• Previous Article
    Preface for special session entitled "Recent Advances of Differential Equations with Applications in Life Sciences"
  • DCDS-S Home
  • This Issue
  • Next Article
    Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors
2017, 10(5): 935-942. doi: 10.3934/dcdss.2017048

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

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

Received  September 2016 Revised  February 2017 Published  June 2017

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

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.

Citation: Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048
References:
[1]

D. G. AronsonD. Ludwig and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258. doi: 10.1007/BF00276310.

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons Ltd, 2003. doi: 10.1002/0470871296.

[3]

E. J. CrampinE. 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.

[4]

J. Gjorgjieva and J. Jacobsen, Turing patterns on growing spheres: The exponential case, Discrete Continuous Dynam. Systems-A, Suppl., (2007), 436-445.

[5]

P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 1991.

[6]

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.

[7]

J. A. LangaA. 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.

[8]

J. A. LangaJ. RobinsonA. 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.

[9]

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.

[10]

A. MadzvamuseE. 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.

[11]

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.

[12]

J. D. Murray, Mathematical Biology Springer-Verlag, Berlin, London, 1993. doi: 10.1007/b98869.

[13]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York and London, 1992.

[14]

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.

[15]

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.

[16]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Amer. Math. Soc. , Providence, 1995.

[17]

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.

[18]

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.

show all references

References:
[1]

D. G. AronsonD. Ludwig and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258. doi: 10.1007/BF00276310.

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons Ltd, 2003. doi: 10.1002/0470871296.

[3]

E. J. CrampinE. 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.

[4]

J. Gjorgjieva and J. Jacobsen, Turing patterns on growing spheres: The exponential case, Discrete Continuous Dynam. Systems-A, Suppl., (2007), 436-445.

[5]

P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 1991.

[6]

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.

[7]

J. A. LangaA. 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.

[8]

J. A. LangaJ. RobinsonA. 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.

[9]

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.

[10]

A. MadzvamuseE. 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.

[11]

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.

[12]

J. D. Murray, Mathematical Biology Springer-Verlag, Berlin, London, 1993. doi: 10.1007/b98869.

[13]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York and London, 1992.

[14]

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.

[15]

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.

[16]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Amer. Math. Soc. , Providence, 1995.

[17]

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.

[18]

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.

[1]

Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221

[2]

Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609

[3]

Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic & Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865

[4]

Xin Li, Xingfu Zou. On a reaction-diffusion model for sterile insect release method with release on the boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2509-2522. doi: 10.3934/dcdsb.2012.17.2509

[5]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393

[6]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with \begin{document}$ p(x) $\end{document}-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[7]

Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006

[8]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343

[9]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693

[10]

Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559

[11]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[12]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837

[13]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253

[14]

C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603

[15]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153

[16]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

[17]

Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255

[18]

Arti Mishra, Benjamin Ambrosio, Sunita Gakkhar, M. A. Aziz-Alaoui. A network model for control of dengue epidemic using sterile insect technique. Mathematical Biosciences & Engineering, 2018, 15 (2) : 441-460. doi: 10.3934/mbe.2018020

[19]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5003-5036. doi: 10.3934/dcds.2015.35.5003

[20]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (2)
  • HTML views (4)
  • Cited by (0)

Other articles
by authors

[Back to Top]