• Previous Article
    Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations
  • DCDS-S Home
  • This Issue
  • Next Article
    Solutions of nonlinear periodic Dirac equations with periodic potentials
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 & 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.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester 2003. doi: 10.1002/0470871296.  Google Scholar

[3]

F. ChenL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

L. ZhouS. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester 2003. doi: 10.1002/0470871296.  Google Scholar

[3]

F. ChenL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

L. ZhouS. 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.  Google Scholar

[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.  Google Scholar

[1]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

[2]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162

[3]

Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148

[4]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263

[5]

Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021007

[6]

Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis, Bo Zhang. Simulation of post-hurricane impact on invasive species with biological control management. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4059-4071. doi: 10.3934/dcds.2020038

[7]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[8]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[9]

Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328

[10]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[11]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[12]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[13]

Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

[14]

Qian Liu, Shuang Liu, King-Yeung Lam. Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3683-3714. doi: 10.3934/dcds.2020050

[15]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[16]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050

[17]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[18]

Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure & Applied Analysis, 2021, 20 (2) : 495-510. doi: 10.3934/cpaa.2020277

[19]

Claude-Michel Brauner, Luca Lorenzi. Instability of free interfaces in premixed flame propagation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 575-596. doi: 10.3934/dcdss.2020363

[20]

Aurelia Dymek. Proximality of multidimensional $ \mathscr{B} $-free systems. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021013

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (165)
  • HTML views (589)
  • Cited by (0)

[Back to Top]