Advanced Search
Article Contents
Article Contents

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

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

    * Corresponding author: Jerome Goddard II, jgoddard@aum.edu 
The second author is supported by NSF grant DMS-1853372 and the third author is supported by NSF grant DMS-1853352
Abstract Full Text(HTML) Figure(4) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35J25, 35J60; Secondary: 35J66.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Illustration of $ 1-\alpha(s) $ and $ f(s) $

    Figure 2.  Bifurcation diagram for the solution set of (3)

    Figure 3.  Bifurcation diagram for the solution set of (3) for $ \gamma \gg 1 $ and $ \epsilon \approx 0 $.

    Figure 4.  Graphs of $ \kappa $ vs $ B_1(\kappa) $ and $ \frac{\kappa^2}{\gamma^2 (g(0))^2} $

  • [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. CroninN. FonsekaJ. Goddard IIJ. 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. CroninJ. 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. FonsekaJ. GoddardQ. MorrisR. 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.
    [7] J. Goddard IIQ. MorrisC. 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. HarmanJ. Goddard IIR. 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.
  • 加载中



Article Metrics

HTML views(550) PDF downloads(207) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint