Maximal total population of species in a diffusive logistic model.
Diffusive logistic equation
Gradient-based algorithm
Optimal control
Population dynamics
Rearrangements
Journal
Journal of mathematical biology
ISSN: 1432-1416
Titre abrégé: J Math Biol
Pays: Germany
ID NLM: 7502105
Informations de publication
Date de publication:
07 10 2022
07 10 2022
Historique:
received:
11
10
2021
accepted:
26
09
2022
revised:
07
07
2022
entrez:
7
10
2022
pubmed:
8
10
2022
medline:
12
10
2022
Statut:
epublish
Résumé
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.
Identifiants
pubmed: 36207613
doi: 10.1007/s00285-022-01817-0
pii: 10.1007/s00285-022-01817-0
doi:
Types de publication
Journal Article
Research Support, U.S. Gov't, Non-P.H.S.
Langues
eng
Sous-ensembles de citation
IM
Pagination
47Subventions
Organisme : National Science Foundation
ID : DMS 1818948
Organisme : National Science Foundation
ID : DMS-2208373
Informations de copyright
© 2022. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
Références
Bai X, He X, Li F (2016) An optimization problem and its application in population dynamics. Proc Am Math Soc 144(5):2161–2170. https://doi.org/10.1090/proc/12873
doi: 10.1090/proc/12873
Berestycki H, Hamel F, Roques L (2005) Analysis of the periodically fragmented environment model: I—species persistence. J Math Biol 51(1):75–113. https://doi.org/10.1007/s00285-004-0313-3
doi: 10.1007/s00285-004-0313-3
Brock F (2007) Chapter 1—rearrangements and applications to symmetry problems in PDE. In: Chipot M (ed) Stationary partial differential equations. Handbook of differential equations: stationary partial differential equations, vol 4. North-Holland, Amsterdam, pp 1–60. https://doi.org/10.1016/S1874-5733(07)80004-0
doi: 10.1016/S1874-5733(07)80004-0
Burton G (1987) Rearrangements of functions, maximization of convex functionals, and vortex rings. Math Ann 276(2):225–253. https://doi.org/10.1007/BF01450739
doi: 10.1007/BF01450739
Burton G (1989) Variational problems on classes of rearrangements and multiple configurations for steady vortices. Annales de l’Institut Henri Poincare (C) Non Linear Anal 6:295–319
doi: 10.1016/s0294-1449(16)30320-1
Cantrell RS, Cosner C (1989) Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc R Soc Edinb Sect A: Math 112(3–4):293–318. https://doi.org/10.1017/S030821050001876X
doi: 10.1017/S030821050001876X
Cantrell RS, Cosner C (1991) The effects of spatial heterogeneity in population dynamics. J Math Biol 29(4):315–338. https://doi.org/10.1007/BF00167155
doi: 10.1007/BF00167155
Chugunova M, Jadamba B, Kao C-Y, Klymko C, Thomas E, Zhao B (2016). Study of a mixed dispersal population dynamics model. In: Topics in numerical partial differential equations and scientific computing. Springer, New York, NY, pp 51–77. https://doi.org/10.1007/978-1-4939-6399-7_3
Cosner C, Cuccu F, Porru G (2013) Optimization of the first eigenvalue of equations with indefinite weights. Adv Nonlinear Stud 13(1):79–95. https://doi.org/10.1515/ans-2013-0105
doi: 10.1515/ans-2013-0105
Ding W, Finotti H, Lenhart S, Lou Y, Ye Q (2010) Optimal control of growth coefficient on a steady-state population model. Nonlinear Anal Real World Appl 11(2):688–704. https://doi.org/10.1016/j.nonrwa.2009.01.015
doi: 10.1016/j.nonrwa.2009.01.015
Gilbarg D, Trudinger NS (2015) Elliptic partial differential equations of second order. Springer, Berlin
Goss-Custard J, Stillman R, Caldow R, West A, Guillemain M (2003) Carrying capacity in overwintering birds: when are spatial models needed? J Appl Ecol 40(1):176–187. https://doi.org/10.1046/j.1365-2664.2003.00785.x
doi: 10.1046/j.1365-2664.2003.00785.x
Hardy GH, Littlewood JE, Pólya G, Pólya G et al (1952) Inequalities. Cambridge University Press, Cambridge
He X, Ni W-M (2016a) Global dynamics of the Lotka–Volterra competition–diffusion system: diffusion and spatial heterogeneity I. Commun Pure Appl Math 69(5):981–1014
doi: 10.1002/cpa.21596
He X, Ni W-M (2016b) Global dynamics of the Lotka–Volterra competition–diffusion system with equal amount of total resources, II. Calc Var Partial Differ Equ 55(2):25
doi: 10.1007/s00526-016-0964-0
He X, Ni W-M (2017) Global dynamics of the Lotka–Volterra competition–diffusion system with equal amount of total resources, III. Calc Var Partial Differ Equ 56(5):132. https://doi.org/10.1007/s00526-017-1234-5
doi: 10.1007/s00526-017-1234-5
Heo J, Kim Y (2021) On the fragmentation phenomenon in the population optimization problem. Proc Am Math Soc 149(12):5211–5221
doi: 10.1090/proc/15633
Hintermüller M, Kao C-Y, Laurain A (2012) Principal eigenvalue minimization for an elliptic problem with indefinite weight and robin boundary conditions. Appl Math Optim 65(1):111–146. https://doi.org/10.1007/S00245-011-9153-X
doi: 10.1007/S00245-011-9153-X
Kao C-Y, Mohammadi SA (2020) Maximal total population of species in a diffusive logistic model. https://doi.org/10.5281/zenodo.5525494
Kao C-Y, Lou Y, Yanagida E (2008) Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Math Biosci Eng 5(2):315. https://doi.org/10.3934/mbe.2008.5.315
doi: 10.3934/mbe.2008.5.315
Kawohl B (2006) Rearrangements and convexity of level sets in PDE, vol 1150. Springer, Berlin
Lam K-Y, Liu S, Lou Y (2020) Selected topics on reaction–diffusion–advection models from spatial ecology. arXiv preprint arXiv:2004.07978 . https://doi.org/10.5206/mase/10644
Lamboley J, Laurain A, Nadin G, Privat Y (2016) Properties of optimizers of the principal eigenvalue with indefinite weight and robin conditions. Calc Var Partial Differ Equ 55(6):1–37. https://doi.org/10.1007/s00526-016-1084-6
doi: 10.1007/s00526-016-1084-6
Lê A (2006) Eigenvalue problems for the p-Laplacian. Nonlinear Anal: Theory Methods Appl 64(5):1057–1099. https://doi.org/10.1016/j.na.2005.05.056
doi: 10.1016/j.na.2005.05.056
Liang S, Lou Y (2012) On the dependence of population size upon random dispersal rate. Discrete Contin Dyn Syst B 17(8):2771–2788
doi: 10.3934/dcdsb.2012.17.2771
Lieb EH, Loss M (2001) Analysis, vol 14. American Mathematical Soc, Providence, RI
Lou Y (2006) On the effects of migration and spatial heterogeneity on single and multiple species. J Differ Equ 223(2):400–426. https://doi.org/10.1016/j.jde.2005.05.010
doi: 10.1016/j.jde.2005.05.010
Lou Y (2008) Some challenging mathematical problems in evolution of dispersal and population dynamics. In: Friedman A (ed) Tutorials in mathematical biosciences IV. Lecture Notes in Mathematics. Springer, Berlin, pp 171–205. https://doi.org/10.1007/978-3-540-74331-6_5
doi: 10.1007/978-3-540-74331-6_5
Lou Y, Yanagida E (2006) Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics. Jpn J Ind Appl Math 23(3):275–292. https://doi.org/10.1007/BF03167595
doi: 10.1007/BF03167595
Mazari I, Ruiz-Balet D (2021) A fragmentation phenomenon for a nonenergetic optimal control problem: optimization of the total population size in logistic diffusive models. SIAM J Appl Math 81(1):153–172. https://doi.org/10.1137/20M132818X
doi: 10.1137/20M132818X
Mazari I, Nadin G, Privat Y (2020) Optimal location of resources maximizing the total population size in logistic models. Journal de mathématiques pures et appliquées 134:1–35. https://doi.org/10.1016/j.matpur.2019.10.008
doi: 10.1016/j.matpur.2019.10.008
Mazari I, Nadin G, Privat Y (2021) Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate. Commun Partial Differ Equ. https://doi.org/10.1080/03605302.2021.2007533
doi: 10.1080/03605302.2021.2007533
Nagahara K, Yanagida E (2018) Maximization of the total population in a reaction–diffusion model with logistic growth. Calc Var Partial Differ Equ 57(3):80. https://doi.org/10.1007/s00526-018-1353-7
doi: 10.1007/s00526-018-1353-7
Skellam JG (1951) Random dispersal in theoretical populations. Biometrika 38(1/2):196–218. https://doi.org/10.1093/biomet/38.1-2.196
doi: 10.1093/biomet/38.1-2.196
Sperner E (1981) Spherical symmetrization and eigenvalue estimates. Math Z 176:75–86. https://doi.org/10.1007/BF01258906
doi: 10.1007/BF01258906
Yousefnezhad M, Mohammadi S (2016) Stability of a predator–prey system with prey taxis in a general class of functional responses. Acta Math Sci 36(1):62–72. https://doi.org/10.21136/AM.2018.0227-17
doi: 10.21136/AM.2018.0227-17
Yousefnezhad M, Mohammadi SA, Bozorgnia F (2018) A free boundary problem for a predator–prey model with nonlinear prey-taxis. Appl Math 63(2):125–147. https://doi.org/10.21136/AM.2018.0227-17
doi: 10.21136/AM.2018.0227-17