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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[27]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[27]

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

[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.

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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