Dynamic behaviors of a stage structure single species model with cannibalism. (English) Zbl 1426.34056
Summary: A stage structure single species model with cannibalism takes the form \[\begin{aligned}\frac{dx}{dt}&=\alpha y-\gamma x-\Omega x-\theta xy, \\ \frac{dy}{dt}&=\Omega x-\beta y \end{aligned}\] is revisited in this paper, where \(\alpha,\gamma,\Omega,\theta\) and \(\beta\) are all positive constants. We first show by numeric simulation that one of the main result of S. G. Gao is incorrect [“Stability for stage structure single species model with cannibalism”, J. Anshan Normal Univ. 4, 41–43 (2002)]. Then by constructing some suitable Lyapunov function, sufficient conditions which ensure the globally asymptotically stability of the boundary equilibrium of above system is obtained.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 92D25 | Population dynamics (general) |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
References:
| [1] | F. D. Chen, W. L. Chen, Y. M. Wu and Z. Z. Ma, Permanence of a stage-structured predator-prey system, Appl. Math. Comput., 219(2013), 8856-8862. · Zbl 1288.92016 |
| [2] | F. D. Chen, X. D. Xie and Z. Li, Partial survival and extinction of a delayed predator-prey model with stage structure, Appl. Math. Comput., 219(2012), 4157- 4162. · Zbl 1311.92154 |
| [3] | Z. H. Ma, Z. Z. Li, S. Wang, T. Li and F. Zhang, Permanence of a predator-prey system with stage structure and time delay, Appl. Math. Comput., 201(2008), 65-71. · Zbl 1143.92329 |
| [4] | T. T. Li, F. D. Chen, J. H. Chen and Q. X. Lin, Stability of a mutualism model in plant-pollinator system with stage-structure and the Beddington-DeAngelis functional response, J. Nonlinear Funct. Anal., 2017(2017), Article ID 50. |
| [5] | Z. Li and F. D. Chen, Extinction in periodic competitive stage-structured LotkaVolterra model with the e¤ects of toxic substances, J. Comput. Appl. Math., 231(2009), 143-153. · Zbl 1165.92322 |
| [6] | Z. Li, M. A. Han and F. D. Chen, Global stability of stage-structured predator-prey model with modi…ed Leslie-Gower and Holling-type II schemes, Int. J. Biomath., 5(2012), 1250057, 13 pp. · Zbl 1297.92066 |
| [7] | Z. Li, M. Han and F. D. Chen, Global stability of a predator-prey system with stage structure and mutual interference, Discrete Contin. Dyn. Syst., Ser. B, 19(2014), 173-187. · Zbl 1287.34071 |
| [8] | F. D. Chen, X. D. Xie and X. X. Chen, Dynamic behaviors of a stage-structured cooperation model, Commun. Math. Biol. Neurosci., 2015(2015), Article ID 4. |
| [9] | X. Lin, X. Xie, F. D. Chen and T. T. Li, Convergences of a stage-structured predator-prey model with modi…ed Leslie-Gower and Holling-type II schemes, Adv. Di¤erence Equ. 2016, Paper No. 181, 19 pp. · Zbl 1419.34158 |
| [10] | F. D. Chen, H. N. Wang, Y. H. Lin and W. L. Chen, Global stability of a stagestructured predator-prey system, Appl. Math. Comput., 223(2013), 45-53. · Zbl 1329.92101 |
| [11] | L. Q. Pu, Z. S. Miao and R. Y. Han, Global stability of a stage-structured predatorprey model, Commun. Math. Biol. Neurosci., 2015(2015), Article ID 5. |
| [12] | R. Y. Han, L. Y. Yang and Y. L. Xue, Global attractivity of a single species stagestructured model with feedback control and in…nite delay, Commun. Math. Biol. Neurosci., 2015(2015), Article ID 6. |
| [13] | Z. H. Ma and S. F. Wang, Permanence of a food-chain system with stage structure and time delay, Commun. Math. Biol. Neurosci., 2017(2017), Article ID 15. |
| [14] | H. L. Wu and F. D. Chen, Harvesting of a single-species system incorporating stage structure and toxicity, Discrete Dyn. Nat. Soc., Volume 2009, Article ID 290123, 16 pages. · Zbl 1197.34085 |
| [15] | S. Khajanchi and S. Banerjee, Role of constant prey refuge on stage structure predator-prey model with ratio dependent functional response, Appl. Math. Comput., 314(2017), 193-198. · Zbl 1426.34098 |
| [16] | Q. Lin, X. Xie and F. Chen, Dynamical analysis of a logistic model with impulsive Holling type-II harvesting, Adv. Di¤erence Equ. 2018, Paper No. 112, 22 pp. · Zbl 1445.92242 |
| [17] | Y. Xue, L. Pu and L. Y. Yang, Global stability of a predator-prey system with stage structure of distributed-delay type, Commun. Math. Biol. Neurosci, 2015(2015), Article ID 12. |
| [18] | Y. Lu, K. A. Pawelek and S. Q. Liu, A stage-structured predator-prey model with predation over juvenile prey, Appl. Math. Comput., 297(2017), 115-130. · Zbl 1411.34081 |
| [19] | R. K. Naji and S. JasimMajeed, The dynamical analysis of a prey-predator model with a refuge-stage structure prey population, International Journal of Di¤erential Equations, Volume 2016, Article ID 2010464, 10 pages. · Zbl 1459.92090 |
| [20] | W. S. Yang, X. P. Li and Z. J. Bai, Permanence of periodic Holling typeIV predator-prey system with stage structure for prey, Math. Comput. Model., 48(2008), 677-684. · Zbl 1156.34327 |
| [21] | C. Q. Lei, Dynamic behaviors of a non-selective harvesting single species stage structure system incorporating partial closure for the populations, Adv. Di¤er. Equ., 2018(2018): 245. · Zbl 1446.37096 |
| [22] | L. S. Chen, Mathematical Ecological Modeling and Research Method, Second Edition, Science Press, 2017. (in Chinese) |
| [23] | K. Yang, Z. S. Miao, F. D. Chen and X. D. Xie, In‡uence of single feedback control variable on an autonomous Holling-II type cooperative system, J. Math. Anal. Appl., 435(2016), 874-888. · Zbl 1334.34108 |
| [24] | Y. Liu, X. D. Xie and Q. F. Lin, Permanence, partial survival, extinction, and global attractivity of a nonautonomous harvesting Lotka-Volterra commensalism model incorporating partial closure for the populations, Adv. Di¤erence Equ. 2018, Paper No. 211, 16 pp. · Zbl 1448.92224 |
| [25] | R. X. Wu, L. Li and X. Y. Zhou, A commensal symbiosis model with Holling type functional response, J. Math. Comput. Sci., 16(2016), 364-371. |
| [26] | Q. Lin, X. Xie, F. D. Chen and Q. F. Lin, Dynamical analysis of a logistic model with impulsive Holling type-II harvesting, Adv. Di¤erence Equ. 2018, Paper No. 112, 22 pp. · Zbl 1445.92242 |
| [27] | F. Chen, X. Chen and S. Y. Huang, Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the e¤ect of toxic substances, Open Math., 14(2016), 1157-1173. · Zbl 1357.34085 |
| [28] | X. Xie, Y. Xue, R. X. Wu and L. Zhao, Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton, Adv. Di¤erence Equ. 2016, Paper No. 258, 13 pp. · Zbl 1418.92142 |
| [29] | F. Chen, X. Xie, Z. S. Miao and L. Q. Pu, Extinction in two species nonautonomous nonlinear competitive system, Appl. Math. Comput., 274(2016), 119-124. · Zbl 1410.34135 |
| [30] | S. |
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