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An optimization problem and its application in population dynamics


Authors: Xueli Bai, Xiaoqing He and Fang Li
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
Published electronically: October 6, 2015
MathSciNet review: 3460175
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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.


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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
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
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
Email: fli@cpde.ecnu.edu.cn

DOI: https://doi.org/10.1090/proc/12873
Keywords: Total population, optimization, spatial heterogeneity
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
Article copyright: © Copyright 2015 American Mathematical Society

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