American Institute of Mathematical Sciences

Advanced
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

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 and Continuous Dynamical Systems - B, 2007, 8 (1) : 61-72. doi: 10.3934/dcdsb.2007.8.61
[1]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure and Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721
[2]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519
[3]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[4]

Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001
[5]

Ting-Hao Hsu, Gail S. K. Wolkowicz. A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1257-1277. doi: 10.3934/dcdsb.2019219
[6]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63
[7]

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
[8]

Michel Pierre, Didier Schmitt. Examples of finite time blow up in mass dissipative reaction-diffusion systems with superquadratic growth. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022039
[9]

Mostafa Bendahmane. Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Networks and Heterogeneous Media, 2008, 3 (4) : 863-879. doi: 10.3934/nhm.2008.3.863
[10]

Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189
[11]

Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003
[12]

Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 759-779. doi: 10.3934/dcds.2021136
[13]

Yu-Shuo Chen, Jong-Shenq Guo, Masahiko Shimojo. Recent developments on a singular predator-prey model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1811-1825. doi: 10.3934/dcdsb.2020040
[14]

Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010
[15]

Nguyen Huu Du, Nguyen Hai Dang. Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2693-2712. doi: 10.3934/cpaa.2014.13.2693
[16]

Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114
[17]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[18]

Juntang Ding, Xuhui Shen. Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4243-4254. doi: 10.3934/dcdsb.2018135
[19]

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
[20]

Feiying Yang, Wantong Li, Renhu Wang. Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4083-4105. doi: 10.3934/cpaa.2021146

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy