American Institute of Mathematical Sciences

Advanced
August  2004, 4(3): 555-562. doi: 10.3934/dcdsb.2004.4.555

A monotone-iterative method for finding periodic solutions of an impulsive competition system on tumor-normal cell interaction

Jiawei Dou 1, , Lan-sun Chen 2, and  Kaitai Li 3,

1. 

College of Mathematics and Information Science, Shanxi Normal University, Xi'an 710062, China
2. 

Institute of Mathematics, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, China
3. 

Research Center for Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  October 2002 Revised  August 2003 Published  May 2004

In this paper, a monotone-iterative scheme is established for finding positive periodic solutions of a competition model of tumor-normal cell interaction. The model describes the evolution of a population with normal and tumor cells in a periodically changing environment. This population is under periodical chemotherapeutic treatment. Competition among the two kinds of cells is considered. The mathematical problem involves a coupled system of Lotka-Volterra together with periodically pulsed conditions. The existence of positive periodic solutions is proved by the monotone iterative technique and in a special case, the uniqueness of a periodic solution is obtained by proving that any two periodic solutions have the same average. Moreover, we also show that the system is permanent under the conditions which guarantee the existence of the periodic solution. Some computer simulations are carried out to demonstrate the main results.
Keywords: competition system, Impulsive, tumor-normal cell interaction., monotone-iterative method, periodic solution.
Mathematics Subject Classification: 34A37, 92D2.
Citation: Jiawei Dou, Lan-sun Chen, Kaitai Li. A monotone-iterative method for finding periodic solutions of an impulsive competition system on tumor-normal cell interaction. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 555-562. doi: 10.3934/dcdsb.2004.4.555
[1]

Yuanshi Wang, Hong Wu. Transition of interaction outcomes in a facilitation-competition system of two species. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1463-1475. doi: 10.3934/mbe.2017076
[2]

Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060
[3]

M. Guedda, R. Kersner, M. Klincsik, E. Logak. Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1589-1600. doi: 10.3934/dcdsb.2014.19.1589
[4]

Janet Dyson, Rosanna Villella-Bressan, G. F. Webb. The evolution of a tumor cord cell population. Communications on Pure & Applied Analysis, 2004, 3 (3) : 331-352. doi: 10.3934/cpaa.2004.3.331
[5]

Miao Yu, Haoyang Lu, Weipeng Shang. A new iterative identification method for damping control of power system in multi-interference. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020104
[6]

Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201
[7]

Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure & Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014
[8]

João Fialho, Feliz Minhós. High order periodic impulsive problems. Conference Publications, 2015, 2015 (special) : 446-454. doi: 10.3934/proc.2015.0446
[9]

Yangjin Kim, Hans G. Othmer. Hybrid models of cell and tissue dynamics in tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1141-1156. doi: 10.3934/mbe.2015.12.1141
[10]

Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
[11]

Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems & Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103
[12]

Dan Liu, Shigui Ruan, Deming Zhu. Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. Mathematical Biosciences & Engineering, 2012, 9 (2) : 347-368. doi: 10.3934/mbe.2012.9.347
[13]

Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007
[14]

Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1187-1213. doi: 10.3934/mbe.2017061
[15]

Morten Brøns. An iterative method for the canard explosion in general planar systems. Conference Publications, 2013, 2013 (special) : 77-83. doi: 10.3934/proc.2013.2013.77
[16]

Lingling Lv, Zhe Zhang, Lei Zhang, Weishu Wang. An iterative algorithm for periodic sylvester matrix equations. Journal of Industrial & Management Optimization, 2018, 14 (1) : 413-425. doi: 10.3934/jimo.2017053
[17]

Daniel Vasiliu, Jianjun Paul Tian. Periodic solutions of a model for tumor virotherapy. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1587-1597. doi: 10.3934/dcdss.2011.4.1587
[18]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043
[19]

Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure & Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779
[20]

Ahuod Alsheri, Ebraheem O. Alzahrani, Asim Asiri, Mohamed M. El-Dessoky, Yang Kuang. Tumor growth dynamics with nutrient limitation and cell proliferation time delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3771-3782. doi: 10.3934/dcdsb.2017189

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences