# American Institute of Mathematical Sciences

December  2020, 13(12): 3401-3415. doi: 10.3934/dcdss.2020245

## 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

* Corresponding author: r_shivaj@uncg.edu

Received  January 2019 Published  December 2020 Early access  January 2020

Fund Project: This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-1516519 & DMS-1516560

We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely,
 $\begin{equation*} \label{abs} \left\lbrace \begin{matrix}-\Delta u = \lambda u(1-u) ;\; x\in\Omega\\ \frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u = 0; \; x\in\partial \Omega \end{matrix} \right. \end{equation*}$
where
 $\Omega$
is a bounded domain in
 $\mathbb{R}^N$
;
 $N > 1$
with smooth boundary
 $\partial \Omega$
or
 $\Omega = (0,1)$
,
 $\frac{\partial u}{\partial \eta}$
is the outward normal derivative of
 $u$
on
 $\partial \Omega$
,
 $\lambda$
is a domain scaling parameter,
 $\gamma$
is a measure of the exterior matrix (
 $\Omega^c$
) hostility, and
 $A\in (0,1)$
and
 $\epsilon>0$
are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for
 $u and increasing for $ u>A $. We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of $ \lambda $. When $ \Omega = (0,1) $we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter $ \gamma $varies. Our results indicate that when $ \gamma $is large there is no Allee effect for any $ \lambda $. We employ a method of sub-supersolutions to obtain existence and multiplicity results when $ N>1 $, and the quadrature method to study the case $ N = 1 $. Citation: Nalin Fonseka, Ratnasingham Shivaji, Jerome Goddard, Ⅱ, Quinn A. Morris, Byungjae Son. On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3401-3415. doi: 10.3934/dcdss.2020245 ##### References:  [1] W. C. Allee, The Social Life of Animals, W. W. Norton & Company, Inc., New York, 1938. Google Scholar [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. Google Scholar [3] J. T. Cronin, Movement and spatial population structure of a prairie planthopper, Ecology, 84 (2003), 1179-1188. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar show all references ##### References:  [1] W. C. Allee, The Social Life of Animals, W. W. Norton & Company, Inc., New York, 1938. Google Scholar [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. Google Scholar [3] J. T. Cronin, Movement and spatial population structure of a prairie planthopper, Ecology, 84 (2003), 1179-1188. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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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