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November  2019, 12(7): 2063-2084. doi: 10.3934/dcdss.2019133

## A Leslie-Gower predator-prey model with a free boundary

 a. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China b. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada

* Corresponding author: Zhiming Guo, guozm@gzhu.edu.cn

Dedicated to Professor Norman Dancer on the occasion of his 70th birthday

Received  December 2017 Revised  June 2018 Published  December 2018

Fund Project: The work of ME and LW was supported by NSERC Discovery Grants from the Natural Sciences and Engineering Research Council of Canada (NSERC). YL and ZG acknowledge support from the National Natural Science Foundation of China (No.11771104), Program for Changjiang Scholars and Innovative Research Team in University (IRT-16R16).YL was supported by the Innovation Research for the Postgraduates of Guangzhou University under Grant No.2017GDJC-D05.

In this paper, we consider a Leslie-Gower predator-prey model in one-dimensional environment. We study the asymptotic behavior of two species evolving in a domain with a free boundary. Sufficient conditions for spreading success and spreading failure are obtained. We also derive sharp criteria for spreading and vanishing of the two species. Finally, when spreading is successful, we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem that follows from the original model.

Citation: Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133
##### References:
 [1] M. A. Aziz-Alaoui and M. Daher-Okiye, Boundedness and Global Stability or a Predator-prey Model with Modified Leslie-Gower and Holling-Type Ⅱ Schemes, Applied Mathematics Letters, 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6. [2] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester 2003. doi: 10.1002/0470871296. [3] F. Chen, L. Chen and X. Xie, On a Leslie-Gower predator-preymodel incorporating a prey refuge, Nonlinear Analysis: Real World Applications, 10 (2009), 2905-2908.  doi: 10.1016/j.nonrwa.2008.09.009. [4] X. F. Chen and 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. [5] 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.  doi: 10.1137/090771089. [6] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitior, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105. [7] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J.Dyn. Diff. Equat., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0. [8] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201. [9] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X. [10] Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004. [11] W. J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differential Equations, 261 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022. [12] J. Wang, The selection for dispersal: A diffusive competition model with a free boundary, Z. Angew. Math. Phys., 66 (2015), 2143-2160.  doi: 10.1007/s00033-015-0519-9. [13] M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013. [14] 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. [15] M. X. 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.  doi: 10.1016/j.nonrwa.2015.01.004. [16] M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5. [17] R. Z. Yang and J. J. Wei, The effect of delay on a diffusive predator-prey system with modified leslie-gower functional response, Bull. Malays. Math. Sci. Soc., 40 (2017), 51-73.  doi: 10.1007/s40840-015-0261-7. [18] J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003. [19] Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Applicable Analysis, 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806. [20] J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3. [21] L. Zhou, S. Zhang and Z. H. Liu, A free boundary problem of a predator-prey model with advection in heterogeneous environment, Appl. Math. Comput., 289 (2016), 22-36.  doi: 10.1016/j.amc.2016.05.008. [22] P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J.Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.

show all references

##### References:
 [1] M. A. Aziz-Alaoui and M. Daher-Okiye, Boundedness and Global Stability or a Predator-prey Model with Modified Leslie-Gower and Holling-Type Ⅱ Schemes, Applied Mathematics Letters, 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6. [2] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester 2003. doi: 10.1002/0470871296. [3] F. Chen, L. Chen and X. Xie, On a Leslie-Gower predator-preymodel incorporating a prey refuge, Nonlinear Analysis: Real World Applications, 10 (2009), 2905-2908.  doi: 10.1016/j.nonrwa.2008.09.009. [4] X. F. Chen and 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. [5] 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.  doi: 10.1137/090771089. [6] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitior, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105. [7] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J.Dyn. Diff. Equat., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0. [8] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201. [9] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X. [10] Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004. [11] W. J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differential Equations, 261 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022. [12] J. Wang, The selection for dispersal: A diffusive competition model with a free boundary, Z. Angew. Math. Phys., 66 (2015), 2143-2160.  doi: 10.1007/s00033-015-0519-9. [13] M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013. [14] 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. [15] M. X. 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.  doi: 10.1016/j.nonrwa.2015.01.004. [16] M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5. [17] R. Z. Yang and J. J. Wei, The effect of delay on a diffusive predator-prey system with modified leslie-gower functional response, Bull. Malays. Math. Sci. Soc., 40 (2017), 51-73.  doi: 10.1007/s40840-015-0261-7. [18] J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003. [19] Y. Zhang and M. X. Wang, A free boundary problem of the ratio-dependent prey-predator model, Applicable Analysis, 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806. [20] J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3. [21] L. Zhou, S. Zhang and Z. H. Liu, A free boundary problem of a predator-prey model with advection in heterogeneous environment, Appl. Math. Comput., 289 (2016), 22-36.  doi: 10.1016/j.amc.2016.05.008. [22] P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J.Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.
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