February  2004, 4(1): 241-266. doi: 10.3934/dcdsb.2004.4.241

A model of tumor and lymphocyte interactions

1. 

Department of Biopharmaceutical Sciences, University of California, San Francisco, San Francisco, CA 94143, United States

Received  November 2002 Revised  September 2003 Published  November 2003

The interactions between a solid tumor and the immune system are described both prior to and after neovascularization by a predator-prey model, and predictions about tumor behavior in a host are made. Trajectory analysis of phase-plane portraits as well as standard perturbation analysis show that most system steady states are unstable but that stability is theoretically possible. Reasonable parameter value estimation enables meaningful analysis of system behavior, and Mathematica is used to simulate model dynamics. The model accounts for many observed tumor behaviors, and regions of uncontrolled tumor growth, tumor extinction in finite time, and irreversible lymphocyte decline are found either analytically or numerically. A better understanding of tumor-immune dynamics is obtained, allowing for improved research on treatment specifically in the area of immunotherapy.
Citation: Amy H. Lin. A model of tumor and lymphocyte interactions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 241-266. doi: 10.3934/dcdsb.2004.4.241
[1]

Peter A. Braza. A dominant predator, a predator, and a prey. Mathematical Biosciences & Engineering, 2008, 5 (1) : 61-73. doi: 10.3934/mbe.2008.5.61

[2]

Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey. Mathematical Biosciences & Engineering, 2005, 2 (4) : 703-717. doi: 10.3934/mbe.2005.2.703

[3]

J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059

[4]

R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423

[5]

Hongxiao Hu, Liguang Xu, Kai Wang. A comparison of deterministic and stochastic predator-prey models with disease in the predator. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2837-2863. doi: 10.3934/dcdsb.2018289

[6]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

[7]

Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

[8]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877

[9]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

[10]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[11]

Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082

[12]

Fei Xu, Ross Cressman, Vlastimil Křivan. Evolution of mobility in predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3397-3432. doi: 10.3934/dcdsb.2014.19.3397

[13]

Jing-An Cui, Xinyu Song. Permanence of predator-prey system with stage structure. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 547-554. doi: 10.3934/dcdsb.2004.4.547

[14]

Christian Kuehn, Thilo Gross. Nonlocal generalized models of predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 693-720. doi: 10.3934/dcdsb.2013.18.693

[15]

Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128

[16]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

[17]

Dongmei Xiao, Kate Fang Zhang. Multiple bifurcations of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 417-433. doi: 10.3934/dcdsb.2007.8.417

[18]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185

[19]

Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265

[20]

Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]