Permanence and partial extinction in a delayed three-species food chain model with stage structure and time-varying coefficients
Authors
Huanyan Xi
- Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China
Lihong Huang
- Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China
Yuncheng Qiao
- Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China
Huaiyu Li
- Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China
Chuangxia Huang
- Department of Mathematics and Statistics, Changsha University of Science and Technology, 410114, Changsha, P. R. China
Abstract
By taking full consideration of maturity (\(\tau_{1}\) represents the maturity of predator and \(\tau_{2}\) represents the maturity of top predator)
and the effects of environmental parameters, a new delayed three-species food chain model with stage structure and time-varying coefficients is
established. With the help of the comparison theorem and the technique of mathematical analysis, the positivity and boundedness of solutions of
the model are investigated. Furthermore, some sufficient conditions on the permanence and partial extinction of the system are derived.
Some interesting findings show that the delays have great impacts on the permanence for the system. More precisely, if \(\tau_{2}\in(n, +\infty)\),
then the system is partially extinct: on one hand, if \(\tau_{1}\in(0,n_{1})\) and \(\tau_{2}\in(n, +\infty)\), then the prey and predator species
will coexist, i.e., both the prey and predator species are always permanent, yet the top predator species will go extinct eventually.
On the other hand, if \(\tau_{1}\in(n_{4},+\infty)\) and \(\tau_{2}\in(n, +\infty)\), where \(n_{4}\) is greater than \(n_{1}\),
then all predator species will become extinct eventually. Numerical simulations are great well agreement with the theoretical results.
Keywords
- Food chain model
- delay
- stage structure
- permanence
- extinction
References
[1] W. G. Aiello, H. I. Freedman, A time delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139–153.
[2] H. Baek, H. H. Lee, Permanence of a three-species food chain system with impulsive perturbations, Kyungpook Math. J., 48 (2008), 503–514.
[3] A. A. Berryman, The Orgins and Evolution of Predator-Prey Theory, Ecology, 73 (1992), 1530–1535.
[4] H. Boudjellaba, T. Sari, Stability loss delay in harvesting competing populations, J. Differential Equations, 152 (1999), 394–408.
[5] K. Charles Kendeigh, Animal Ecology, Prentice-Hall, New York, (1961).
[6] F. Chen, H. Wang, Y. Lin, W. Chen, Global stability of a stage-structured predator-prey system, Appl. Math. Comput., 223 (2013), 45–53.
[7] J. Cui, L. Chen, W. Wang, The effect of dispersal on population growth with stage-structure, Comput. Math. Appl., 39 (2000), 91–102.
[8] J. M. Cushing, Periodic time-dependent predator-prey system, SIAM J. Appl. Math., 32 (1977), 82–95.
[9] K. Das, K. Reddy, M. Srinivas, N. Gazi, Chaotic dynamics of a three species prey-predator competition model with noise in ecology, Appl. Math. Comput., 231 (2014), 117–133.
[10] L. Deng, X. Wang, M. Peng, Hopf bifurcation analysis for a ratio-dependent predator-prey system with two delays and stage structure for the predator, Appl. Math. Comput., 231 (2014), 214–230.
[11] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433–463.
[12] W. Fengying,W. Ke, Permanence of variable coefficients predator-prey system with stage structure, Appl. Math. Comput., 180 (2006), 594–598.
[13] X. He, Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198 (1996), 355–370.
[14] H. Hu, L. Huang, Stability and Hopf bifurcation in a delayed predator-prey system with stage structure for prey, Nonlinear Anal. Real World Appl., 11 (2010), 2757–2769.
[15] G. Hu, X. Li, Stability and Hopf bifurcation for a delayed predator-prey model with disease in the prey, Chaos Solitons Fractals, 45 (2012), 229–237.
[16] C. Huang, J. Cao, Convergence dynamics of stochastic Cohen-Grossberg neural networks with unbounded distributed delays, IEEE Trans. Neural Netw., 22 (2011), 561–572.
[17] C. Huang, J. Cao, J. Cao, Stability analysis of switched cellular neural networks: A mode-dependent average dwell time approach, Neural Netw., 82 (2016), 84–99.
[18] C. Huang, C. Peng, X. Chen, F. Wen, Dynamics analysis of a class of delayed economic model, Abstr. Appl. Anal., 2013 (2013), 12 pages.
[19] C. Huang, Z. Yang, T. Yi, X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101–2114.
[20] T. K. Kar, S. Jana, Stability and bifurcation analysis of a stage structured predator-prey model with time delay, Appl. Math. Comput., 219 (2012), 3779–3792.
[21] Y. Kuang, J. W.-H. So, Analysis of a delayed two-stage population with space-limited recruitment, SIAM J. Appl. Math., 55 (1995), 1675–1696.
[22] S. Liu, B. Kouche, N.-E. Tatar, Permanence extinction and global asymptotic stability in a stage structured system with distributed delays, J. Math. Anal. Appl., 301 (2005), 187–207.
[23] Z.-H. Ma, Z.-Z. Li, S.-F. Wang, T. Li, F.-P. Zhang, Permanence of a predator-prey system with stage structure and time delay, Appl. Math. Comput., 201 (2008), 65–71.
[24] W. Mbava, J. Y. T. Mugisha, J. W. Gonsalves, Prey, predator and super-predator model with disease in the super-predator, Appl. Math. Comput., 297 (2017), 92–114.
[25] B. Patra, A. Maiti, G. Samanta, Effect of time-delay on a ratio-dependent food chain model, Nonlinear Anal. Model. Control, 14 (2009), 199–216.
[26] S. Ruan, Y. Tang, W. Zhang, Versal unfoldings of predator-prey systems with ratio-dependent functional response, J. Differential Equations, 249 (2010), 1410–1435.
[27] C. Shen, M. You, Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion, Appl. Math. Comput., 217 (2010), 1825–1830.
[28] Z. Teng, Y. Yu, The extinction in nonautonomous prey-predator Lotka-Volterra systems, Acta Math. Appl. Sinica, 15 (1999), 401–408.
[29] B. Tian, S. Zhong, Z. Liu, Extinction and persistence of a nonautonomous stochastic food-chain system with impulsive perturbations, Int. J. Biomath., 2016 (2016), 26 pages.
[30] K.Wang, Permanence and global asymptotical stability of a predator-prey model with mutual interference, Nonlinear Anal. Real World Appl., 12 (2011), 1062–1071.
[31] W. Wang, G. Mulone, F. Salemi, V. Salone, Permanence and stability of a stage-structured predator-prey model, J. Math. Anal. Appl., 262 (2001), 499–528.
[32] R. Xu, M. A. J. Chaplain, F. A. Davidson, Global stability of Lotka-Volterra type predator-prey model with stage structure and time delay, Appl. Math. Comput., 159 (2004), 863–880.
[33] C. Xu, S. Yuan, T. Zhang, Global dynamics of a predator-prey model with defense mechanism for prey, Appl. Math. Lett., 62 (2016), 42–48.
[34] P. Yongzhen, M. Guo, C. Li, A delay digestion process with application in a three-species ecosystem, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4365–4378.
[35] H. Zhang, L. Chen, R. Zhu, Permanence and extinction of a periodic predator-prey delay system with functional response and stage structure for prey, Appl. Math. Comput., 184 (2007), 931–944.