Abstract
In this paper, a generalized population model with delays and harvesting term is studied, which includes some well-known models, such as Gilpin–Ayala competitive model and Logarithmic model. By a suitable Lyapunov function and the Banach fixed point theorem, we obtain the existence and uniqueness of globally attractive pseudo almost periodic solution of the model and prove its permanence. Some examples and numerical simulations are given to illustrate the feasibility and usefulness of the model and results.
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References
Alzabut JO, Stamov GT, Sermutlu E (2011) Positive almost periodic solutions for a delay Logarithmic population model. Math Comput Model 53(1):161–167. https://doi.org/10.1016/j.mcm.2010.07.029
Amdouni M, Chérif F (2018) The pseudo almost periodic solutions of the new class of Lotka-Volterra recurrent neural networks with mixed delays. Chaos Solit Fract 113:79–88. https://doi.org/10.1016/j.chaos.2018.05.004
Amdouni M, Chérif F, Alzabut J (2021) Pseudo almost periodic solutions and global exponential stability of a new class of nonlinear generalized Gilpin-Ayala competitive model with feedback control with delays. Comput Appl Math 40(3):91. https://doi.org/10.1007/s40314-021-01464-zcom
Ammar B, Chérif F, Alimi AM (2012) Existence and uniqueness of pseudo almost-periodic solutions of recurrent neural networks with time-varying coefficients and mixed delays. IEEE Trans Neural Netw Learn Syst 23(1):109–118. https://doi.org/10.1109/TNNLS.2011.2178444
Ayala FJ, Gilpin ME, Eherenfeld JG (1973) Competition between species: theoretical models and experimental tests. Theor Popul Biol 4:331–356. https://doi.org/10.1016/0040-5809(73)90014-2
Brauer F, Castilli-Chavez C (2001) Mathematical Models in Population Biology and Epidemiology. Springer, Berlin
Cantrell RS, Cosner C (2003) Spatial Ecology Via Reaction-Diffusion Equations. John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England
Chen F (2006) Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model. Nonlinear Anal Real World Appl 7(4):895–915. https://doi.org/10.1016/j.nonrwa.2005.04.007
Chérif F (2015) Pseudo almost periodic solution of nicholson’s blowflies model with mixed delays. Appl Math Model 39(17):5152–5163. https://doi.org/10.1016/j.apm.2015.03.043
Cieutat P, Fatajou S, N’Guérékata GM (2010) Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations. Appl Anal 89:11–27. https://doi.org/10.1080/00036810903397503
Coppel WA (1978) Dichotomies in Stability Theory. Lecture Notes in Mathematics. Springer, Berlin
Diagana T (2013) Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces. Springer, New York
Fan M, Wang K (2000) Global periodic solutions of a generalized n-species Gilpin-Ayala competition model. Comput Math Appl 40(10–11):1141–1151. https://doi.org/10.1016/S0898-1221(00)00228-5
Fink AM (1974) Almost Periodic Differential Equations. Springer, New York
Gilpin ME, Ayala FJ (1973) Global models of growth and competition. Proc. Nat. Acad. Sci. 70(12):3590–3593. https://doi.org/10.1073/pnas.70.12.3590
Gopalsamy K (1986) Permanence in Lotka-Volterra equations: linked prey-predator systems. Math Biosci 82(2):165–191. https://doi.org/10.1016/0025-5564(86)90136-7
Gopalsamy K (1992) Stability and Oscillation in Delay Differential Equations of Population Dynamic. Kluwer Academic Publisher, Boston
Hale J (1977) Theory of Functional Differential Equations. Springer, NewYork
He CY (1992) Almost Periodic Differential Equations. Higher Education Publishing House, Beijing ((Chinese))
Li Z, Chen F (2009) Almost periodic solutions of a discrete almost periodic logistic equation. Math Comput Model 50:254–259. https://doi.org/10.1016/j.mcm.2008.12.017
Li Y, Yang K (2001) Periodic solutions of periodic delay Lotka-Volterra equations and systems. J Math Anal Appl 255(1):260–280. https://doi.org/10.1006/jmaa.2000.7248
Liu ZJ (2002) Positive periodic solutions for delay multispecies Logrithmic population model. Chin. J. Eng. Math. 19(4):11–16. https://doi.org/10.1038/sj.cr.7290130
Liu PY, Li HX (2020) Global stability of autononmous and nonautonomous hepatitis B virus models in patchy environment. J. Appl. Anal. Appl. Comput 10(5):1771–1799. https://doi.org/10.11948/20190191
Liu M, Wang K (2012) Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays. Commun Nonlinear Sci Numer Simul 17:3115–3123. https://doi.org/10.1016/j.cnsns.2011.09.021
Luo ZG, Luo LP (2013) Existence and stability of positive periodic solutions for a neutral multispecies Logarithmic population model with feedback control and impulse. Abstr Appl Anal 2013:741043. https://doi.org/10.1155/2013/741043
Stamov GT, Petrov N (2008) Lyapunov-Razumikhin method for existence of almost periodic solutions of impulsive differential-difference equations. Nonlinear Stud 15:151–163
Wang Q (2020) Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Commun. Pure Appl. Anal. 19(6):3245–3255. https://doi.org/10.3934/cpaa.2020142
Wu Y, Xia YH, Deng SF (2021) Existence and stability of pseudo almost periodic solutions for a delayed multispecies logarithmic population model with feedback control. Qual. Theory Dyn. Syst. 20(1):6. https://doi.org/10.1007/s12346-020-00445-7
Xing YF, Li HX (2021) Almost periodic solutions for a SVIR epidemic model with relapse. Math Biosci Eng 18(6):7191–7217. https://doi.org/10.3934/mbe.2021356
Xiong J, Li X, Wang H (2019) Global asymptotic stability of a Lotka-Volterra competition model with stochasticity in inter-specific competition. Appl Math Lett 89:58–63. https://doi.org/10.1016/j.aml.2018.09.018
Yuan R (2007) On almost periodic solutions of logistic delay differential equations with almost periodic time dependence. J Math Anal Appl 330:780–798. https://doi.org/10.1016/j.jmaa.2006.08.027
Zhang CY (1992) Almost Periodic Functions and Ergodicity. Kluwer Academic Publishers, London
Zhang CY (1994) Pseudo almost-periodic solutions of some differential-equations. J Math Anal Appl 181(1):62–76. https://doi.org/10.1006/jmaa.1994.1005
Zhao K, Yuan Y (2010) Four positive periodic solutions to a periodic Lotka-Volterra predatory-prey system with harvesting terms. Nonlinear Anal Real World Appl 11(4):2448–2455. https://doi.org/10.1016/j.nonrwa.2009.08.001
Zheng FX, Li HX (2022) Pseudo almost automorphic mild solutions to non-autonomous differential equations in the “strong topology’’. Banach J. Math. Anal. 16(1):1–32. https://doi.org/10.1007/s43037-021-00165-3
Acknowledgements
This paper is supported by the National Natural Science Foundation of China (No. 11971329)
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Xing, Y., Li, HX. Pseudo almost periodic solutions and global exponential stability of a generalized population model with delays and harvesting term. Comp. Appl. Math. 41, 350 (2022). https://doi.org/10.1007/s40314-022-02049-0
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DOI: https://doi.org/10.1007/s40314-022-02049-0
Keywords
- Pseudo almost periodic solution
- Logarithmic model
- Gilpin–Ayala competitive model
- Global exponential stability