Random dispersal vs. non-local dispersal

Previous Article
Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density
DCDS Home
This Issue
Next Article
Higher integrability for gradients of solutions to degenerate parabolic systems

April 2010, 26(2): 551-596. doi: 10.3934/dcds.2010.26.551

Random dispersal vs. non-local dispersal

Chiu-Yen Kao ^1, , Yuan Lou ^2, and Wenxian Shen ^3,

1.	Department of Mathematics, The Ohio State University, Columbus, OH 43210
2.	Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210
3.	Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849

Received November 2008 Revised July 2009 Published October 2009

Abstract
Related Papers

Random dispersal is essentially a local behavior which describes the movement of organisms between adjacent spatial locations. However, the movements and interactions of some organisms can occur between non-adjacent spatial locations. To address the question about which dispersal strategy can convey some competitive advantage, we consider a mathematical model consisting of one reaction-diffusion equation and one integro-differential equation, in which two competing species have the same population dynamics but different dispersal strategies: the movement of one species is purely by random walk while the other species adopts a non-local dispersal strategy. For spatially periodic and heterogeneous environments we show that (i) for fixed random dispersal rate, if the nonlocal dispersal distance is sufficiently small, then the non-local disperser can invade the random disperser but not vice versa; (ii) for fixed nonlocal dispersal distance, if the random dispersal rate becomes sufficiently small, then the random disperser can invade the nonlocal disperser but not vice versa. These results suggest that for spatially periodic heterogeneous environments, the competitive advantage may belong to the species with much lower effective rate of dispersal. This is in agreement with previous results for the evolution of random dispersal [9, 13] that the slower disperser has an advantage. Nevertheless, if random dispersal strategy with either zero Dirichlet or zero Neumann boundary condition is compared with non-local dispersal strategy with hostile surroundings, the species with much lower effective rate of dispersal may not have the competitive advantage. Numerical results will be presented to shed light on the global dynamics of the system for general values of non-local interaction distance and also to point to future research directions.

Keywords: Reaction-diffusion, Competition, Non-local dispersal, Random dispersal, Integral kernel..

Mathematics Subject Classification: Primary: 35K57; Secondary: 45K05, 47G20, and 92D2.

Citation: Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Random dispersal vs. non-local dispersal. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 551-596. doi: 10.3934/dcds.2010.26.551

[1]	Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[2]	Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126
[3]	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
[4]	Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701
[5]	Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[6]	Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
[7]	Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[8]	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
[9]	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
[10]	Song Liang, Yuan Lou. On the dependence of population size upon random dispersal rate. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2771-2788. doi: 10.3934/dcdsb.2012.17.2771
[11]	Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875
[12]	Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301
[13]	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
[14]	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
[15]	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
[16]	Badr Saad T. Alkahtani, Ilknur Koca. A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 429-442. doi: 10.3934/dcdss.2020024
[17]	Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203
[18]	Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036
[19]	Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[20]	Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences