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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

 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  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 & 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.  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
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