MONOTONE METHODS AND STABILITY RESULTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY

Yueding Yuan; Zhiming Guo

doi:10.11948/2018.1342

2018 Volume 8 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(5): 1342-1368

Next Article Previous Article

Yueding Yuan, Zhiming Guo. MONOTONE METHODS AND STABILITY RESULTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1342-1368. doi: 10.11948/2018.1342

Citation:

Yueding Yuan, Zhiming Guo. MONOTONE METHODS AND STABILITY RESULTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1342-1368. doi: 10.11948/2018.1342

MONOTONE METHODS AND STABILITY RESULTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY

Yueding Yuan^1,2,
Zhiming Guo³

1 School of Mathematics and Statistics, Central South University, Changsha 410083, China;
2 School of Mathematics and Computer Sciences, Yichun University, Yichun 336000, China;
3 School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

Fund Project:

Abstract

Abstract

In this paper, we study the applications of the monotone iteration method for investigating the existence and stability of solutions to nonlocal reaction-diffusion equations with time delay. We emphasize the importance of the idea of monotone iteration schemes for investigating the stability of solutions to such equations. We show that every steady state of such equations obtained by using the monotone iteration method is priori stable and all stable steady states can be obtained by using such method. Finally, we apply our main results to three population models.
- Monotone method /
- nonlocal reaction term /
- delay /
- existence and uniqueness /
- stability
MSC: 34K20;35A01;35A16;35K20

FullText(HTML)

References

[1]	S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary cinditions, I, Comm. Pure Appl. Math., 1959, 12(4), 623-727. Google Scholar
[2]	H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 1971, 21(2), 125-146. Google Scholar
[3]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Review, 1976, 18(4), 620-709. Google Scholar
[4]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd., Chichester, UK, 2003. Google Scholar
[5]	S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reactiondiffusion population model, J. Differential Equations, 2016, 260(1), 218-240. Google Scholar
[6]	S. Chen and J. Yu, Stability analysis of a reaction-diffusion equation with spatiotemporal delay and Dirichlet boundary condition, J. Dyn. Diff. Equat., 2016, 28(3-4), 857-866. Google Scholar
[7]	L. Collatz, Funktionalanalysis und Numerische Mathematik, Springer, Berlin, 1964. Google Scholar
[8]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998, 19. Google Scholar
[9]	W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects, J. Math. Anal. Appl., 1997, 206(2), 547-566. Google Scholar
[10]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964. Google Scholar
[11]	S. A. Gourley and Y. Kuang, wavefronts and global stability in a time-delayed population model with stage structure, R. Soc. Land. Proc., Ser. A:Math. Phys. Eng. Sci., 2003, 459(2034), 1563-1579. DOI:10.1098/rspa.2002.1094. Google Scholar
[12]	S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in:Nonlinear Dynamics and Evolution Equations (H. Brunner, X.-Q. Zhao and X. Zou, eds.), Fields Inst. Commun., 2006, 48, 137-200. Google Scholar
[13]	S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 2015, 259(4), 1409-1448. Google Scholar
[14]	S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 2016, 260(1), 781-817. Google Scholar
[15]	Z. Guo, Z. Yang and X. Zou, Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition:a non-monotone case, Commun. Pure Appl. Anal., 2012, 11(5), 1825-1838. Google Scholar
[16]	W. Huang and Y. Wu, A note on monotone iteration method for traveling waves of reaction-diffusion systems with time delay, J. Appl. Anal. Comput., 2014, 4(3), 283-294. Google Scholar
[17]	T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980. Google Scholar
[18]	H. B. Keller, Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rat. Mech. Anal., 1969, 35(5), 363-381. Google Scholar
[19]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralćeva, Linear and quasilinear equations of parabolic type, AMS Translations of Mathematical Monographs, 1968, 23. Google Scholar
[20]	O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Euasilinear Elliptic Equations, Academic Press, New York, 1968. Google Scholar
[21]	D. Liang, J. W. H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numerical computations, Diff. Eqns. Dynam. Syst., 2003, 11, 117-139. Google Scholar
[22]	X. Lu, Persistence and extinction in a competition-diffusion system with time delays, Canad. Appl. Math. Quart., 1994, 2(2), 231-246. Google Scholar
[23]	X. Lu and W. Feng, Dynamics and numerical simulations of food-chain populations, Appl. Math. Comput., 1994, 65(1-3), 335-344. Google Scholar
[24]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 1977, 197(4300), 287-289. DOI:10.1126/science.267326. Google Scholar
[25]	R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 1990, 321(1), 1-44. Google Scholar
[26]	C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, New York, 1966. Google Scholar
[27]	C. V. Pao, On a coupled reaction-diffusion system with time delays, SIAM J. Math. Anal., 1987, 18(4), 1026-1039. Google Scholar
[28]	C. V. Pao, Numerical methods for coupled systems of nonlinear parabolic boundary value problems, J. Math. Anal. Appl., 1990, 151(2), 581-608. Google Scholar
[29]	C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Google Scholar
[30]	C. V. Pao, Coupled nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 1995, 196(1), 237-265. Google Scholar
[31]	C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 1996, 198(3), 751-779. Google Scholar
[32]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1967. Google Scholar
[33]	R. Redlinger, Existence theorems for semilinear parabolic systems with functionals, Nonlinear Anal., 1984, 8(6), 667-682. Google Scholar
[34]	D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 1972, 21(11), 979-1000. Google Scholar
[35]	A. Schiaffino and A. Tesei, Monotone methods and attractivity results for Volterra integro-partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 1981, 89(1-2), 135-142. Google Scholar
[36]	L. Shu, P. Weng and Y. Tian, Traveling wavefronts of a delayed lattice reactiondiffusion model, J. Appl. Anal. Comput., 2015, 5(1), 64-76. Google Scholar
[37]	J. W. H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 2001, 457, 1841-1853. DOI:10.1098/rspa.2001.0789. Google Scholar
[38]	H. R. Thieme and X. Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. RWA., 2001, 2(2), 145-160. Google Scholar
[39]	H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Eqns., 2003, 195(2), 430-470. Google Scholar
[40]	J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. Google Scholar
[41]	S. L. Wu, C. H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 2015, 258(4), 1058-1105. Google Scholar
[42]	D. Xu and X. Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canad. Appl. Math. Quart., 2003, 11(3), 303-319. Google Scholar
[43]	T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition:a non-monotone case, J. Differential Equations, 2008, 245(11), 3376-3388. Google Scholar
[44]	T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dyn. Diff. Equat., 2013, 25(4), 959-979. Google Scholar
[45]	T. Yi and X. Zou, Dirichlet problem of a delayed reaction-diffusion equation on a semi-infinite interval, J. Dyn. Diff. Equat., 2016, 28(3-4), 1007-1030. Google Scholar
[46]	Y. Yuan, Z. Guo and M. Tang, A nonlocal diffusion population model with age structure and Dirichlet boundary condition, Commun. Pure Appl. Anal., 2015, 14(5), 2095-2115. Google Scholar
[47]	Y. Yuan and Z. Guo, Global dynamics of a nonlocal population model with age structure in a bounded domain:A non-monotone case, Sci. China Math., 2015, 58(10), 2145-2166. Google Scholar
[48]	L. Zhang, Z.-C. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period, J. Differential Equations, 2015, 258(9), 3011-3036. Google Scholar
[49]	X. Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with time delay, Canad. Appl. Math. Quart., 2009, 17(1), 271-281. Google Scholar