# American Institute of Mathematical Sciences

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

## Positive periodic solution for generalized Basener-Ross model

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

* Corresponding author: Zhibo Cheng

Received  September 2019 Revised  November 2019 Published  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.

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:
 [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

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
 [1] Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374 [2] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [3] Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395 [4] Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004 [5] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [6] Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148 [7] Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178 [8] Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294 [9] Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026 [10] Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021006 [11] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [12] Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021009 [13] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293 [14] Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087 [15] Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 [16] Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313 [17] Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371 [18] Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021005 [19] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [20] Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021010

2019 Impact Factor: 1.27

Article outline