An optimization problem and its application in population dynamics
HTML articles powered by AMS MathViewer
- by Xueli Bai, Xiaoqing He and Fang Li PDF
- Proc. Amer. Math. Soc. 144 (2016), 2161-2170 Request permission
Abstract:
This paper is concerned with a diffusive logistic model in population ecology. As observed by Y. Lou, in a spatially heterogeneous environment, this model can always support a total population at equilibrium greater than the total carrying capacity. In other words, the ratio of the total population at equilibrium to the total carrying capacity is always larger than $1$. Our goal is to find the supremum of this ratio taken over all possible choices of spatial distributions of resources and the species’ dispersal rate. A conjecture proposed by W.-M. Ni is that, in the one-dimensional case, the supremum is $3$. We settle this conjecture and then apply our result to study the global dynamics of a heterogeneous Lotka-Volterra competition-diffusion system.References
- Robert Stephen Cantrell and Chris Cosner, Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. MR 2191264, DOI 10.1002/0470871296
- Jack Dockery, Vivian Hutson, Konstantin Mischaikow, and Mark Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol. 37 (1998), no. 1, 61–83. MR 1636644, DOI 10.1007/s002850050120
- Xiaoqing He and Wei-Ming Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: Heterogeneity vs. homogeneity, J. Differential Equations 254 (2013), no. 2, 528–546. MR 2990042, DOI 10.1016/j.jde.2012.08.032
- Xiaoqing He and Wei-Ming Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: The general case, J. Differential Equations 254 (2013), no. 10, 4088–4108. MR 3032297, DOI 10.1016/j.jde.2013.02.009
- Xiaoqing He and Wei-Ming Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity, I, Comm. Pure Appl. Math., to appear.
- Xiaoqing He and Wei-Ming Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, II, submitted.
- Xiaoqing He and Wei-Ming Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, preprint.
- V. Hutson, Y. Lou, and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations 185 (2002), no. 1, 97–136. MR 1935633, DOI 10.1006/jdeq.2001.4157
- King-Yeung Lam and Wei-Ming Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math. 72 (2012), no. 6, 1695–1712. MR 3022283, DOI 10.1137/120869481
- Fang Li, Liping Wang, and Yang Wang, On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B 15 (2011), no. 3, 669–686. MR 2774133, DOI 10.3934/dcdsb.2011.15.669
- Yuan Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations 223 (2006), no. 2, 400–426. MR 2214941, DOI 10.1016/j.jde.2005.05.010
- Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in mathematical biosciences. IV, Lecture Notes in Math., vol. 1922, Springer, Berlin, 2008, pp. 171–205. MR 2392287, DOI 10.1007/978-3-540-74331-6_{5}
- T. Mori and S. Yotsutani, private communication.
- Wei-Ming Ni, The mathematics of diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. MR 2866937, DOI 10.1137/1.9781611971972
Additional Information
- Xueli Bai
- Affiliation: Department of Applied Mathematics, Northwestern Polytechnical University, 127 West Youyi Road, 710072, Xi’an, Shaanxi, People’s Republic of China
- MR Author ID: 910254
- Email: mybxl110@163.com
- Xiaoqing He
- Affiliation: Center for PDE, East China Normal University, 500 Dongchuan Road, Minhang 200241, Shanghai, People’s Republic of China
- MR Author ID: 997876
- Email: xqhe@cpde.ecnu.edu.cn
- Fang Li
- Affiliation: Center for PDE, East China Normal University, 500 Dongchuan Road, Minhang 200241, Shanghai, People’s Republic of China
- MR Author ID: 1206479
- Email: fli@cpde.ecnu.edu.cn
- Received by editor(s): January 9, 2015
- Received by editor(s) in revised form: June 24, 2015
- Published electronically: October 6, 2015
- Additional Notes: The first author was supported by Shanghai Postdoctoral Science Foundation (No. 13R21412600), Postdoctoral Science Foundation of China (No. 2014M551359) and Chinese NSF (No. 11501207).
The third author was supported by Chinese NSF (No. 11201148), Shanghai Pujiang Program (No. 13PJ1402400). - Communicated by: Yingfei Yi
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2161-2170
- MSC (2010): Primary 35B09, 35Q92; Secondary 35B30, 35B40
- DOI: https://doi.org/10.1090/proc/12873
- MathSciNet review: 3460175