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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

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].
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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