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

Mohan Mallick; Sarath Sasi; R. Shivaji; S. Sundar

doi:10.3934/cpaa.2021195

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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

Mohan Mallick ^1, , Sarath Sasi ^2,, , R. Shivaji ^3, and S. Sundar ^4,

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 $

$ \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) $

Keywords: Reaction diffusion, nonlinear boundary conditions, bifurcation, population dynamics, Allee effect.

Mathematics Subject Classification: Primary: 35K57; Secondary: 35B32, 92D40.

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. Cantrell, C. 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. Fonseka, J. Goddard II, R. 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. Fonseka, R. Shivaji, J. Goddard II, Q. 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. Goddard, E. 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 II, Q. Morris, C. 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 II, R. 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. Gordon, E. 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. Lee, S. 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. Lee, R. 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. Cantrell, C. 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. Fonseka, J. Goddard II, R. 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. Fonseka, R. Shivaji, J. Goddard II, Q. 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. Goddard, E. 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 II, Q. Morris, C. 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 II, R. 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. Gordon, E. 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. Lee, S. 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. Lee, R. 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 $.

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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 $.

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

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

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