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On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model
1. | The University of North Carolina at Greensboro, PO Box 26170, Greensboro, NC 27402-6170, USA |
2. | Auburn University at Montgomery, 7400 East Drive, Montgomery, AL 36117, USA |
3. | Appalachian State University, 121 Bodenheimer Drive, Boone, NC 28608, USA |
4. | University of Maine, 5752 Neville Hall, Room 333, Orono, ME 04469, USA |
$ \begin{equation*} \label{abs} \left\lbrace \begin{matrix}-\Delta u = \lambda u(1-u) ;\; x\in\Omega\\ \frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u = 0; \; x\in\partial \Omega \end{matrix} \right. \end{equation*} $ |
$ \Omega $ |
$ \mathbb{R}^N $ |
$ N > 1 $ |
$ \partial \Omega $ |
$ \Omega = (0,1) $ |
$ \frac{\partial u}{\partial \eta} $ |
$ u $ |
$ \partial \Omega $ |
$ \lambda $ |
$ \gamma $ |
$ \Omega^c $ |
$ A\in (0,1) $ |
$ \epsilon>0 $ |
$ u<A $ |
$ u>A $ |
$ \lambda $ |
$ \Omega = (0,1) $ |
$ \gamma $ |
$ \gamma $ |
$ \lambda $ |
$ N>1 $ |
$ N = 1 $ |
References:
[1] |
W. C. Allee, The Social Life of Animals, W. W. Norton & Company, Inc., New York, 1938. |
[2] |
R. S. Cantrell and C. Cosner,
Density dependent behavior at habitat boundaries and the Allee effect, Bulletin of Mathematical Biology, 69 (2007), 2339-2360.
doi: 10.1007/s11538-007-9222-0. |
[3] |
J. T. Cronin,
Movement and spatial population structure of a prairie planthopper, Ecology, 84 (2003), 1179-1188.
|
[4] |
R. Dhanya, E. Ko and R. Shivaji,
A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl., 424 (2015), 598-612.
doi: 10.1016/j.jmaa.2014.11.012. |
[5] |
R. Dhanya, R. Shivaji and B. Son,
A three solution theorem for a singular differential equation with nonlinear boundary conditions, Topol. Methods Nonlinear Anal., 74 (2011), 6202-6208.
doi: 10.1016/j.na.2011.06.001. |
[6] |
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.
|
[7] |
J. Goddard Ⅱ, Q. Morris, S. Robinson and R. Shivaji,
An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Bound. Value Probl., 2018 (2018), 1-17.
doi: 10.1186/s13661-018-1090-z. |
[8] |
J. Goddard Ⅱ, Q. Morris, R. Shivaji and B. Son,
Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differential Equations, 2018 (2018), 1-12.
|
[9] |
J. Goddard Ⅱ and R. Shivaji,
Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1019-1040.
doi: 10.1017/S0308210516000408. |
[10] |
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. |
[11] |
F. J. Odendaal, P. Turchin and F. R. Stermitz,
Influence of host-plant density and male harassment on the distribution of female euphydryas anicia (nymphalidae), Oecologia, 78 (1989), 283-288.
doi: 10.1007/BF00377167. |
[12] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
![]() ![]() |
[13] |
M. A. Rivas and S. Robinson, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 45, 25 pp.
doi: 10.1051/cocv/2018039. |
[14] |
A. M. Shapiro,
The role of sexual behavior in density-related dispersal of pierid butterflies, The American Naturalist, 104 (1970), 367-372.
doi: 10.1086/282670. |
[15] |
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. |
[16] |
R. Shivaji,
A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications, Lecture Notes in Pure and Applied Mathematics, 109 (1987), 561-566.
|
show all references
References:
[1] |
W. C. Allee, The Social Life of Animals, W. W. Norton & Company, Inc., New York, 1938. |
[2] |
R. S. Cantrell and C. Cosner,
Density dependent behavior at habitat boundaries and the Allee effect, Bulletin of Mathematical Biology, 69 (2007), 2339-2360.
doi: 10.1007/s11538-007-9222-0. |
[3] |
J. T. Cronin,
Movement and spatial population structure of a prairie planthopper, Ecology, 84 (2003), 1179-1188.
|
[4] |
R. Dhanya, E. Ko and R. Shivaji,
A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl., 424 (2015), 598-612.
doi: 10.1016/j.jmaa.2014.11.012. |
[5] |
R. Dhanya, R. Shivaji and B. Son,
A three solution theorem for a singular differential equation with nonlinear boundary conditions, Topol. Methods Nonlinear Anal., 74 (2011), 6202-6208.
doi: 10.1016/j.na.2011.06.001. |
[6] |
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.
|
[7] |
J. Goddard Ⅱ, Q. Morris, S. Robinson and R. Shivaji,
An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Bound. Value Probl., 2018 (2018), 1-17.
doi: 10.1186/s13661-018-1090-z. |
[8] |
J. Goddard Ⅱ, Q. Morris, R. Shivaji and B. Son,
Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differential Equations, 2018 (2018), 1-12.
|
[9] |
J. Goddard Ⅱ and R. Shivaji,
Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1019-1040.
doi: 10.1017/S0308210516000408. |
[10] |
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. |
[11] |
F. J. Odendaal, P. Turchin and F. R. Stermitz,
Influence of host-plant density and male harassment on the distribution of female euphydryas anicia (nymphalidae), Oecologia, 78 (1989), 283-288.
doi: 10.1007/BF00377167. |
[12] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
![]() ![]() |
[13] |
M. A. Rivas and S. Robinson, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 45, 25 pp.
doi: 10.1051/cocv/2018039. |
[14] |
A. M. Shapiro,
The role of sexual behavior in density-related dispersal of pierid butterflies, The American Naturalist, 104 (1970), 367-372.
doi: 10.1086/282670. |
[15] |
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. |
[16] |
R. Shivaji,
A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications, Lecture Notes in Pure and Applied Mathematics, 109 (1987), 561-566.
|










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