Some remarks on a singular reaction-diffusion system arising in predator-prey modeling

Previous Article
An age and spatially structured model of tumor invasion with haptotaxis
DCDS-B Home
This Issue
Next Article
Polytopic Lyapunov functions for persistence analysis of competing species

July 2007, 8(1): 61-72. doi: 10.3934/dcdsb.2007.8.61

Some remarks on a singular reaction-diffusion system arising in predator-prey modeling

Sebastién Gaucel ^1, and Michel Langlais ^2,

1.	Unité MIA MathRisq, INRA, Domaine de Vilvert, F-78352 Jouy-en-Josas cedex, France
2.	Université Victor Segalen Bordeaux 2, case 26, UMR CNRS 5251 IMB & INRIA Futurs Anubis, 146, rue Léo Saignat, 33076 Bordeaux Cedex, France

Received September 2006 Revised November 2006 Published April 2007

Abstract
Related Papers

This note is dedicated to the question of global existence for solutions to a two component singular system of reaction-diffusion equations modeling predator-prey interactions in insular environments. Depending on a 2D parameter space, positive orbits of the underlying ODE system undergo interesting dynamics, e.g., finite time existence and global existence may coexist. These results are partially extended to the reaction-diffusion system in the case of identical diffusivities. Our analysis relies on an auxiliary non singular reaction-diffusion system whose solutions may or may not blow up in finite time. Numerical simulations illustrate our analysis, including a numerical evidence of spatio-temporal oscillations.

Keywords: Global existence, oscillations, predator-prey model in insular environment, invasion and persistence., blow-up time, singular reaction-diffusion systems.

Mathematics Subject Classification: Primary: 35K57, 35B05; Secondary: 34D23 92D2.

Citation: Sebastién Gaucel, Michel Langlais. Some remarks on a singular reaction-diffusion system arising in predator-prey modeling. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 61-72. doi: 10.3934/dcdsb.2007.8.61

[1]	Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[2]	Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721
[3]	Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519
[4]	Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001
[5]	Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63
[6]	Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182
[7]	Mostafa Bendahmane. Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Networks & Heterogeneous Media, 2008, 3 (4) : 863-879. doi: 10.3934/nhm.2008.3.863
[8]	Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189
[9]	Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010
[10]	Nguyen Huu Du, Nguyen Hai Dang. Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2693-2712. doi: 10.3934/cpaa.2014.13.2693
[11]	Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114
[12]	Juntang Ding, Xuhui Shen. Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4243-4254. doi: 10.3934/dcdsb.2018135
[13]	Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[14]	Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69
[15]	Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653
[16]	Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523
[17]	Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
[18]	Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
[19]	Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[20]	Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences