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  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<A $
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. DhanyaE. 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. DhanyaR. 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 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.   Google Scholar

[7]

J. Goddard ⅡQ. MorrisS. 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 ⅡQ. MorrisR. 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. OdendaalP. 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. DhanyaE. 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. DhanyaR. 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 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.   Google Scholar

[7]

J. Goddard ⅡQ. MorrisS. 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 ⅡQ. MorrisR. 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. OdendaalP. 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

Figure 1.  Habitat $ \Omega $ and the exterior matrix $ \Omega^c $
Figure 2.  An example that illustrates U-shaped density dependent dispersal ($ 1-\alpha(u) $) on the boundary
Figure 3.  Eigencurve $ B(\kappa) $ and principal eigenvalue of (1.5)
Figure 4.  Bifurcation diagrams for (1.4)
Figure 5.  Shape of a positive solution
Figure 6.  Plot that illustrates the existence of $ \epsilon_{\lambda} $
Figure 7.  The graph of $ H(q) $
Figure 8.  Evolution of bifurcation diagrams for (1.8) as $ \gamma $ varies when $ \epsilon = 0.1 $ and $ A = 0.5 $
Figure 9.  Bifurcation diagrams for (1.8) for several values of $ \gamma $, when $ \epsilon = 0.01 $ and $ A = 0.8. $
Figure 10.  Picture that illustrates that if $ \lambda < E_1(\gamma,D) $ then $ \sigma_1(\lambda,\gamma,D)>0 $
Figure 11.  The plot illustrates the existence of $ \kappa_1(\lambda,\gamma,D) $
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