Open Access
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Article Number 62
Number of page(s) 26
DOI https://doi.org/10.1051/mmnp/2020034
Published online 03 December 2020
  1. P.A. Abrams and L.R. Ginzburg, The nature of predation: prey dependent, ratio dependent or neither? Trends Ecol. Evol. 15 (2000) 337–341. [CrossRef] [PubMed] [Google Scholar]
  2. G. Bunting, Y.H. Du and K. Krakowski, Spreading speed revisited: analysis of a free boundary model. Netw. Heterog. Media 7 (2012) 583–603. [CrossRef] [Google Scholar]
  3. J.-F. Cao, W.-T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment. J. Math. Anal. Appl. 449 (2017) 1015–1035. [CrossRef] [Google Scholar]
  4. D.L. DeAngelis, R.A. Goldstein and R.V. O’Neill, A model for tropic interaction. Ecol. 56 (1975) 881–892. [CrossRef] [Google Scholar]
  5. Y.H. Du, Z.M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment. J. Funct. Anal. 265 (2013) 2089–2142. [CrossRef] [Google Scholar]
  6. Y.H. Du and Z.G. Lin, Spreading–vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42 (2010) 377–405. [CrossRef] [MathSciNet] [Google Scholar]
  7. Y.H. Du and B.D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries. J. Eur. Math. Soc. 17 (2015) 2673–2724. [CrossRef] [Google Scholar]
  8. Y.H. Du, M.X. Wang and M.L. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary. J. Math. Pures Appl. 107 (2017) 253–287. [CrossRef] [Google Scholar]
  9. R. Feng, Spatiotemporal complexity of a three-species ratio-dependent food chain model. Nonlin. Dynam. 76 (2014) 1661–1676. [CrossRef] [Google Scholar]
  10. S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control. Math. Biosci. 181 (2003) 55–83. [CrossRef] [Google Scholar]
  11. O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. Academic Press, New York, London (1968). [CrossRef] [Google Scholar]
  12. X.W. Liu and B.D. Lou, Asymptotic behavior of solutions to diffusion problems with Robin and free boundary conditions. MMNP 8 (2013) 18–32. [Google Scholar]
  13. J.B. Liao, J.H. Chen, Z.X. Ying, D.E. Hiebeler and I. Nijs, An extended patch-dynamic framework for food chains in fragmented landscapes. Sci. Rep. 6 (2016) 33100. [CrossRef] [PubMed] [Google Scholar]
  14. J.B. Liao, D. Bearup and B. Blasius, Diverse responses of species to landscape fragmentation in a simple food chain. J. Anim. Ecol. 86 (2017) 1169–1178. [CrossRef] [PubMed] [Google Scholar]
  15. H. Monobe and C.-H. Wu, On a free boundary problem for a reaction–diffusion–advection logistic model in heterogeneous environment. J. Differ. Equ. 261 (2016) 6144–6177. [CrossRef] [Google Scholar]
  16. J.D. Murray, Mathematical Biology I: An introduction, 3rd ed. Spring-Verlag, New York (2002). [CrossRef] [Google Scholar]
  17. C.V. Pao, Dynamics of food-chain models with density-dependent diffusion and ratio-dependent reaction function. J. Math. Anal. Appl. 433 (2016) 355–374. [CrossRef] [Google Scholar]
  18. R. Peng, J.P. Shi and M.X. Wang, Stationary pattern of a ratio-dependent food chain model with diffusion. SIAM J. Appl. Math. 67 (2007) 1479–1503. [CrossRef] [Google Scholar]
  19. R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession. Discrete Contin. Dyn. Syst. 33 (2013) 2007–2031. [CrossRef] [Google Scholar]
  20. M.X. Wang, Stationary patterns for a prey–predator model with prey-dependent and ratio-dependent functional responses and diffusion. Phys. D 196 (2004) 172–192. [CrossRef] [Google Scholar]
  21. M.X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient. J. Differ. Equ. 258 (2015) 1252–1266. [CrossRef] [Google Scholar]
  22. M.X. Wang. A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. J. Funct. Anal. 270 (2016) 483–508. [CrossRef] [Google Scholar]
  23. M.X. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete Contin. Dyn. Syst. Ser. B 24 (2019) 415–421. [Google Scholar]
  24. M.X. Wang and Q.Y. Zhang, Dynamics for the diffusive Leslie–Gower model with double free boundaries. Discrete Contin. Dyn. Syst. 38 (2018) 2591–2607. [CrossRef] [Google Scholar]
  25. M.X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries. Nonlin. Anal. 159 (2017) 458–467. [CrossRef] [Google Scholar]
  26. M.X. Wang and Y. Zhang, Dynamics for a diffusive prey–predator model with different free boundaries. J. Differ. Equ. 264 (2018) 3527–3558. [CrossRef] [Google Scholar]
  27. M.X. Wang and J.F. Zhao, Free boundary problems for a Lotka–Volterra competition system. J. Dynam. Differ. Equ. 26 (2014) 655–672. [CrossRef] [Google Scholar]
  28. M.X. Wang and J.F. Zhao, A free boundary problem for the predator-prey model with double free boundaries. J. Dynam. Differ. Equ. 29 (2017) 957–979. [CrossRef] [Google Scholar]
  29. K. Wonlyul and A. Inkyung, Dynamics of a simple food chain model with a ratio-dependent functional response. Nonlin. Anal. Real World Appl. 12 (2011) 1670–1680. [CrossRef] [Google Scholar]
  30. C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries. J. Differ. Equ. 259 (2015) 873–897. [CrossRef] [Google Scholar]
  31. D.W. Zhang and B.X. Dai, A free boundary problem for the diffusive intraguild predation model with intraspecific competition. J. Math. Anal. Appl. 474 (2019) 381–412. [CrossRef] [Google Scholar]
  32. D.W. Zhang and B.X. Dai, The diffusive intraguild predation model with intraspecific competition and double free boundaries. Appl. Anal. (2020) DOI: 10.1080/00036811.2020.1716971. [Google Scholar]
  33. Y. Zhang and M.X. Wang, A free boundary problem of the ratio-dependent prey–predator model. Appl. Anal. 94 (2015) 2147–2167. [CrossRef] [Google Scholar]
  34. J.F. Zhao and M.X. Wang, A free boundary problem of a predator–prey model with higher dimension and heterogeneous environment. Nonlin. Anal. Real World Appl. 16 (2014) 250–263. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.