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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 |
$ \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} $ |
$ \mu $ |
$ m $ |
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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