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

Jumpei Inoue; Kousuke Kuto

doi:10.3934/dcdsb.2020186

Previous Article
Bistability of sequestration networks
DCDS-B Home
This Issue
Next Article
The Keller-Segel system with logistic growth and signal-dependent motility

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

Received January 2020 Revised April 2020 Published 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 & Continuous Dynamical Systems - B, 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, World Scientific, 2006. doi: 10.1142/9789812774446. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	J. Inoue, Limiting profile of the optimal distribution in a stationary logistic equation, submitted. Google Scholar
[16]	K.-Y. Lam and Y. Lou, Persistence, competition and evolution, in The Dynamics of Biological Systems, Springer Verlag 2019,205–238. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634. doi: 10.1360/N012015-00233. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	K. Taira, Diffusive logistic equations in population dynamics, Adv. Differential Equations, 7 (2002), 237-256. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, World Scientific, 2006. doi: 10.1142/9789812774446. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	J. Inoue, Limiting profile of the optimal distribution in a stationary logistic equation, submitted. Google Scholar
[16]	K.-Y. Lam and Y. Lou, Persistence, competition and evolution, in The Dynamics of Biological Systems, Springer Verlag 2019,205–238. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634. doi: 10.1360/N012015-00233. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	K. Taira, Diffusive logistic equations in population dynamics, Adv. Differential Equations, 7 (2002), 237-256. Google Scholar
[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. Google Scholar

[1]	Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
[2]	Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029
[3]	David Aleja, Julián López-Gómez. Some paradoxical effects of the advection on a class of diffusive equations in Ecology. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3031-3056. doi: 10.3934/dcdsb.2014.19.3031
[4]	Joseph A. Iaia. Localized radial solutions to a semilinear elliptic equation in $\mathbb{R}^n$. Conference Publications, 1998, 1998 (Special) : 314-326. doi: 10.3934/proc.1998.1998.314
[5]	Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122
[6]	Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943
[7]	Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[8]	Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885
[9]	Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
[10]	Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447
[11]	Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41
[12]	Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198
[13]	Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166
[14]	Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067
[15]	Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232
[16]	Michael E. Filippakis, Donal O'Regan, Nikolaos S. Papageorgiou. Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507
[17]	Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113
[18]	A. M. Micheletti, Monica Musso, A. Pistoia. Super-position of spikes for a slightly super-critical elliptic equation in $R^N$. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 747-760. doi: 10.3934/dcds.2005.12.747
[19]	Alfonso Castro, Shu-Zhi Song. Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball. Discrete & Continuous Dynamical Systems - S, 2019 doi: 10.3934/dcdss.2020127
[20]	Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors