October  2021, 26(10): 5509-5517. doi: 10.3934/dcdsb.2020356

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  October 2021 Early access  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, 2021, 26 (10) : 5509-5517. doi: 10.3934/dcdsb.2020356
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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} $
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