69 (2016) · Zbl 1489.92125 · doi:10.1016/j.cnsns.2015.06.008 |
| [15] | Ouyang, M.; Li, X., Permanence and asymptotical behavior of stochastic prey-predator system with Markovian switching, Applied Mathematics and Computation, 266, 539-559 (2015) · Zbl 1410.34144 · doi:10.1016/j.amc.2015.05.083 |
| [16] | Mougi, A.; Kondoh, M., Stability of competition-antagonism-mutualism hybrid community and the role of community network structure, Journal of Theoretical Biology, 360, 54-58 (2014) · Zbl 1343.92544 · doi:10.1016/j.jtbi.2014.06.030 |
| [17] | Kuang, Y., Delay Differential Equations: with Applications in Population Dynamics. Delay Differential Equations: with Applications in Population Dynamics, Mathematics in Science and Engineering, 191 (1993), Boston, Mass, USA: Academic Press, Boston, Mass, USA · Zbl 0777.34002 |
| [18] | Mao, X.; Yuan, C., Stochastic Differential Equations with Markovian Switching (2006), London, UK: Imperial College Press, London, UK · Zbl 1126.60002 · doi:10.1142/p473 |
| [19] | Jana, D.; Agrawal, R.; Upadhyay, R. K., Dynamics of generalist predator in a stochastic environment: effect of delayed growth and prey refuge, Applied Mathematics and Computation, 268, 1072-1094 (2015) · Zbl 1410.34136 · doi:10.1016/j.amc.2015.06.098 |
| [20] | Zou, L.; Xiong, Z.; Shu, Z., The dynamics of an eco-epidemic model with distributed time delay and impulsive control strategy, Journal of the Franklin Institute, 348, 9, 2332-2349 (2011) · Zbl 1252.34095 · doi:10.1016/j.jfranklin.2011.06.023 |
| [21] | Liu, M.; Mandal, P. S., Dynamical behavior of a one-prey two-predator model with random perturbations, Communications in Nonlinear Science and Numerical Simulation, 28, 1-3, 123-137 (2015) · Zbl 1510.92169 · doi:10.1016/j.cnsns.2015.04.010 |
| [22] | Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 59, 2, 249-256 (1978) · doi:10.2307/1936370 |
| [23] | Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and Applications, 334, 1, 69-84 (2007) · Zbl 1113.92052 · doi:10.1016/j.jmaa.2006.12.032 |
| [24] | Li, X.; Gray, A.; Jiang, D.; Mao, X., Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, Journal of Mathematical Analysis and Applications, 376, 1, 11-28 (2011) · Zbl 1205.92058 · doi:10.1016/j.jmaa.2010.10.053 |
| [25] | Li, X.; Mao, X.; Shen, Y., Approximate solutions of stochastic differential delay equations with Markovian switching, Journal of Difference Equations and Applications, 16, 2-3, 195-207 (2010) · Zbl 1187.60046 · doi:10.1080/10236190802695456 |
| [26] | Kloeden, P. E.; Shardlow, T., The Milstein scheme for stochastic delay differential equations without using anticipative calculus, Stochastic Analysis and Applications, 30, 2, 181-202 (2012) · Zbl 1247.65005 · doi:10.1080/07362994.2012.628907 |
| [27] | Liu, M.; Wang, K.; Wu, Q., Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bulletin of Mathematical Biology, 73, 9, 1969-2012 (2011) · Zbl 1225.92059 · doi:10.1007/s11538-010-9569-5 |
| [28] | Schreiber, S. J., Persistence for stochastic difference equations: a mini-review, Journal of Difference Equations and Applications, 18, 8, 1381-1403 (2012) · Zbl 1258.39010 · doi:10.1080/10236198.2011.628662 |
| [29] | Vrkoč, I.; Křivan, V., Asymptotic stability of tri-trophic food chains sharing a common resource, Mathematical Biosciences, 270, 90-94 (2015) · Zbl 1364.92034 · doi:10.1016/j.mbs.2015.10.005 |
| [30] | Hu, Y.; Yan, M.; Xiang, Z., An impulsively controlled three-species prey-predator model with stage structure and birth pulse for predator, Discrete Dynamics in Nature and Society, 2015 (2015) · Zbl 1418.92098 · doi:10.1155/2015/380492 |
| [31] | Liu, M.; Bai, C.; Deng, M.; Du, B., Analysis of stochastic two-prey one-predator model with Lévy jumps, Physica A: Statistical Mechanics and Its Applications, 445, 176-188 (2016) · Zbl 1400.92437 · doi:10.1016/j.physa.2015.10.066 |
| [32] | Liu, Q., Asymptotic properties of a stochastic \(n\)-species Gilpin-Ayala competitive model with Lévy jumps and Markovian switching, Communications in Nonlinear Science and Numerical Simulation, 26, 1-3, 1-10 (2015) · Zbl 1440.92055 · doi:10.1016/j.cnsns.2015.01.007 |
| [33] | Chen, C.; Kang, Y., Dynamics of a stochastic multi-strain SIS epidemic model driven by Lévy noise, Communications in Nonlinear Science and Numerical Simulation, 42, 379-395 (2017) · Zbl 1473.92037 · doi:10.1016/j.cnsns.2016.06.012 |
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.
