This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space R^n. The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed.
| Published in | American Journal of Applied Mathematics (Volume 8, Issue 5) |
| DOI | 10.11648/j.ajam.20200805.17 |
| Page(s) | 284-292 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2020. Published by Science Publishing Group |
Free Boundary, Predator-Prey Model, Spreading-Vanishing Dichotomy, Spreading Speed, Heterogeneous Environment
| [1] | G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603. |
| [2] | R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. |
| [3] | R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley and Sons, 2003. |
| [4] | X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. |
| [5] | H. Cheng and R. Yuan, A free boundary problem of some Lesile-Gower predator-prey model with higher dimensional environment, submitted, (2019). |
| [6] | H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 18 (2015), 565-579. |
| [7] | H. Cheng and R. Yuan, Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 5422-5454. |
| [8] | Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366. |
| [9] | Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035. |
| [10] | Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. |
| [11] | Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364. |
| [12] | Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305. |
| [13] | Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. |
| [14] | Y. Du and Z. Lin, The diffusive competition model with a free boundary: invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132. |
| [15] | Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724. |
| [16] | Y. Du and L. Ma, Logistic type equations on R^n by a sequeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. |
| [17] | Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396. |
| [18] | A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15. |
| [19] | J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895. |
| [20] | J.-S. Guo and C.-H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27. |
| [21] | D. Hilhorst, M. Iida, M. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180. |
| [22] | D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285. |
| [23] | H. Huang and M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050. |
| [24] | Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492. |
| [25] | O. g. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural’tseva, Linear and quasi-linear equations of parabolic type, American Mathematical Soc., 1968. |
| [26] | Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. |
| [27] | Y. Liu, Z. Guo, M. E. Smaily and L. Wang, A Leslie-Gower predator-prey model with a free boundary, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2063-2084. |
| [28] | M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. |
| [29] | M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498. |
| [30] | M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280. |
| [31] | 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. |
| [32] | M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. |
| [33] | M. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327. |
| [34] | M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508. |
| [35] | M. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82. |
| [36] | M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672. |
| [37] | Y. Zhang and M. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167. |
| [38] | J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal., 16 (2014), 250-263. |
| [39] | P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. |
APA Style
Shiwen Niu, Hongmei Cheng. (2020). A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment. American Journal of Applied Mathematics, 8(5), 284-292. https://doi.org/10.11648/j.ajam.20200805.17
ACS Style
Shiwen Niu; Hongmei Cheng. A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment. Am. J. Appl. Math. 2020, 8(5), 284-292. doi: 10.11648/j.ajam.20200805.17
AMA Style
Shiwen Niu, Hongmei Cheng. A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment. Am J Appl Math. 2020;8(5):284-292. doi: 10.11648/j.ajam.20200805.17
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajam.20200805.17,
author = {Shiwen Niu and Hongmei Cheng},
title = {A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment},
journal = {American Journal of Applied Mathematics},
volume = {8},
number = {5},
pages = {284-292},
doi = {10.11648/j.ajam.20200805.17},
url = {https://doi.org/10.11648/j.ajam.20200805.17},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200805.17},
abstract = {This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space R^n. The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed.},
year = {2020}
}
TY - JOUR T1 - A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment AU - Shiwen Niu AU - Hongmei Cheng Y1 - 2020/10/26 PY - 2020 N1 - https://doi.org/10.11648/j.ajam.20200805.17 DO - 10.11648/j.ajam.20200805.17 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 284 EP - 292 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20200805.17 AB - This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space R^n. The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed. VL - 8 IS - 5 ER -
Email: