Some monotone properties for solutions to a reaction-diffusion model

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August 2019, 24(8): 4445-4455. doi: 10.3934/dcdsb.2019126

Some monotone properties for solutions to a reaction-diffusion model

Rui Li ^1,, and Yuan Lou ^2,

1.	Institute for Mathematical Sciences, Renmin University of China, Beijing, 100876, China
2.	Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

^* Corresponding author: Rui Li

Received August 2018 Revised November 2018 Published August 2019 Early access June 2019

Motivated by the recent investigation of a predator-prey model in heterogeneous environments [20], we show that the maximum of the unique positive solution of the scalar equation

$ \begin{equation} \begin{cases} \mu\Delta\theta+(m(x)-\theta)\theta = 0 \quad &\text{in} \quad \Omega,\\ \frac{\partial \theta}{\partial n} = 0 \quad &\text{on} \quad \partial\Omega \end{cases} \end{equation} $

is a strictly monotone decreasing function of the diffusion rate

$ \mu $

for several classes of function

$ m $

, which substantially improves a result in [20]. However, the minimum of the positive solution of (1) is not always monotone increasing in the diffusion rate [15].

Keywords: Reaction-diffusion, steady state, diffusion rate, monotone property.

Mathematics Subject Classification: 34D23, 92D25.

Citation: Rui Li, Yuan Lou. Some monotone properties for solutions to a reaction-diffusion model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4445-4455. doi: 10.3934/dcdsb.2019126

References:

[1]	I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017), v+117pp. doi: 10.1090/memo/1161.
[2]	X. L. Bai, X. Q. 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.
[3]	R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273. doi: 10.1016/0022-0396(78)90033-5.
[4]	R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments, Proc. Roy. Soc. Edinburgh, 112A (1989), 293–318. doi: 10.1017/S030821050001876X.
[5]	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.
[6]	R. S. Cantrell and C. Cosner, Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219–252. doi: 10.1137/0153014.
[7]	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.
[8]	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.
[9]	D. DeAngelis, W.-M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239–254. doi: 10.1007/s00285-015-0879-y.
[10]	J. Y. He, Stability of Semi-Trivial Solutions for a Predator-Prey Model in Heterogeneous Environment, M.S. Thesis (in Chinese), East China Normal University, April 2018.
[11]	X. Q. He, K.-Y. Lam, Y. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model: Homogeneous vs. heterogenous environments, J. Math. Biol., 2019, 1–32. doi: 10.1007/s00285-018-1321-z.
[12]	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.
[13]	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.
[14]	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.
[15]	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), Art. 25, 20 pp. doi: 10.1007/s00526-016-0964-0.
[16]	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.
[17]	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.
[18]	Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in Mathematical Biosciences. IV, 171–205, Lecture Notes in Math., 1922, Math. Biosci. Subser., Springer, Berlin, 2008. doi: 10.1007/978-3-540-74331-6_5.
[19]	Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634.
[20]	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.
[21]	W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.
[22]	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), Art. 80, 14 pp. doi: 10.1007/s00526-018-1353-7.
[23]	H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454. doi: 10.2977/prims/1195188180.
[24]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2nd ed., Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5.

show all references

References:

[1]	I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017), v+117pp. doi: 10.1090/memo/1161.
[2]	X. L. Bai, X. Q. 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.
[3]	R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273. doi: 10.1016/0022-0396(78)90033-5.
[4]	R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments, Proc. Roy. Soc. Edinburgh, 112A (1989), 293–318. doi: 10.1017/S030821050001876X.
[5]	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.
[6]	R. S. Cantrell and C. Cosner, Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219–252. doi: 10.1137/0153014.
[7]	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.
[8]	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.
[9]	D. DeAngelis, W.-M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239–254. doi: 10.1007/s00285-015-0879-y.
[10]	J. Y. He, Stability of Semi-Trivial Solutions for a Predator-Prey Model in Heterogeneous Environment, M.S. Thesis (in Chinese), East China Normal University, April 2018.
[11]	X. Q. He, K.-Y. Lam, Y. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model: Homogeneous vs. heterogenous environments, J. Math. Biol., 2019, 1–32. doi: 10.1007/s00285-018-1321-z.
[12]	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.
[13]	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.
[14]	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.
[15]	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), Art. 25, 20 pp. doi: 10.1007/s00526-016-0964-0.
[16]	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.
[17]	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.
[18]	Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in Mathematical Biosciences. IV, 171–205, Lecture Notes in Math., 1922, Math. Biosci. Subser., Springer, Berlin, 2008. doi: 10.1007/978-3-540-74331-6_5.
[19]	Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634.
[20]	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.
[21]	W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.
[22]	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), Art. 80, 14 pp. doi: 10.1007/s00526-018-1353-7.
[23]	H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454. doi: 10.2977/prims/1195188180.
[24]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2nd ed., Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5.

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