# American Institute of Mathematical Sciences

• Previous Article
On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit
• DCDS-B Home
• This Issue
• Next Article
Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations
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

 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.

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.
 [1] Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212 [2] Kazuaki Taira. A mathematical study of diffusive logistic equations with mixed type boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021166 [3] Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029 [4] Jesús Ildefonso Díaz, Jesús Hernández. Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022018 [5] Zaizheng Li, Zhitao Zhang. Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology. Electronic Research Archive, 2021, 29 (6) : 3761-3774. doi: 10.3934/era.2021060 [6] David Aleja, Julián López-Gómez. Some paradoxical effects of the advection on a class of diffusive equations in Ecology. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3031-3056. doi: 10.3934/dcdsb.2014.19.3031 [7] 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 [8] Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122 [9] Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943 [10] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [11] Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885 [12] 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 and Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713 [13] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447 [14] Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067 [15] Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166 [16] 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 [17] Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198 [18] Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232 [19] 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 and Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507 [20] 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 and Continuous Dynamical Systems, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113

2020 Impact Factor: 1.327