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November  2020, 25(11): 4361-4382. doi: 10.3934/dcdsb.2020101

Positive periodic solution for generalized Basener-Ross model

Zhibo Cheng , and  Xiaoxiao Cui

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

* Corresponding author: Zhibo Cheng

Received  September 2019 Revised  November 2019 Published  November 2020 Early access  April 2020

Fund Project: The first author is supported by National Natural Science Foundation of China (11501170), China Postdoctoral Science Foundation funded project (2016M590886), Young backbone teachers of colleges and universities in Henan Province (2017GGJS057), Fundamental Research Funds for the Universities of Henan Province (NSFRF170302)

This paper is devoted to the existence of at least one positive periodic solution for generalized Basener-Ross model with time-dependent coefficients. Our proof is based on Manásevich-Mawhin continuation theorem, Leray-Schauder alternative principle, fixed point theorem in cones. Moreover, we obtain that there are at least two positive periodic solutions for this model.

Keywords: Basener-Ross model, positive periodic solutions, Manásevich-Mawhin continuation theorem, Leray-Schauder alternative principle, fixed point theorem in cones.
Mathematics Subject Classification: Primary: 34B16, 34B18; Secondary: 34C25.
Citation: Zhibo Cheng, Xiaoxiao Cui. Positive periodic solution for generalized Basener-Ross model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4361-4382. doi: 10.3934/dcdsb.2020101
References:
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F. Güngör and P. J. Torres, Integrability of the Basener-Ross model with time-dependent coefficients, SeMA J., 76 (2019), 485-493.  doi: 10.1007/s40324-019-00187-w.  Google Scholar

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A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theoret. Biol., 236 (2005), 276-290.  doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar

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R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p$-Laplacian-like operators, J. Differential Equations, 145 (1998), 367-393.  doi: 10.1006/jdeq.1998.3425.  Google Scholar

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D. O'Regan, Existence Theory for Nonlinear Ordinary Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-017-1517-1.  Google Scholar

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J. Ren, D. Zhu and H. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1843-1865.  doi: 10.3934/dcdsb.2018240.  Google Scholar

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Y. Wang, H. Lian and W. Ge, Periodic solutions for a second order nonlinear functional differential equation, Appl. Math. Lett., 20 (2007), 110-115.  doi: 10.1016/j.aml.2006.02.028.  Google Scholar

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show all references

References:
[1]

P. Amarasekare, Effects of temperature on consumer-resource interactions, J. Animal Ecology, 84 (2015), 665-679.  doi: 10.1111/1365-2656.12320.  Google Scholar

[2]

B. Basener and D. S. Ross, Booming and crashing populations and Easter Island, SIAM J. Appl. Math., 65 (2004/05), 684-701.  doi: 10.1137/S0036139903426952.  Google Scholar

[3]

M. Chen, M. Fan, X. Yuan and H. Zhu, Effect of seasonal changing temperature on the growth of phytoplankton, Math. Biosci. Eng., 14 (2017), 1091-1117.  doi: 10.3934/mbe.2017057.  Google Scholar

[4]

Z. Cheng and F. Li, Positive periodic solutions for a kind of second-order neutral differential equations with variable coefficient and delay, Mediterr. J. Math., 15 (2018), Art. 134, 19 pp. doi: 10.1007/s00009-018-1184-y.  Google Scholar

[5]

Z. Cheng and J. Ren, Periodic solution for second order damped differential equations with attractive-repulsive singularities, Rocky Mountain J. Math., 48 (2018), 753-768.  doi: 10.1216/RMJ-2018-48-3-753.  Google Scholar

[6]

A. Granas, R. B. Guenther and J. W. Lee, Some general existence principles in the Carathéodory theory of nonlinear differential systems, J. Math. Pures Appl., 70 (1991), 153-196.   Google Scholar

[7]

F. Güngör and P. J. Torres, Integrability of the Basener-Ross model with time-dependent coefficients, SeMA J., 76 (2019), 485-493.  doi: 10.1007/s40324-019-00187-w.  Google Scholar

[8]

A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theoret. Biol., 236 (2005), 276-290.  doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar

[9]

R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p$-Laplacian-like operators, J. Differential Equations, 145 (1998), 367-393.  doi: 10.1006/jdeq.1998.3425.  Google Scholar

[10]

D. O'Regan, Existence Theory for Nonlinear Ordinary Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-017-1517-1.  Google Scholar

[11]

J. Ren, D. Zhu and H. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1843-1865.  doi: 10.3934/dcdsb.2018240.  Google Scholar

[12]

Y. Wang, H. Lian and W. Ge, Periodic solutions for a second order nonlinear functional differential equation, Appl. Math. Lett., 20 (2007), 110-115.  doi: 10.1016/j.aml.2006.02.028.  Google Scholar

[13]

Y. Xu, D. Zhu and J. Ren, On a reaction-diffusion-advection system: Fixed boundary or free boundary, Electron. J. Qual. Theory Differ. Equ., (2018), Paper No. 26, 31 pp. doi: 10.14232/ejqtde.2018.1.26.  Google Scholar

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