February  2022, 21(2): 705-726. doi: 10.3934/cpaa.2021195

Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions

1. 

Department of Mathematics, SRM University AP, Andhra Pradesh-522502, India

2. 

Department of Mathematics, IIT Palakkad, Kerala-678557, India

3. 

Department of Mathematics and Statistics, University of North Carolina at Greensboro, NC 27412, USA

4. 

Department of Mathematics, IIT Madras, Chennai-600036, India

* Corresponding author

Received  May 2021 Revised  October 2021 Published  February 2022 Early access  December 2021

Fund Project: The third author was supported by the NSF award DMS- 1853352 to perform this research work

We study the structure of positive solutions to steady state ecological models of the form:
$ \begin{array}{l} \left\{ \begin{split} -\Delta u& = \lambda uf(u)\; \; && {\rm{in}}\; \; \Omega,\\ \alpha(u)&\frac{\partial u}{\partial \eta}+[1-\alpha(u)]u = 0 &&\;\;\;{\rm{on}}\; \; \partial\Omega, \end{split} \right. \end{array} $
where
$ \Omega $
is a bounded domain in
$ \mathbb{R}^n; $
$ n>1 $
with smooth boundary
$ \partial\Omega $
or
$ \Omega = (0,1) $
,
$ \frac{\partial}{\partial\eta} $
represents the outward normal derivative on the boundary,
$ \lambda $
is a positive parameter,
$ f:[0,\infty)\to \mathbb{R} $
is a
$ C^2 $
function such that
$ \tfrac{f(s)}{k-s}>0 $
for some
$ k>0 $
, and
$ \alpha:[0,k]\to[0,1] $
is also a
$ C^2 $
function. Here
$ f(u) $
represents the per capita growth rate,
$ \alpha(u) $
represents the fraction of the population that stays on the patch upon reaching the boundary, and
$ \lambda $
relates to the patch size and the diffusion rate. In particular, we will discuss models in which the per capita growth rate is increasing for small
$ u $
, and models where grazing is involved. We will focus on the cases when
$ \alpha'(s)\geq 0 $
;
$ [0,k] $
, which represents negative density dependent dispersal on the boundary. We employ the method of sub-super solutions, bifurcation theory, and stability analysis to obtain our results. We provide detailed bifurcation diagrams via a quadrature method for the case
$ \Omega = (0,1) $
.
Citation: Mohan Mallick, Sarath Sasi, R. Shivaji, S. Sundar. Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2022, 21 (2) : 705-726. doi: 10.3934/cpaa.2021195
References:
[1]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains, J. Differ. Equ., 231 (2006), 768-804.  doi: 10.1016/j.jde.2006.08.018.  Google Scholar

[3]

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

[4]

R. S. CantrellC. Cosner and S. Martĺnez, Steady state solutions of a logistic equation with nonlinear boundary conditions, Rocky Mountain J. Math., 41 (2011), 445-455.  doi: 10.1216/RMJ-2011-41-2-445.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[7]

E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 429-452.  doi: 10.1112/plms/s3-53.3.429.  Google Scholar

[8]

L. Evans, Partial Differential Equations, Graduate studies in mathematics. American Mathematical Society, 2010. doi: 10.1090/gsm/019.  Google Scholar

[9]

N. FonsekaJ. Goddard IIR. Shivaji and B. Son, A diffusive weak allee effect model with u-shaped emigration and matrix hostility, Discrete Contin. Dyn. Syst. Ser. S, 26 (2021), 5509-5517.  doi: 10.3934/dcdsb.2020356.  Google Scholar

[10]

N. FonsekaR. ShivajiJ. Goddard IIQ. A. Morris 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: 10.1103/physrevd.13.3410.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, springer, 2015.  Google Scholar

[13]

J. GoddardE. K. Lee and R. Shivaji, Population models with nonlinear boundary conditions, Electron. J. Differ. Equ.[electronic only], 2010 (2010), 135-149.   Google Scholar

[14]

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

[15]

J. Goddard, II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differ. Equ., 2018 (2018), 12 pp.  Google Scholar

[16]

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), 17 pp. doi: 10.1186/s13661-018-1090-z.  Google Scholar

[17]

J. Goddard II and R. Shivaji, Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary, J. Math. Anal. Appl., 414 (2014), 561-573.  doi: 10.1016/j.jmaa.2014.01.016.  Google Scholar

[18]

J. Goddard IIR. Shivaji and E. K. Lee, Diffusive logistic equation with non-linear boundary conditions, J. Math. Anal. Appl., 375 (2011), 365-370.  doi: 10.1016/j.jmaa.2010.09.057.  Google Scholar

[19]

P. V. GordonE. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal., 15 (2014), 51-57.  doi: 10.1016/j.nonrwa.2013.05.005.  Google Scholar

[20]

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

[21]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970/1971), 1-13.  doi: 10.1512/iumj.1970.20.20001.  Google Scholar

[22]

A. Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056.  Google Scholar

[23]

E. LeeS. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.  doi: 10.1016/j.jmaa.2011.03.048.  Google Scholar

[24]

E. K. LeeR. Shivaji and J. Ye, Positive solutions for elliptic equations involving nonlinearities with falling zeroes, Appl. Math. Lett., 22 (2009), 846-851.  doi: 10.1016/j.aml.2008.08.020.  Google Scholar

[25]

M. K. Mallick., Steady State Reaction Diffusion Equations with Falling Zero Reaction Terms and Nonlinear Boundary Conditions, PhD thesis, Chennai India, 2019. Google Scholar

[26]

M. H. Protter and H. F. Weinberger., Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[27]

J. Serrin., A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[28]

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

show all references

References:
[1]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains, J. Differ. Equ., 231 (2006), 768-804.  doi: 10.1016/j.jde.2006.08.018.  Google Scholar

[3]

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

[4]

R. S. CantrellC. Cosner and S. Martĺnez, Steady state solutions of a logistic equation with nonlinear boundary conditions, Rocky Mountain J. Math., 41 (2011), 445-455.  doi: 10.1216/RMJ-2011-41-2-445.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[7]

E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 429-452.  doi: 10.1112/plms/s3-53.3.429.  Google Scholar

[8]

L. Evans, Partial Differential Equations, Graduate studies in mathematics. American Mathematical Society, 2010. doi: 10.1090/gsm/019.  Google Scholar

[9]

N. FonsekaJ. Goddard IIR. Shivaji and B. Son, A diffusive weak allee effect model with u-shaped emigration and matrix hostility, Discrete Contin. Dyn. Syst. Ser. S, 26 (2021), 5509-5517.  doi: 10.3934/dcdsb.2020356.  Google Scholar

[10]

N. FonsekaR. ShivajiJ. Goddard IIQ. A. Morris 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: 10.1103/physrevd.13.3410.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, springer, 2015.  Google Scholar

[13]

J. GoddardE. K. Lee and R. Shivaji, Population models with nonlinear boundary conditions, Electron. J. Differ. Equ.[electronic only], 2010 (2010), 135-149.   Google Scholar

[14]

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

[15]

J. Goddard, II, Q. Morris, R. Shivaji and B. Son, Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions, Electron. J. Differ. Equ., 2018 (2018), 12 pp.  Google Scholar

[16]

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), 17 pp. doi: 10.1186/s13661-018-1090-z.  Google Scholar

[17]

J. Goddard II and R. Shivaji, Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary, J. Math. Anal. Appl., 414 (2014), 561-573.  doi: 10.1016/j.jmaa.2014.01.016.  Google Scholar

[18]

J. Goddard IIR. Shivaji and E. K. Lee, Diffusive logistic equation with non-linear boundary conditions, J. Math. Anal. Appl., 375 (2011), 365-370.  doi: 10.1016/j.jmaa.2010.09.057.  Google Scholar

[19]

P. V. GordonE. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal., 15 (2014), 51-57.  doi: 10.1016/j.nonrwa.2013.05.005.  Google Scholar

[20]

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

[21]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970/1971), 1-13.  doi: 10.1512/iumj.1970.20.20001.  Google Scholar

[22]

A. Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056.  Google Scholar

[23]

E. LeeS. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.  doi: 10.1016/j.jmaa.2011.03.048.  Google Scholar

[24]

E. K. LeeR. Shivaji and J. Ye, Positive solutions for elliptic equations involving nonlinearities with falling zeroes, Appl. Math. Lett., 22 (2009), 846-851.  doi: 10.1016/j.aml.2008.08.020.  Google Scholar

[25]

M. K. Mallick., Steady State Reaction Diffusion Equations with Falling Zero Reaction Terms and Nonlinear Boundary Conditions, PhD thesis, Chennai India, 2019. Google Scholar

[26]

M. H. Protter and H. F. Weinberger., Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[27]

J. Serrin., A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[28]

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

Figure 1.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 1-\frac{s}{10}-1.5\frac{s}{1+s^2} $ with $ \alpha(1) = 1 $.
Figure 2.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 1-\frac{s}{10}-1.5\frac{s}{1+s^2} $ with $ \alpha(1)<1 $.
Figure 3.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 5-s-\frac{1}{1+5s} $ with $ \alpha(1) = 1 $.
Figure 4.  Bifurcation diagram for (6.1)-(6.3) when $ f(s) = 5-s-\frac{1}{1+5s} $ with $ \alpha(1)<1 $.
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