# American Institute of Mathematical Sciences

April  2021, 26(4): 2055-2065. doi: 10.3934/dcdsb.2020280

## The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration

 1 School of Mathematical Sciences and Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China 2 Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai, Shandong 264209, China

* Corresponding author: Xing Liang

Dedicated to Professor Sze-Bi Hsu on the occasion of his 70th birthday

Received  May 2020 Revised  August 2020 Published  April 2021 Early access  September 2020

Fund Project: Liang's research is supported by the National Natural Science Foundation of China (11971454) and the Fundamental Research Funds for the Central Universities; Zhang's research is supported by the National Natural Science Foundation of China(11901138) and Natural Science Foundation of Shandong Province (ZR2019QA006)

This paper focuses on an optimization problem arising in population biology. We investigate the effect of the resources distribution and the migration rate on the total population size of some species, which migrates among patches with the identical probability and grows logistically in each patch. We aim to maximize the total population size by the distribution of resources and the rate of migration.

Citation: Xing Liang, Lei Zhang. The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2055-2065. doi: 10.3934/dcdsb.2020280
##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522. [2] X. Bai, X. He and F. Li, An optimization problem and its application in population dynamics, Proc. Amer. Math. Soc., 144 (2016), 2161-2170.  doi: 10.1090/proc/12873. [3] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262. [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2004. doi: 10.1002/0470871296. [5] W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Anal. Real World Appl., 11 (2010), 688-704.  doi: 10.1016/j.nonrwa.2009.01.015. [6] F. G. Frobenius, Über matrizen aus nicht negativen elementen, S.-B. Deutsch. Akad. Wiss. Berlin, v. 1912,456–477. [7] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [8] Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634.  doi: 10.1360/N012015-00233. [9] I. Mazari, G. Nadin and Y. Privat, Optimal location of resources maximizing the total population size in logistic models, J. Math. Pure. Appl., 134 (2020), 1-35.  doi: 10.1016/j.matpur.2019.10.008. [10] K. Nagahara and E. Yanagida, Maximization of the total population in a reaction-diffusion model with logistic growth, Calc. Var. Partial Differential Equations, 57, (2018), Paper No. 80, 14 pp. doi: 10.1007/s00526-018-1353-7. [11] O. Perron, Zur theorie der matrices, Math. Ann., 64 (1907), 248-263.  doi: 10.1007/BF01449896. [12] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, 2008. [13] X.-Q. Zhao, Dynamical Systems in Population Biology, $2^{nd}$ edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.

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##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522. [2] X. Bai, X. He and F. Li, An optimization problem and its application in population dynamics, Proc. Amer. Math. Soc., 144 (2016), 2161-2170.  doi: 10.1090/proc/12873. [3] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262. [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2004. doi: 10.1002/0470871296. [5] W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Anal. Real World Appl., 11 (2010), 688-704.  doi: 10.1016/j.nonrwa.2009.01.015. [6] F. G. Frobenius, Über matrizen aus nicht negativen elementen, S.-B. Deutsch. Akad. Wiss. Berlin, v. 1912,456–477. [7] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [8] Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634.  doi: 10.1360/N012015-00233. [9] I. Mazari, G. Nadin and Y. Privat, Optimal location of resources maximizing the total population size in logistic models, J. Math. Pure. Appl., 134 (2020), 1-35.  doi: 10.1016/j.matpur.2019.10.008. [10] K. Nagahara and E. Yanagida, Maximization of the total population in a reaction-diffusion model with logistic growth, Calc. Var. Partial Differential Equations, 57, (2018), Paper No. 80, 14 pp. doi: 10.1007/s00526-018-1353-7. [11] O. Perron, Zur theorie der matrices, Math. Ann., 64 (1907), 248-263.  doi: 10.1007/BF01449896. [12] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, 2008. [13] X.-Q. Zhao, Dynamical Systems in Population Biology, $2^{nd}$ edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.
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