Abstract
In this paper, we investigate the maximization of the total population of a single species which is governed by a stationary diffusive logistic equation with a fixed amount of resources. For large diffusivity, qualitative properties of the maximizers like symmetry will be addressed. Our results are in line with previous findings which assert that for large diffusion, concentrated resources are favorable for maximizing the total population. Then, an optimality condition for the maximizer is derived based upon rearrangement theory. We develop an efficient numerical algorithm applicable to domains with different geometries in order to compute the maximizer. It is established that the algorithm is convergent. Our numerical simulations give a real insight into the qualitative properties of the maximizer and also lead us to some conjectures about the maximizer.
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Data Availability
Numerical results generated during the current study are available in the Zenodo repository, https://doi.org/10.5281/zenodo.5525494 (Kao and Mohammadi 2020).
Code Availability
The codes are avaliable in the Zenodo repository, https://doi.org/10.5281/zenodo.5525494 (Kao and Mohammadi 2020).
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Chiu-Yen Kao’s work is supported in part by NSF grants DMS-1818948 and DMS-2208373.
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Kao, CY., Mohammadi, S.A. Maximal total population of species in a diffusive logistic model. J. Math. Biol. 85, 47 (2022). https://doi.org/10.1007/s00285-022-01817-0
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DOI: https://doi.org/10.1007/s00285-022-01817-0