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Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 14
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Articles

Traveling waves for a periodic Lotka–Volterra predator–prey system

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Pages 2619-2638 | Received 22 Jul 2017, Accepted 18 Apr 2018, Published online: 30 May 2018

References

  • Murray JD . Mathematical biology II. Spatial models and biomedical applications. 3rd ed. New York (NY): Springer-Verlag; 2003.
  • Volpert AI , Volpert VA , Volpert VA . Traveling wave solutions of parabolic systems, Translations of mathematical monographs. vol. 140. Providence: AMS; 1994.
  • Ye Q , Li Z , Wang MX , et al . Introduction to reaction-diffusion equations. 2nd ed. Beijing: Science Press; 2011.
  • Smoller J . Shock waves and reaction diffusion equations. New York (NY): Springer-Verlag; 1994.
  • Gilding BH , Kersner R . Travelling Waves in Nonlinear Diffusion-Convection Reaction. Basel: Birkhäuser Verlag; 2004.
  • Zhao XQ . Dynamincal systems in population biology. New York (NY): Springer; 2003.
  • Dunbar SR . Traveling wave solutions of diffusive Lotka-Volterra equations. J Math Biol. 1983;17:11–32.
  • Dunbar SR . Traveling wave solutions in diffusive predator-prey systems:periodic orbits and point-to-periodic heteroclic orbits. SIAM J Appl Math. 1986;46:1057–1078.
  • Gardner R , Smoller J . The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index. J Differ Equ. 1983;47:133–161.
  • Fu SC , Tsai JC . Wave propagation in predator-prey systems. Nonlinearity. 2015;28:4389–4423.
  • Fu SC . Traveling waves for a diffusive SIR model with delay. J Math Anal Appl. 2016;435:20–37.
  • Huang WZ . Traveling wave solutions for a class of predator-prey systems. J Dyn Differ Equ. 2012;24:633–644.
  • Huang WZ . A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems. J Differ Equ. 2016;260:2190–2224.
  • Lin G . Invasion traveling wave solutions of a predator-prey system. Nonlinear Anal. 2014;96:47–58.
  • Huang YL . G, Lin. Traveling wave solutions in a diffusive system with two preys and one predator, J Math Anal Appl. 2014;418:163–184.
  • Huang J , Zou X . Traveling wave solutions in delayed reaction diffusion systems with partial monotonicity. Acta Math Appl Sinica. 2006;22:243–256.
  • Lin G , Li WT , Ma M . Travelling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete Contin Dyn Syst Ser B. 2010;19:393–414.
  • Pan S . Convergence and traveling wave solutions for a predator-prey system with distributed delays. Mediterr J Math. 2017;14:15pp. Art. 103.
  • Chen YY , Guo JS , Yao CH . Traveling wave solutions for a continuous and discrete diffusive predator-prey model. J Math Anal Appl. 2017;455:212–239.
  • Wang MX . Spreading and vanishing in the diffusive prey-predator model with a free boundary. Commun. Nonlinear Sci Numer Simul. 2015;23:311–327.
  • Wang MX . On some free boundary problems of the prey-predator model. J Differ Equ. 2014;256:3365–3394.
  • Lin G . Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator. Nonlinear Anal. 2011;74:2448–2461.
  • Pan S . Asymptotic spreading in a Lotka-Volterra predator-prey system. J Math Anal Appl. 2013;407:230–236.
  • Pan S . Invasion speed of a predator-prey system. Appl Math Lett. 2017;74:46–51.
  • Alikakos N , Bates P , Chen X . Periodic traveling waves and locating oscillating patterns in multidimensional domains. Trans Am Math Soc. 1999;351:2777–2805.
  • Bao XX , Wang ZC . Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system. J Differ Equ. 2013;255:2402–2435.
  • Bates P , Chen F . Periodic traveling waves for a nonlocal integro-differential model. Electron J Differ Equ. 1999;26:1–19.
  • Liang X , Yi Y , Zhao XQ . Spreading speeds and traveling waves for periodic evolution systems. J Differ Equ. 2006;231:57–77.
  • Liang X , Zhao XQ . Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun Pure Appl Math. 2007;60:1–40.
  • Nadin G . Traveling fronts in space-time periodic media. J Math Pures Appl. 2009;92:232–262.
  • Shen W . Traveling waves in time periodic lattice differential equations. Nonlinear Anal. 2003;54:319–339.
  • Shen W . Traveling waves in time almost periodic structures governed by bistable nonlinearities. I and II, J Differ Equ. 1999;159:1–101.
  • Shen W . Dynamical systems and traveling waves in almost periodic structures. J Differ Equ. 2001;169:493–548.
  • Xin X . Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity. J Dyn Differ Equ. 1991;3:541–573.
  • Zhao G , Ruan S . Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion. J Math Pures Appl. 2011;95:627–671.
  • Zhao G , Ruan S . Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems. J Differ Equ. 2014;257:1078–1147.
  • Bo WJ , Lin G , Ruan S . Traveling wave solutions for time periodic reaction-diffusion systems. 2016. Discrete Contin Dyn Syst Ser A, in press.
  • Wang ZC , Zhang L , Zhao XQ . Time periodic traveling waves for a periodic and diffusive SIR epidemic model. J Dyn Differ Equ. 2018;30:379–403.
  • Teng Z , Yu Y , Feng B . Stability of positive periodic solutions to periodic predator-prey systems. Acta Math Appl Sinica. 1998;21:589–596.
  • Teng Z , Chen L . Global asymptotic stability of periodic Lotka-Volterra systems with delays. Nonlinear Anal. 2001;45:1081–1095.
  • Lunardi A . Analtic semigroups and optimal regularity in parabolic problems. Boston (MA): Birkhauser; 1995.
  • Teng Z . Uniform persistence of the periodic predator-prey Lotka-Volterra systems. Appl Anal. 1999;72:339–352.
  • Teng Z . Nonautonomous Lotka-Volterra systems with delays. J Differ Equ. 2002;179:538–561.
  • Fife PC , Tang M . Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances. J Differ Equ. 1981;40:168–185.
  • Berestycki H , Hamel F , Nadin G . Asymptotic spreading in heterogeneous diffusive excitable media. J Funct Anal. 2008;255:2146–2189.
  • Lin G , Ruan S . Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays. J Dyn Differ Equ. 2014;26:583–605.

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