    ## 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

* Corresponding author: Jerome Goddard II, jgoddard@aum.edu

Received  July 2020 Revised  October 2020 Published  December 2020

Fund Project: The second author is supported by NSF grant DMS-1853372 and the third author is supported by NSF grant DMS-1853352

We study positive solutions to steady state reaction diffusion equations of the form:
 $\begin{equation*} \; \; \begin{matrix} -\Delta u = \lambda f(u);\; \Omega \\ \; \; \alpha(u)\frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[1-\alpha(u)]u = 0; \; \partial \Omega\end{matrix} \end{equation*}$
where
 $u$
is the population density,
 $f(u) = \frac{1}{a}u(u+a)(1-u)$
represents a weak Allee effect type growth of the population with
 $a\in (0,1)$
,
 $\alpha(u)$
is the probability of the population staying in the habitat
 $\Omega$
when it reaches the boundary, and positive parameters
 $\lambda$
and
 $\gamma$
represent the domain scaling and effective exterior matrix hostility, respectively. In particular, we analyze the case when
 $\alpha(s) = \frac{1}{[1+(A - s)^2 + \epsilon]}$
for all
 $s \in [0,1]$
, where
 $A\in (0,1)$
and
 $\epsilon\geq 0$
. In this case
 $1-\alpha(s)$
represents a U-shaped relationship between density and emigration. Existence, nonexistence, and multiplicity results for this model are established via the method of sub-super solutions.
Citation: Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020356
##### References:
  H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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  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.  Google Scholar  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.  Google Scholar  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  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. Google Scholar  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  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.  Google Scholar  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.  Google Scholar  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  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. Google Scholar

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##### References:
  H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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  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.  Google Scholar  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.  Google Scholar  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  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. Google Scholar  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  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.  Google Scholar  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.  Google Scholar  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  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. Google Scholar Bifurcation diagram for the solution set of (3) for $\gamma \gg 1$ and $\epsilon \approx 0$. Graphs of $\kappa$ vs $B_1(\kappa)$ and $\frac{\kappa^2}{\gamma^2 (g(0))^2}$
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