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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Lou, Some reaction diffusion models in spatial ecology, Scientia Sinica Mathematica, 45 (2015), 1619-1634.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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