A free boundary problem of the ratio-dependent prey-predator model

References

• Allendorf FW, Lunduist LL. Introduction: population, biology, evolution and control of invasive species. Con-serv. Biol. 2013;17:24–30.
   Web of Science ®Google Scholar
• Du YH, Lin ZG. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 2010;42:377–405.
   Web of Science ®Google Scholar
• Kolar CS, Lodge DM. Progress invasion biology: predicting invaders. Trends Ecol. Evol. 2001;16:199–204.
   PubMed Web of Science ®Google Scholar
• Lockwood JL, Hoopes MF, Marchetti MP. Invasion ecology. Oxford: Blackwell; 2007.
   Google Scholar
• Du YH, Guo ZM. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, II. J. Differ. Equ. 2011;250:4336–4366.
   Web of Science ®Google Scholar
• Arditi R, Ginzburg LR. Coupling in predator-prey dynamics: ration dependence. J. Theoret. Biol. 1989;139:311–326.
   Web of Science ®Google Scholar
• Kuang Y, Beretta E. Global qualitative analysis of a ratio-dependent predator-prey systems. J. Math. Biosci. 1998;88:389–406.
   Web of Science ®Google Scholar
• Jost C, Arino O, Arditi R. About deterministic extinction in ratio-dependent prey-prdator model. Bull. Math. Biol. 1999;61:19–32.
   Web of Science ®Google Scholar
• Hus SB, Hwang TW, Kuang Y. Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 2001;43:377–396.
   PubMed Web of Science ®Google Scholar
• Pang PYH, Wang MX. Qualitative analysis of a ratio-dependent predator-prey system with diffusion. Proc. Roy. Soc. Edinburgh Sect. A. 2003;133:919–942.
   Google Scholar
• Rubinstein LI. The Stefan problem. Providence (RI): American Mathematical Society; 1971.
   Google Scholar
• Chen XF, Friedman A. A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal. 2000;32:778–800.
   Web of Science ®Google Scholar
• Du YH, Guo ZM. The Stefan problem for the Fisher-KPP equation. J. Differ. Equ. 2012;253:996–1035.
   Web of Science ®Google Scholar
• Du YH, Guo ZM, Peng R. A diffusive logistic model with a free boundary in time-periodic environment. J. Funct. Anal. 2013;265:2089–2142.
   Web of Science ®Google Scholar
• Peng R, Zhao XQ. The diffusive logistic model with a free boundary and seasonal succession. Discrete Cont. Dyn. Syst. A. 2013;33:2007–2031.
   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;24:873–895.
   Web of Science ®Google Scholar
• Wang MX, Zhao JF. A free boundary problem for a predator-prey model with double free boundaries. arXiv:1312.7751, [math.DS].
   Google Scholar
• Du YH, Liang X. Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model. Ann. Inst. Henri Poincare (C) Non Linear Anal. In press. Available from: http://dx.doi.org/10.1016/j.anihpc.2013.11.004.
   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;19:3105–3132.
   Web of Science ®Google Scholar
• Du YH, Lou BD. Spreading and vanishing in nonlinear diffusion problems with free boundaries. arXiv:1301.5373v1, [math.AP].
   Google Scholar
• Du YH, Matsuzawa H, Zhou ML. Sharp estimate of the spreading speed determined by nonlinear free boundary problems. SIAM J. Math. Anal. 2014;46:375–396.
   Web of Science ®Google Scholar
• Zhao JF, Wang MX. A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment. Nonlinear Anal.: Real World Appl. 2014;16:250–263.
   Web of Science ®Google Scholar
• Ladyzenskaja OA, Solonnikov VA, Ural’ceva NN. Linear and quasilinear equations of parabolic type. Providence (RI): American Mathematical Society; 1968.
   Google Scholar
• Wang MX. On some free boundary problems of the prey-predator model. J. Differ. Equ. 2014;256:3365–3394.
   Web of Science ®Google Scholar
• Wang MX, Zhao JF. Free boundary problems for a Lotka-Volterra competition system. J. Dyn. Differ. Equ. 2014. doi:10.1007/s10884-014-9363-4.
   Web of Science ®Google Scholar
• Ricci R, Tarzia DA. Asymptotic behavior of the solutions the dead-core problem. Nonlinear Anal. 1989;13:405–411.
   Web of Science ®Google Scholar
• Du YH, Ma L. Logistic type equation on ℝN by a squeezing method involving boundary blow-up solutions. J. Lond. Math. Soc. 2001;64:107-124.
   Web of Science ®Google Scholar
• Wang MX. A free boundary problem of prey-predator model with variable intrinsic growth rate for predator. arXiv:1312.7783v3, [math.DS].
   Google Scholar