Dynamics for a nonlocal competition system with a free boundary

References

• Britton NF . Aggregation and the competitive exclusion principle. J Theor Biol. 1989;136(1):57–66.
   PubMed Web of Science ®Google Scholar
• Britton NF . Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model. SIAM J Appl Math. 1990;50(6):1663–1688.
   Web of Science ®Google Scholar
• Billingham J . Dynamics of a strongly nonlocal reaction-diffusion population model. Nonlinearity. 2003;17(1):313–346.
   Web of Science ®Google Scholar
• Gourley SA , Chaplain MAJ , Davidson FA . Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation. Dyn Syst. 2001;16(2):173–192.
   Web of Science ®Google Scholar
• Gourley SA , Britton NF . A predator-prey reaction-diffusion system with nonlocal effects. J Math Biol. 1996;34(3):297–333.
   Web of Science ®Google Scholar
• Alfaro M , Coville J . Rapid travling waves in the nonlocal Fisher equation connect two unstable states. Appl Math Lett. 2012;25(12):2095–2099.
   Web of Science ®Google Scholar
• Berestycki H , Nadin G , Perthame B , et al . The non-local Fisher-KPP equation: traveling waves and steady states. Nonlinearity. 2009;22(12):2813–2844.
   Web of Science ®Google Scholar
• Choudhury SR . Turing instability in competition models with delay I: linear Theroy. SIAM J Appl Math. 1994;54(5):1425–1450.
   Web of Science ®Google Scholar
• Deng K . On a nonlocal reaction-diffusion population model. Discrete Contin Dyn Syst Ser B. 2008;9(1):65–73.
   Web of Science ®Google Scholar
• Fang J , Zhao XQ . Monotone wavefronts of the nonlocal Fisher-KPP equation. Nonlinearity. 2011;24(11):3043–3054.
   Web of Science ®Google Scholar
• Gourley SA . Traveling front solutions of a nonlocal Fisher equation. J Math Biol. 2000;41(3):272–284.
   PubMed Web of Science ®Google Scholar
• Hamel F , Ryzhik L . On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds. Nonlinearity. 2014;27(11):2735–2753.
   Web of Science ®Google Scholar
• Wang ZC , Li WT . Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays. Z Angew Math Phys. 2007;58(4):571–591.
   Web of Science ®Google Scholar
• Bunting G , Du YH , Krakowski K . Spreading speed revisited: analysis of a free boundary model (special issue dedicated to H. Matano). Netw Heterog Media. 2012;7:583–603.
   Web of Science ®Google Scholar
• Du YH , Lin ZG . The diffusive competition model with a free boundary: invasion of a superior or inferior competitor. Discrete Contin Dyn Syst Ser B. 2014;19(10):3105–3132.
   Web of Science ®Google Scholar
• Du YH , Wang MX , Zhou ML . Semi-wave and spreading speed for the diffusive competition model with a free boundary. J Math Pures Appl. 2017;107(3):253–287.
   Web of Science ®Google Scholar
• Wang J , Zhang L . Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment. J Math Anal Appl. 2015;423(1):377–398.
   Web of Science ®Google Scholar
• Zhou L , Zhang S , Liu ZH . An evolutional free-boundary problem of a reaction-diffusion-advection system. Proc R Soc Edinburgh Sect A. 2017;147(3):615–648.
   Web of Science ®Google Scholar
• Chen QL , Li FQ , Wang F . The diffusive competition problem with a free boundary in heterogeneous time-periodic environment. J Math Anal Appl. 2016;433(2):1594–1631.
   Web of Science ®Google Scholar
• Du YH , Liang X . Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model. Ann Inst H Poincare Anal Non Lineaire. 2015;32(2):279–305.
   Web of Science ®Google Scholar
• Wang MX , Zhang Y . The time-periodic diffusive competition models with a free boundary and sign-changing growth rates. Z Angew Math Phys. 2016;67(5):132.
   Web of Science ®Google Scholar
• Guo JS , Wu CH . On a free boundary problem for a two-species weak competition system. J Dyn Differ Equ. 2012;24(4):873–895.
   Web of Science ®Google Scholar
• Wang MX , Zhao JF . Free boundary problems for a Lotka-Volterra competition system. J Dyn Differ Equ. 2014;26(3):655–672.
   Web of Science ®Google Scholar
• Zhou L , Zhang S , Liu ZH . Pattern formations for a strong interacting free boundary problem. Acta Appl Math. 2017;148:121–142.
   Web of Science ®Google Scholar
• Guo JS , Wu CH . Dynamics for a two-species competition-diffusion model with two free boundaries. Nonlinearity. 2015;28(1):1–27.
   Web of Science ®Google Scholar
• Wang MX , Zhang Y . Note on a two-species competition-diffusion model with two free boundaries. Nonlinear Anal. 2017;159:458–467.
   Web of Science ®Google Scholar
• Wang MX . On some free boundary problems of the prey-predator model. J Differ Equ. 2014;256(10):3365–3394.
   Web of Science ®Google Scholar
• Wang MX . Spreading and vanishing in the diffusive prey-predator model with a free boundary. Commun Nonlinear Sci Numer Simulat. 2015;23(1–3):311–327.
   Web of Science ®Google Scholar
• Wang MX , Zhao JF . A free boundary problem for the predator-prey model with double free boundaries. J Dyn Differ Equ. 2017;29:957–979.
   Web of Science ®Google Scholar
• Zhao JF , Wang MX . A free boundary problem for a predator-prey model with higher dimension and heterogeneous environment. Nonlinear Anal Real World Appl. 2014;16(7):250–263.
   Google Scholar
• Zhou L , Zhang S , Liu ZH . A free boundary problem of a predator-prey model with advection in heterogeneous environment. Appl Math Comput. 2016;289(c):22–36.
   Google Scholar
• Chang CH , Chen CC . Traveling wave solutions of a free boundary problem for a two-species competitive model. Commun Pures Appl Anal. 2013;12(2):1065–1074.
   Web of Science ®Google Scholar
• Wu CH . Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete Contin Dyn Syst Ser B. 2013;18(9):2441–2455.
   Web of Science ®Google Scholar
• Cantrell RS , Cosner C . Spatial ecology via reaction-diffusion equations, wiley series in mathematical and computational biology. Wiley; 2003.
   Google Scholar
• Kan-On Y . Fisher wave fronts for the Lotka-Volterra competition model with diffussion. Nonliear Anal. 1997;28(1):145–164.
   Web of Science ®Google Scholar
• Kishimoto K , Weinberger HF . The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains. J Differ Equ. 1985;58(1):15–21.
   Web of Science ®Google Scholar
• Okubo A , Maini PK , Williamson MH , Murray JD . On the spatial spread of the grey squirrel in Britain. Proc R Soc London Ser B. 1989;238:113–125.
   PubMed Web of Science ®Google Scholar
• Pao CV . Nonlinear parabolic and elliptic equations. New York (NY): Plenum Press; 1992.
   Google Scholar
• Du YH , Lin ZG . Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J Math Anal. 2010;42(3):337–405.
   Google Scholar
• Du YH , Lou BD . Spreading and vanishing in nonlinear diffusion problems with free boundaries. J Eur Math Soc. 2015;17(10):2673–2724.
   Web of Science ®Google Scholar
• Du YH , Guo ZM . Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary II. J Differ Equ. 2011;250(12):4336–4366.
   Web of Science ®Google Scholar
• Du YH , Matsuzawa H , Zhou ML . Sharp estimate of the spreading speed determined by nonlinear free boundary problems. SIMA J Math Anal. 2014;46(1):375–396.
   Web of Science ®Google Scholar
• Ding WW , Du YH , Liang X . Spreading in space-time preiodic media governed by a monostable equation with free boundaries. Part I: Continuous initial functions, J Differ Equ. 2017;262(10):4988–5021.
   Web of Science ®Google Scholar
• Ding WW , Du YH , Liang X . Spreading in space-time preiodic media governed by a monostable equation with free boundaries, Part II: spreading speed. 2016. submitted, arXiv:1611.01908.
   Google Scholar
• Kaneko Y , Yamada Y . A free boundary problem for a reaction diffusion equation appearing in ecology. Adv Math Sci Appl. 2011;21(2):467–492.
   Google Scholar
• Peng R , Zhao XQ . The diffusive logistic model with a free boundary and seasonal succession. Discrete Contin Dyn Syst Ser A. 2013;33(5):2007–2031.
   Web of Science ®Google Scholar
• Wang MX . The diffusive logistic equation with a free boundary and sign-changing coefficient. J Differ Equ. 2015;258(4):1252–1266.
   Web of Science ®Google Scholar
• Wang MX . A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. J Funct Anal. 2016;270(2):483–508.
   Web of Science ®Google Scholar
• Zhou L , Zhang S , Liu ZH . A reaction-diffusive-advection equation with a free boundary and sign-changing coefficient. Acta Appl Math. 2016;143:189–216.
   Web of Science ®Google Scholar
• Ladyzenskaja OA , Solonnikov VA , Uralceva NN . Linear and quasi-linear equations of parabolic type. New York (NY): Academic Press; 1968.
   Google Scholar
• Wang MX , Zhang Y . Dynamics for a diffusive prey-predator model with different free boundaries. 2015. arXiv:1511.06479.
   Google Scholar