# American Institute of Mathematical Sciences

2006, 3(2): 283-296. doi: 10.3934/mbe.2006.3.283

## Competition and Dispersal Delays in Patchy Environments

 1 Department of Mathematics and Statistics, University of Victoria, PO BOX 3045 STN CSC, Victoria, B.C. V8W 3P4, Canada, Canada

Received  July 2005 Revised  January 2006 Published  February 2006

Dispersal delays are introduced into a competition model for two species that disperse among $n$ identical patches. The model is formulated as a system of integro-differential equations with an arbitrary distribution of dispersal times between patches. By identifying steady states and analyzing local stability, conditions for competitive exclusion, coexistence or extinction are determined in terms of the system parameters. These are confirmed by numerical simulations with a delta function distribution, showing that all solutions approach a steady state and that high dispersal is generally a disadvantage to a species. However, if the two species have identical local dynamics, then small dispersal rates (with certain parameter restrictions) can be an advantage to the dispersing species. If the number of species is increased to three, then oscillatory coexistence with dispersal delay is possible.
Citation: Nancy Azer, P. van den Driessche. Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences & Engineering, 2006, 3 (2) : 283-296. doi: 10.3934/mbe.2006.3.283
 [1] Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699 [2] Donald L. DeAngelis, Bo Zhang. Effects of dispersal in a non-uniform environment on population dynamics and competition: A patch model approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3087-3104. doi: 10.3934/dcdsb.2014.19.3087 [3] Hua Nie, Sze-Bi Hsu, Jianhua Wu. Coexistence solutions of a competition model with two species in a water column. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2691-2714. doi: 10.3934/dcdsb.2015.20.2691 [4] Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046 [5] Theodore E. Galanthay. Mathematical study of the effects of travel costs on optimal dispersal in a two-patch model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1625-1638. doi: 10.3934/dcdsb.2015.20.1625 [6] Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218 [7] E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323 [8] Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012 [9] Jing-Jing Xiang, Yihao Fang. Evolutionarily stable dispersal strategies in a two-patch advective environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1875-1887. doi: 10.3934/dcdsb.2018245 [10] Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531 [11] Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082 [12] Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255 [13] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [14] Robert Stephen Cantrell, Brian Coomes, Yifan Sha. A tridiagonal patch model of bacteria inhabiting a Nanofabricated landscape. Mathematical Biosciences & Engineering, 2017, 14 (4) : 953-973. doi: 10.3934/mbe.2017050 [15] Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1 [16] Roberto Garra. Confinement of a hot temperature patch in the modified SQG model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2407-2416. doi: 10.3934/dcdsb.2018258 [17] Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 [18] S.A. Gourley, Yang Kuang. Two-Species Competition with High Dispersal: The Winning Strategy. Mathematical Biosciences & Engineering, 2005, 2 (2) : 345-362. doi: 10.3934/mbe.2005.2.345 [19] Tung Nguyen, Nar Rawal. Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3799-3816. doi: 10.3934/dcdsb.2018080 [20] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

2018 Impact Factor: 1.313