In this paper we study a ratio-dependent predator-prey model with a free boundary caused by predator-prey interaction over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy; namely, as $ t $ goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and eventually die out. The criteria governing spreading and vanishing are obtained. Finally, when spreading occurs we provide some estimates to the asymptotic spreading speed of the moving boundary $ h(t) $.
Citation: Lingyu Liu, Alexander Wires. A free boundary problem with a Stefan condition for a ratio-dependent predator-prey model[J]. AIMS Mathematics, 2021, 6(11): 12279-12297. doi: 10.3934/math.2021711
In this paper we study a ratio-dependent predator-prey model with a free boundary caused by predator-prey interaction over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy; namely, as $ t $ goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and eventually die out. The criteria governing spreading and vanishing are obtained. Finally, when spreading occurs we provide some estimates to the asymptotic spreading speed of the moving boundary $ h(t) $.
| [1] |
Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105
|
| [2] | P. A. Abrams, L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither? Trends Ecol. Evol., 15 (2000), 337-341. |
| [3] |
N. G. Hairston, F. E. Smith, L. B. Slobodkin, Community structure, population control, and competition, Am. Nat., 94 (1960), 421-425. doi: 10.1086/282146
|
| [4] |
F. L. Robert, Evaluation of natural enemies for biological control: A behavioral approach, Trends Ecol. Evol., 5(1990), 196-199. doi: 10.1016/0169-5347(90)90210-5
|
| [5] |
R. Arditi, L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326. doi: 10.1016/S0022-5193(89)80211-5
|
| [6] | L. I. Rubensteĭn, The Stefan problem, Translations of Mathematical Monographs, Vol. 27, Rhode Island: American Mathematical Society Providence, 1971. |
| [7] |
X. F. Chen, A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693
|
| [8] |
Y. H. Du, L. Ma, Logistic type equations on ${R}^N$ by a squeezing method involving boundary blow-up solutions, J. Lond. Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289
|
| [9] |
Y. H. Du, Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089
|
| [10] |
Y. H. Du, Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differ. Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011
|
| [11] |
Y. H. Du, Z. M. Guo, R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016
|
| [12] | Y. H. Du, B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, Mathematics, 17 (2015), 2673-2724. |
| [13] |
G. Bunting, Y. H. Du, K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583
|
| [14] |
Y. H. Du, Z. M. Guo, The Stefan problem for the Fisher-KPP equation, J. Differ. Equations, 253 (2012), 996-1035. doi: 10.1016/j.jde.2012.04.014
|
| [15] |
M. X. Wang, J. F. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dyn. Differ. Equations, 29 (2017), 957-979. doi: 10.1007/s10884-015-9503-5
|
| [16] |
M. X. Wang, J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Equations, 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4
|
| [17] |
M. X. Wang, Y. G. Zhao, A semilinear parabolic system with a free boundary, Z. Angew. Math. Phys., 66 (2015), 3309-3332. doi: 10.1007/s00033-015-0582-2
|
| [18] |
M. X. Wang, Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal.-Real, 24 (2015), 73-82. doi: 10.1016/j.nonrwa.2015.01.004
|
| [19] |
M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327. doi: 10.1016/j.cnsns.2014.11.016
|
| [20] |
M. X. Wang, On some free boundary problems of the prey-predator model, J. Differ. Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013
|
| [21] |
M. X. Wang, Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differ. Equations, 264 (2018), 3527-3558. doi: 10.1016/j.jde.2017.11.027
|
| [22] |
J. F. Zhao, M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal.-Real, 16 (2014), 250-263. doi: 10.1016/j.nonrwa.2013.10.003
|
| [23] | N. N. Pelen, M. E. Koksal, Necessary and sufficient conditions for the existence of periodic solutions in a predator-prey model, Electron. J. Differ. Equations, 3 (2017), 1-12. |
| [24] | L. Y. Liu, A free boundary problem for a ratio-dependent predator-prey system, 2020. Available from: https://arXiv.org/abs/2006.13770. |
| [25] |
Y. Zhang, M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 1-21. doi: 10.1080/00036811.2014.992104
|
Figures(2)
Lingyu Liu, Alexander Wires. A free boundary problem with a Stefan condition for a ratio-dependent predator-prey model[J]. AIMS Mathematics, 2021, 6(11): 12279-12297. doi: 10.3934/math.2021711
DownLoad: