
-
Previous Article
Necessary and sufficient conditions on weighted multilinear fractional integral inequality
- CPAA Home
- This Issue
-
Next Article
Nonnegative solutions to a doubly degenerate nutrient taxis system
Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions
1. | Department of Mathematics, SRM University AP, Andhra Pradesh-522502, India |
2. | Department of Mathematics, IIT Palakkad, Kerala-678557, India |
3. | Department of Mathematics and Statistics, University of North Carolina at Greensboro, NC 27412, USA |
4. | Department of Mathematics, IIT Madras, Chennai-600036, India |
$ \begin{array}{l} \left\{ \begin{split} -\Delta u& = \lambda uf(u)\; \; && {\rm{in}}\; \; \Omega,\\ \alpha(u)&\frac{\partial u}{\partial \eta}+[1-\alpha(u)]u = 0 &&\;\;\;{\rm{on}}\; \; \partial\Omega, \end{split} \right. \end{array} $ |
$ \Omega $ |
$ \mathbb{R}^n; $ |
$ n>1 $ |
$ \partial\Omega $ |
$ \Omega = (0,1) $ |
$ \frac{\partial}{\partial\eta} $ |
$ \lambda $ |
$ f:[0,\infty)\to \mathbb{R} $ |
$ C^2 $ |
$ \tfrac{f(s)}{k-s}>0 $ |
$ k>0 $ |
$ \alpha:[0,k]\to[0,1] $ |
$ C^2 $ |
$ f(u) $ |
$ \alpha(u) $ |
$ \lambda $ |
$ u $ |
$ \alpha'(s)\geq 0 $ |
$ [0,k] $ |
$ \Omega = (0,1) $ |
References:
[1] |
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. |
[2] |
R. S. Cantrell and C. Cosner,
On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains, J. Differ. Equ., 231 (2006), 768-804.
doi: 10.1016/j.jde.2006.08.018. |
[3] |
R. S. Cantrell and C. Cosner,
Density dependent behavior at habitat boundaries and the Allee effect, Bull. Math. Biol., 69 (2007), 2339-2360.
doi: 10.1007/s11538-007-9222-0. |
[4] |
R. S. Cantrell, C. Cosner and S. Martĺnez,
Steady state solutions of a logistic equation with nonlinear boundary conditions, Rocky Mountain J. Math., 41 (2011), 445-455.
doi: 10.1216/RMJ-2011-41-2-445. |
[5] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[6] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[7] |
E. N. Dancer,
On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 429-452.
doi: 10.1112/plms/s3-53.3.429. |
[8] |
L. Evans, Partial Differential Equations, Graduate studies in mathematics. American Mathematical Society, 2010.
doi: 10.1090/gsm/019. |
[9] |
N. Fonseka, J. Goddard II, R. Shivaji and B. Son,
A diffusive weak allee effect model with u-shaped emigration and matrix hostility, Discrete Contin. Dyn. Syst. Ser. S, 26 (2021), 5509-5517.
doi: 10.3934/dcdsb.2020356. |
[10] |
N. Fonseka, R. Shivaji, J. Goddard II, Q. A. Morris and B. Son,
On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3401-3415.
doi: 10.1103/physrevd.13.3410. |
[11] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1983.
doi: 10.1007/978-3-642-61798-0. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, springer, 2015. |
[13] |
J. Goddard, E. K. Lee and R. Shivaji,
Population models with nonlinear boundary conditions, Electron. J. Differ. Equ.[electronic only], 2010 (2010), 135-149.
|
[14] |
J. Goddard II, Q. Morris, C. Payne and R. Shivaji,
A diffusive logistic equation with U-shaped density dependent dispersal on the boundary, Topol. Methods Nonlinear Anal., 53 (2019), 335-349.
doi: 10.12775/tmna.2018.047. |
[15] |
J. Goddard, II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differ. Equ., 2018 (2018), 12 pp. |
[16] |
J. Goddard, II, Q. A. Morris, S. B. Robinson and R. Shivaji, An exact bifurcation diagram for a reaction-diffusion equation arising in population dynamics, Bound. Value Probl., 2018 (2018), 17 pp.
doi: 10.1186/s13661-018-1090-z. |
[17] |
J. Goddard II and R. Shivaji,
Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary, J. Math. Anal. Appl., 414 (2014), 561-573.
doi: 10.1016/j.jmaa.2014.01.016. |
[18] |
J. Goddard II, R. Shivaji and E. K. Lee,
Diffusive logistic equation with non-linear boundary conditions, J. Math. Anal. Appl., 375 (2011), 365-370.
doi: 10.1016/j.jmaa.2010.09.057. |
[19] |
P. V. Gordon, E. Ko and R. Shivaji,
Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal., 15 (2014), 51-57.
doi: 10.1016/j.nonrwa.2013.05.005. |
[20] |
F. Inkmann,
Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 31 (1982), 213-221.
doi: 10.1512/iumj.1982.31.31019. |
[21] |
T. Laetsch,
The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970/1971), 1-13.
doi: 10.1512/iumj.1970.20.20001. |
[22] |
A. Lê,
Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.
doi: 10.1016/j.na.2005.05.056. |
[23] |
E. Lee, S. Sasi and R. Shivaji,
S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.
doi: 10.1016/j.jmaa.2011.03.048. |
[24] |
E. K. Lee, R. Shivaji and J. Ye,
Positive solutions for elliptic equations involving nonlinearities with falling zeroes, Appl. Math. Lett., 22 (2009), 846-851.
doi: 10.1016/j.aml.2008.08.020. |
[25] |
M. K. Mallick., Steady State Reaction Diffusion Equations with Falling Zero Reaction Terms and Nonlinear Boundary Conditions, PhD thesis, Chennai India, 2019. |
[26] |
M. H. Protter and H. F. Weinberger., Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[27] |
J. Serrin.,
A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
doi: 10.1007/BF00250468. |
[28] |
J. Shi and R. Shivaji,
Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829.
doi: 10.1007/s00285-006-0373-7. |
show all references
References:
[1] |
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. |
[2] |
R. S. Cantrell and C. Cosner,
On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains, J. Differ. Equ., 231 (2006), 768-804.
doi: 10.1016/j.jde.2006.08.018. |
[3] |
R. S. Cantrell and C. Cosner,
Density dependent behavior at habitat boundaries and the Allee effect, Bull. Math. Biol., 69 (2007), 2339-2360.
doi: 10.1007/s11538-007-9222-0. |
[4] |
R. S. Cantrell, C. Cosner and S. Martĺnez,
Steady state solutions of a logistic equation with nonlinear boundary conditions, Rocky Mountain J. Math., 41 (2011), 445-455.
doi: 10.1216/RMJ-2011-41-2-445. |
[5] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[6] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[7] |
E. N. Dancer,
On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 429-452.
doi: 10.1112/plms/s3-53.3.429. |
[8] |
L. Evans, Partial Differential Equations, Graduate studies in mathematics. American Mathematical Society, 2010.
doi: 10.1090/gsm/019. |
[9] |
N. Fonseka, J. Goddard II, R. Shivaji and B. Son,
A diffusive weak allee effect model with u-shaped emigration and matrix hostility, Discrete Contin. Dyn. Syst. Ser. S, 26 (2021), 5509-5517.
doi: 10.3934/dcdsb.2020356. |
[10] |
N. Fonseka, R. Shivaji, J. Goddard II, Q. A. Morris and B. Son,
On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3401-3415.
doi: 10.1103/physrevd.13.3410. |
[11] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1983.
doi: 10.1007/978-3-642-61798-0. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, springer, 2015. |
[13] |
J. Goddard, E. K. Lee and R. Shivaji,
Population models with nonlinear boundary conditions, Electron. J. Differ. Equ.[electronic only], 2010 (2010), 135-149.
|
[14] |
J. Goddard II, Q. Morris, C. Payne and R. Shivaji,
A diffusive logistic equation with U-shaped density dependent dispersal on the boundary, Topol. Methods Nonlinear Anal., 53 (2019), 335-349.
doi: 10.12775/tmna.2018.047. |
[15] |
J. Goddard, II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differ. Equ., 2018 (2018), 12 pp. |
[16] |
J. Goddard, II, Q. A. Morris, S. B. Robinson and R. Shivaji, An exact bifurcation diagram for a reaction-diffusion equation arising in population dynamics, Bound. Value Probl., 2018 (2018), 17 pp.
doi: 10.1186/s13661-018-1090-z. |
[17] |
J. Goddard II and R. Shivaji,
Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary, J. Math. Anal. Appl., 414 (2014), 561-573.
doi: 10.1016/j.jmaa.2014.01.016. |
[18] |
J. Goddard II, R. Shivaji and E. K. Lee,
Diffusive logistic equation with non-linear boundary conditions, J. Math. Anal. Appl., 375 (2011), 365-370.
doi: 10.1016/j.jmaa.2010.09.057. |
[19] |
P. V. Gordon, E. Ko and R. Shivaji,
Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal., 15 (2014), 51-57.
doi: 10.1016/j.nonrwa.2013.05.005. |
[20] |
F. Inkmann,
Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 31 (1982), 213-221.
doi: 10.1512/iumj.1982.31.31019. |
[21] |
T. Laetsch,
The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970/1971), 1-13.
doi: 10.1512/iumj.1970.20.20001. |
[22] |
A. Lê,
Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.
doi: 10.1016/j.na.2005.05.056. |
[23] |
E. Lee, S. Sasi and R. Shivaji,
S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.
doi: 10.1016/j.jmaa.2011.03.048. |
[24] |
E. K. Lee, R. Shivaji and J. Ye,
Positive solutions for elliptic equations involving nonlinearities with falling zeroes, Appl. Math. Lett., 22 (2009), 846-851.
doi: 10.1016/j.aml.2008.08.020. |
[25] |
M. K. Mallick., Steady State Reaction Diffusion Equations with Falling Zero Reaction Terms and Nonlinear Boundary Conditions, PhD thesis, Chennai India, 2019. |
[26] |
M. H. Protter and H. F. Weinberger., Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[27] |
J. Serrin.,
A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
doi: 10.1007/BF00250468. |
[28] |
J. Shi and R. Shivaji,
Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829.
doi: 10.1007/s00285-006-0373-7. |




[1] |
Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks and Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191 |
[2] |
Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 |
[3] |
Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629 |
[4] |
Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 |
[5] |
Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks and Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23 |
[6] |
Chuangxia Huang, Xiaojin Guo, Jinde Cao, Ardak Kashkynbayev. Bistable dynamics on a tick population equation incorporating Allee effect and two different time-varying delays. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022122 |
[7] |
Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062 |
[8] |
Costică Moroşanu, Bianca Satco. Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022042 |
[9] |
Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, 2021, 29 (5) : 3017-3030. doi: 10.3934/era.2021024 |
[10] |
Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 |
[11] |
Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054 |
[12] |
Kwangjoong Kim, Wonhyung Choi, Inkyung Ahn. Reaction-advection-diffusion competition models under lethal boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021250 |
[13] |
Xiaoyuan Chang, Junping Shi. Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4551-4572. doi: 10.3934/dcdsb.2021242 |
[14] |
Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41 |
[15] |
Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19 |
[16] |
Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040 |
[17] |
Wei Wang, Wanbiao Ma, Xiulan Lai. Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3989-4011. doi: 10.3934/dcdsb.2020271 |
[18] |
Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279 |
[19] |
Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 |
[20] |
Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 85-100. doi: 10.3934/mbe.2008.5.85 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]