Abstract
In this paper, a Beddington-DeAngelis amensalism system with weak Allee effect on the first species are introduced and investigated. The existence and stability of all possible trivial, semi-trivial and interior equilibria of the model are studied. By utilizing Sotomayor’s theorem, bifurcation analysis has been proposed and obtain one saddle-node bifurcation. Furthermore, in view of Poincaré transformation, the behaviors near infinity and the nonexistence of close orbits are obtained and lead to the presentation of all possible global phase portraits. The global phase portrait in \(R_{5}\) with two stable node \(E_{2}\) and \(E_{1}^{*}\) is a new case due to the appearance of bistable structure. Finally, some numerical examples are offered to verify and extend the analytical results and visualize the interesting phenomenon.
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References
Allee, W.C.: Animal Aggregations: A Study in General Sociology. University of Chicago Press, Chicago (1931)
Chen, B.G.: Dynamic behaviors of a non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations. Adv. Diff. Equ. 111, 255–261 (2018)
Guan, X.Y., Chen, F.D.: Dynamical analysis of a two species Amensalism model with Beddington-DeAngelis functional response and Allee effect on the second species. Nonlinear Anal. RWA 48, 71–93 (2019)
Halder, S., Bhattacharyya, J., Pal, S.: Comparative studies on a predatorCprey model subjected to fear and Allee effect with type I and type II foraging. J. Appl. Math. Comput. 62, 93–118 (2020)
Ji, W.: On a population model with Allee effects and environmental perturbations. J. Appl. Math. Comput. 64, 749–764 (2020)
Luo, D.M., Wang, Q.R.: Global dynamics of a Holling-II amensalism system with nonlinear growth rate and Allee effect on the first species. Int. J. Bifurcat. Chaos, Accepted for publication
Lin, Q.F., Zhou, X.Y.: On the existence of positive periodic solution of a amensalism model with Holling II functional response. Commun. Math. Biol. Neurosci. 3 (2017)
Liu, Y., Zhao, L., Huang, X.Y., Deng, H.: Stability and bifurcation analysis of two species amensalism model with Michaelis–Menten type harvesting and a cover for the first species. Adv. Differ. Equ. 295 (2018)
Mandal, P.S., Kumar, U., Garain, K., Sharma, R.: Allee effect can simplify the dynamics of a prey-predator model. J. Appl. Math. Comput. 63, 739–770 (2020)
Sharma, S., Samanta, G.P.: A ratio-dependent predator-prey model with Allee effect and disease in prey. J. Appl. Math. Comput. 47, 345–364 (2015)
Sun, G.C.: Qualitative analysis on two populations amensalism model. J. Jiamusi Univ. (Natl. Sci. Ed.) 21(3), 283–286 (2003)
Wang, M.H., Kot, M.: Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 171(1), 83–97 (2001)
Wei, Z., Xia, Y.H., Zhang, T.H.: Stability and bifurcation analysis of an amensalism model with weak Allee effect. Qual. Theory Dyn. Syst. 19(23), 1–15 (2020)
Wu, R.X., Zhao, L., Lin, Q.X.: Stability analysis of a two species amensalism model with Holling II functional response and a cover for the first species. J. Nonlinear Funct. Anal. 2016, 46 (2016)
Wu, R.X.: A two species amensalism model with non-monotonic functional response. Commun. Math. Biol. Neurosci. 2016, 19 (2016)
Xie, X.D., Chen, F.D., He, M.X.: Dynamic behaviors of two species amensalism model with a cover for the first species. J. Math. Comput. Sci. 16, 395–401 (2016)
Yang, L., Zhong, S.: Dynamics of an impulsive diffusive ecological model with distributed delay and additive Allee effect. J. Appl. Math. Comput. 48, 1–23 (2015)
Zhang, J.F.: Bifurcated periodic solutions in an amensalism system with strong generic delay kernel. Math. Methods Appl. Sci. 36(1), 113–124 (2013)
Zhang, Z.: Stability and bifurcation analysis for a amensalism system with delays. Math. Numer. Sin. 30(2), 213–224 (2008)
Zhang, Z.F., Ding, T.R., Huang, W.Z., Dong, Z.X.: Qualitative Theory of Differential Equations (in Chinese). Science Press, Beijing (1992)
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This research was supported by the National Natural Science Foundation of China (Nos. 11671406 and 12071491).
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Luo, D. Global dynamics of a Beddington-DeAngelis amensalism system with weak Allee effect on the first species. J. Appl. Math. Comput. 68, 655–680 (2022). https://doi.org/10.1007/s12190-021-01533-w
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DOI: https://doi.org/10.1007/s12190-021-01533-w
Keywords
- Global dynamics
- Amensalism system
- Stability and bifurcation analysis
- Weak Allee effect
- Beddington-DeAngelis functional response