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Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity
A diffusive weak Allee effect model with U-shaped emigration and matrix hostility
1. | Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA |
2. | Department of Mathematics, Auburn University Montgomery, Montgomery, AL 36124, USA |
3. | Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA |
$ \begin{equation*} \; \; \begin{matrix} -\Delta u = \lambda f(u);\; \Omega \\ \; \; \alpha(u)\frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[1-\alpha(u)]u = 0; \; \partial \Omega\end{matrix} \end{equation*} $ |
$ u $ |
$ f(u) = \frac{1}{a}u(u+a)(1-u) $ |
$ a\in (0,1) $ |
$ \alpha(u) $ |
$ \Omega $ |
$ \lambda $ |
$ \gamma $ |
$ \alpha(s) = \frac{1}{[1+(A - s)^2 + \epsilon]} $ |
$ s \in [0,1] $ |
$ A\in (0,1) $ |
$ \epsilon\geq 0 $ |
$ 1-\alpha(s) $ |
References:
[1] |
H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.
doi: 10.1137/1018114. |
[2] |
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. |
[3] |
J. T. Cronin, N. Fonseka, J. Goddard II, J. Leonard and R. Shivaji,
Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence, Math. Biosci. Eng., 17 (2020), 1718-1742.
doi: 10.3934/mbe.2020090. |
[4] |
J. T. Cronin, J. Goddard II and and R. Shivaji,
Effects of patch matrix-composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933-3975.
doi: 10.1007/s11538-019-00634-9. |
[5] |
N. Fonseka, J. Goddard, Q. Morris, R. Shivaji 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: 10.3934/dcdss.2020245. |
[6] |
N. Fonseka, A. Muthunayake, R. Shivaji and B. Son, Singular reaction diffusion equations where a parameter influences the reaction term and the boundary condition, Topol. Methods Nonlinear Anal., Accepted. Google Scholar |
[7] |
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. |
[8] |
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), Paper No. 170, 17 pp.
doi: 10.1186/s13661-018-1090-z. |
[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] |
R. R. Harman, J. Goddard II, R. Shivaji and J. T. Cronin,
Frequency of occurrence and population-dynamic consequences of different forms of density-dependent emigration, Am. Nat., 195 (2019), 851-867.
doi: 10.1086/708156. |
[11] |
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. |
[12] |
M. A. Rivas and S. B. Robinson, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 45, 25 pp.
doi: 10.1051/cocv/2018039. |
[13] |
D. H. Sattinger,
Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.
doi: 10.1512/iumj.1972.21.21079. |
[14] |
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. |
[15] |
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, Ed. V. Lakshmikantham. |
show all references
References:
[1] |
H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.
doi: 10.1137/1018114. |
[2] |
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. |
[3] |
J. T. Cronin, N. Fonseka, J. Goddard II, J. Leonard and R. Shivaji,
Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence, Math. Biosci. Eng., 17 (2020), 1718-1742.
doi: 10.3934/mbe.2020090. |
[4] |
J. T. Cronin, J. Goddard II and and R. Shivaji,
Effects of patch matrix-composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933-3975.
doi: 10.1007/s11538-019-00634-9. |
[5] |
N. Fonseka, J. Goddard, Q. Morris, R. Shivaji 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: 10.3934/dcdss.2020245. |
[6] |
N. Fonseka, A. Muthunayake, R. Shivaji and B. Son, Singular reaction diffusion equations where a parameter influences the reaction term and the boundary condition, Topol. Methods Nonlinear Anal., Accepted. Google Scholar |
[7] |
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. |
[8] |
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), Paper No. 170, 17 pp.
doi: 10.1186/s13661-018-1090-z. |
[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] |
R. R. Harman, J. Goddard II, R. Shivaji and J. T. Cronin,
Frequency of occurrence and population-dynamic consequences of different forms of density-dependent emigration, Am. Nat., 195 (2019), 851-867.
doi: 10.1086/708156. |
[11] |
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. |
[12] |
M. A. Rivas and S. B. Robinson, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 45, 25 pp.
doi: 10.1051/cocv/2018039. |
[13] |
D. H. Sattinger,
Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.
doi: 10.1512/iumj.1972.21.21079. |
[14] |
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. |
[15] |
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, Ed. V. Lakshmikantham. |


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