On the unboundedness of the ratio of species and resources for the diffusive logistic equation

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May 2021, 26(5): 2441-2450. doi: 10.3934/dcdsb.2020186

On the unboundedness of the ratio of species and resources for the diffusive logistic equation

Jumpei Inoue ^1,, and Kousuke Kuto ^2,

1.	Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 164-8555, Japan
2.	Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 164-8555, Japan

^* Corresponding author: Jumpei Inoue

Dedicated to Professor Yoshio Yamada on the occasion of his 70th birthday

Received January 2020 Revised April 2020 Published May 2021 Early access June 2020

Fund Project: The second author is supported by JSPS KAKENHI Grant-in-Aid Grant Number 19K03581

Concerning a class of diffusive logistic equations, Ni [1,Abstract] proposed an optimization problem to consider the supremum of the ratio of the $ L^1 $ norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is $ 3 $ in the one-dimensional case. In [1], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an $ L^1 $ unbounded sequence of sub-solutions.

Keywords: Diffusive logistic equation, elliptic equations, the sub-super solution method, radial solutions, mathematical ecology.

Mathematics Subject Classification: Primary: 35Q92, 35B30; Secondary: 35B09, 35B40.

Citation: Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

References:

[1]	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.
[2]	R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Royal Soc. Edinburgh A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X.
[3]	R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.
[4]	R. S. Cantrell and C. Cosner, Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219-252. doi: 10.1137/0153014.
[5]	R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145. doi: 10.1007/s002850050122.
[6]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.
[7]	D. L. DeAngelis, B. Zhang, W.-M. Ni and Y. Wang, Carrying capacity of a population diffusing in a heterogeneous environment, Mathematics, 8 (2020), 12 pp. doi: 10.3390/math8010049.
[8]	Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, World Scientific, 2006. doi: 10.1142/9789812774446.
[9]	X. Q. He, K.-Y. Lam, Y. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model: Homogeneous vs. heterogeneous environments, J. Math. Biol., 78 (2019), 1605-1636. doi: 10.1007/s00285-018-1321-z.
[10]	X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.
[11]	X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: The general case, J. Differential Equations, 254 (2013), 4088-4108. doi: 10.1016/j.jde.2013.02.009.
[12]	X. Q. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Comm. Pure. Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596.
[13]	X. Q. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calc. Var. Partial Differential Equations, 55 (2016), 20 pp. doi: 10.1007/s00526-016-0964-0.
[14]	X. Q. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅲ, Calc. Var. Partial Differential Equations, 56 (2017), 26 pp. doi: 10.1007/s00526-017-1234-5.
[15]	J. Inoue, Limiting profile of the optimal distribution in a stationary logistic equation, submitted.
[16]	K.-Y. Lam and Y. Lou, Persistence, competition and evolution, in The Dynamics of Biological Systems, Springer Verlag 2019,205–238.
[17]	R. Li and Y. Lou, Some monotone properties for solutions to a reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4445-4455. doi: 10.3934/dcdsb.2019126.
[18]	S. Liang and Y. Lou, On the dependence of population size upon random dispersal rate, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2771-2788. doi: 10.3934/dcdsb.2012.17.2771.
[19]	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.
[20]	Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences IV, Evolution and Ecology, Lecture Notes in Math., 1922, Math. Biosci. Subser., Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5.
[21]	Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634. doi: 10.1360/N012015-00233.
[22]	Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772. doi: 10.1007/s11784-016-0372-2.
[23]	I. Mazzari, Trait selection and rare mutations; the case of large diffusivities, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6693-6724. doi: 10.3934/dcdsb.2019163.
[24]	I. Mazzari, G. Nadin and Y. Privat, Optimal location of resources maximizing the total population size in logistic models, J. Math. Pure. Appl., in press. doi: 10.1016/j.matpur.2019.10.008.
[25]	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), 14 pp. doi: 10.1007/s00526-018-1353-7.
[26]	W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.
[27]	K. Taira, Diffusive logistic equations in population dynamics, Adv. Differential Equations, 7 (2002), 237-256.
[28]	K. Taira, Logistic Dirichlet problems with discontinuous coefficients, J. Math. Pures. Appl., 82 (2003), 1137-1190. doi: 10.1016/S0021-7824(03)00058-8.

show all references

References:

[1]	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.
[2]	R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Royal Soc. Edinburgh A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X.
[3]	R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.
[4]	R. S. Cantrell and C. Cosner, Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219-252. doi: 10.1137/0153014.
[5]	R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145. doi: 10.1007/s002850050122.
[6]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.
[7]	D. L. DeAngelis, B. Zhang, W.-M. Ni and Y. Wang, Carrying capacity of a population diffusing in a heterogeneous environment, Mathematics, 8 (2020), 12 pp. doi: 10.3390/math8010049.
[8]	Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, World Scientific, 2006. doi: 10.1142/9789812774446.
[9]	X. Q. He, K.-Y. Lam, Y. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model: Homogeneous vs. heterogeneous environments, J. Math. Biol., 78 (2019), 1605-1636. doi: 10.1007/s00285-018-1321-z.
[10]	X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.
[11]	X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: The general case, J. Differential Equations, 254 (2013), 4088-4108. doi: 10.1016/j.jde.2013.02.009.
[12]	X. Q. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Comm. Pure. Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596.
[13]	X. Q. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calc. Var. Partial Differential Equations, 55 (2016), 20 pp. doi: 10.1007/s00526-016-0964-0.
[14]	X. Q. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅲ, Calc. Var. Partial Differential Equations, 56 (2017), 26 pp. doi: 10.1007/s00526-017-1234-5.
[15]	J. Inoue, Limiting profile of the optimal distribution in a stationary logistic equation, submitted.
[16]	K.-Y. Lam and Y. Lou, Persistence, competition and evolution, in The Dynamics of Biological Systems, Springer Verlag 2019,205–238.
[17]	R. Li and Y. Lou, Some monotone properties for solutions to a reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4445-4455. doi: 10.3934/dcdsb.2019126.
[18]	S. Liang and Y. Lou, On the dependence of population size upon random dispersal rate, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2771-2788. doi: 10.3934/dcdsb.2012.17.2771.
[19]	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.
[20]	Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences IV, Evolution and Ecology, Lecture Notes in Math., 1922, Math. Biosci. Subser., Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5.
[21]	Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634. doi: 10.1360/N012015-00233.
[22]	Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772. doi: 10.1007/s11784-016-0372-2.
[23]	I. Mazzari, Trait selection and rare mutations; the case of large diffusivities, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6693-6724. doi: 10.3934/dcdsb.2019163.
[24]	I. Mazzari, G. Nadin and Y. Privat, Optimal location of resources maximizing the total population size in logistic models, J. Math. Pure. Appl., in press. doi: 10.1016/j.matpur.2019.10.008.
[25]	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), 14 pp. doi: 10.1007/s00526-018-1353-7.
[26]	W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.
[27]	K. Taira, Diffusive logistic equations in population dynamics, Adv. Differential Equations, 7 (2002), 237-256.
[28]	K. Taira, Logistic Dirichlet problems with discontinuous coefficients, J. Math. Pures. Appl., 82 (2003), 1137-1190. doi: 10.1016/S0021-7824(03)00058-8.

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