Stability analysis of a model on varying domain with the Robin boundary condition

Xiaofei Cao; Guowei Dai

doi:10.3934/dcdss.2017048

Article Contents

2017, Volume 10, Issue 5: 935-942. Doi: 10.3934/dcdss.2017048

This issue Previous Article Preface for special session entitled "Recent Advances of Differential Equations with Applications in Life Sciences" Next Article Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors

Stability analysis of a model on varying domain with the Robin boundary condition

Xiaofei Cao^1, and
Guowei Dai^2, ,

1.
Department of Mathematics, Southeast University, Nanjing 210096, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

^* Corresponding author: Guowei Dai
^* Corresponding author: Guowei Dai

Received: August 31, 2016

Revised: January 31, 2017

Published: October 2017

The second author is supported by NNSF of China (No. 11401477)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper we develop a non-autonomous reaction-diffusion model with the Robin boundary conditions to describe insect dispersal on an isotropically varying domain. We investigate the stability of the reaction-diffusion model. The stability results of the model describe either insect survival or vanishing.

Keywords:

Mathematics Subject Classification: Primary: 35K57, 37B55; Secondary: 37C75.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	D. G. Aronson , D. Ludwig and H. F. Weinberger , Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979) , 217-258. doi: 10.1007/BF00276310.
	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons Ltd, 2003. doi: 10.1002/0470871296.
	E. J. Crampin , E. A. Gaffney and P. K. Maini , Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biol., 61 (1999) , 1093-1120. doi: 10.1006/bulm.1999.0131.
	J. Gjorgjieva and J. Jacobsen , Turing patterns on growing spheres: The exponential case, Discrete Continuous Dynam. Systems-A, Suppl., (2007) , 436-445.
	P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 1991.
	S. Kondo and R. Asai , A reaction-diffusion wave on the skin of the marine angelfish, Nature, 376 (2002) , 765-768. doi: 10.1038/376765a0.
	J. A. Langa , A. R. Bernal and A. Suárez , On the long time behavior of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method, J. Differential Equations, 249 (2010) , 414-445. doi: 10.1016/j.jde.2010.04.001.
	J. A. Langa , J. Robinson , A. Rodriguez-Bernal and A. Suarez , Permanence and asymptotically stable complete trajectories for nonautonomous lotka-volterra models with diffusion, SIAM J. Math. Anal., 40 (2009) , 2179-2216. doi: 10.1137/080721790.
	Y. Lou , Some challenging mathematical problems in evolution of disperal and population dynamics, Tutorials in Mathematical Biosciences, 1922 (2008) , 171-205. doi: 10.1007/978-3-540-74331-6_5.
	A. Madzvamuse , E. A. Gaffney and P. K. Maini , Stability analysis of non-autonomous reaction-diffusion, J. Math. Biol., 61 (2010) , 133-164. doi: 10.1007/s00285-009-0293-4.
	J. Mierczyn'ski , The principal spectrum for linear nonautonomous parabolic PDEs of second order: Basic properties, J. Differential Equations, 168 (2000) , 453-476. doi: 10.1006/jdeq.2000.3893.
	J. D. Murray, Mathematical Biology Springer-Verlag, Berlin, London, 1993. doi: 10.1007/b98869.
	C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York and London, 1992.
	A. Rodriguez-Bernal and A. Vidal-López , Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problem, Discrete Continuous Dynam. Systems, 18 (2007) , 537-567. doi: 10.3934/dcds.2007.18.537.
	A. Rodriguez-Bernal and A. Vidal-López , Extremal equilibria for reaction-diffusion equations in bounded domains and applications, J. Differential Equations, 244 (2008) , 2983-3030. doi: 10.1016/j.jde.2008.02.046.
	H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Amer. Math. Soc. , Providence, 1995.
	Q. Tang and Z. Lin , The asymptotic analysis of an insect dispersal model on a growing domain, J. Math. Anal. Appl., 378 (2011) , 649-656. doi: 10.1016/j.jmaa.2011.01.057.
	C. Varea, J. L. Aragón and R. A. Barrio, Confined Turing patterns in growing systems ,Phys. Rev. E, 56 (1997), 1250. doi: 10.1103/PhysRevE.56.1250.