American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Coexistence and asymptotic stability in stage-structured predator-prey models
  • MBE Home
  • This Issue
  • Next Article
    Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
2014, 11(4): 807-821. doi: 10.3934/mbe.2014.11.807

On a diffusive predator-prey model with nonlinear harvesting

Peng Feng 1,

1. 

Department of Mathematics, Florida Gulf Coast University, 11501 FGCU Blvd. S., Fort Myers, FL 33965, United States

Received  August 2013 Revised  November 2013 Published  March 2014

In this paper, we study the dynamics of a diffusive Leslie-Gower model with a nonlinear harvesting term on the prey. We analyze the existence of positive equilibria and their dynamical behaviors. In particular, we consider the model with a weak harvesting term and find the conditions for the local and global asymptotic stability of the interior equilibrium. The global stability is established by considering a proper Lyapunov function. In contrast, the model with strong harvesting term has two interior equilibria and bi-stability may occur for this system. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.
Keywords: pattern formation, global stability., turing instability, nonlinear harvesting, Leslie-Gower, Predator-prey.
Mathematics Subject Classification: Primary: 35K40, 35K57, 35B36; Secondary: 35Q9.
Citation: Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807
References:
[1]

M. A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population, Chaos Sol. and Fractals, 14 (2002), 1275-1293. doi: 10.1016/S0960-0779(02)00079-6.

[2]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6.

[3]

J. B. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168. doi: 10.1007/s002850050095.

[4]

Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Eqns., 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007.

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New-York, 1983.

[6]

X. N. Guan, W. M. Wang and Y. L. Cai, Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Analysis: Real World Applications, 12 (2011), 2385-2395. doi: 10.1016/j.nonrwa.2011.02.011.

[7]

R. P. Gupta and P Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295. doi: 10.1016/j.jmaa.2012.08.057.

[8]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2011), 697-699. doi: 10.1016/S0893-9659(01)80029-X.

[9]

T. K. Kar, Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, J. Comput. Appl Math., 185 (2006), 19-33. doi: 10.1016/j.cam.2005.01.035.

[10]

B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with nonconstant harvesting, Disc. Cont. Dyn. Syst. S, 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303.

[11]

P. H. Leslie, Some further notes on the use of matrices on population mathematics, Biometrika, 35 (1948), 213-245.

[12]

R. M. May, Stability and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, 1974.

[13]

A. F. Nindjin, M. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118. doi: 10.1016/j.nonrwa.2005.10.003.

[14]

D. J. Wollkind, J. B. Collings and J. A. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, Bull. Math. Biol., 50 (1988), 379-409. doi: 10.1007/BF02459707.

[15]

D. Xiao, W. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal and Appl., 324 (2006), 14-29. doi: 10.1016/j.jmaa.2005.11.048.

[16]

Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 1990.

[17]

N. Zhang, F. D. Chen, Q. Q. Su and T. Wu, Dynamic behaviors of a harvesting Leslie-Gower predator-prey model, Discrete Dyn. Nat. Soc., (2011), Art. ID 473949, 14 pp. doi: 10.1155/2011/473949.

[18]

Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, J. Math. Anal. Appl., 384 (2011), 400-408. doi: 10.1016/j.jmaa.2011.05.081.

show all references

References:
[1]

M. A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population, Chaos Sol. and Fractals, 14 (2002), 1275-1293. doi: 10.1016/S0960-0779(02)00079-6.

[2]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6.

[3]

J. B. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168. doi: 10.1007/s002850050095.

[4]

Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Eqns., 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007.

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New-York, 1983.

[6]

X. N. Guan, W. M. Wang and Y. L. Cai, Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Analysis: Real World Applications, 12 (2011), 2385-2395. doi: 10.1016/j.nonrwa.2011.02.011.

[7]

R. P. Gupta and P Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295. doi: 10.1016/j.jmaa.2012.08.057.

[8]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2011), 697-699. doi: 10.1016/S0893-9659(01)80029-X.

[9]

T. K. Kar, Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, J. Comput. Appl Math., 185 (2006), 19-33. doi: 10.1016/j.cam.2005.01.035.

[10]

B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with nonconstant harvesting, Disc. Cont. Dyn. Syst. S, 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303.

[11]

P. H. Leslie, Some further notes on the use of matrices on population mathematics, Biometrika, 35 (1948), 213-245.

[12]

R. M. May, Stability and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, 1974.

[13]

A. F. Nindjin, M. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118. doi: 10.1016/j.nonrwa.2005.10.003.

[14]

D. J. Wollkind, J. B. Collings and J. A. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, Bull. Math. Biol., 50 (1988), 379-409. doi: 10.1007/BF02459707.

[15]

D. Xiao, W. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal and Appl., 324 (2006), 14-29. doi: 10.1016/j.jmaa.2005.11.048.

[16]

Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 1990.

[17]

N. Zhang, F. D. Chen, Q. Q. Su and T. Wu, Dynamic behaviors of a harvesting Leslie-Gower predator-prey model, Discrete Dyn. Nat. Soc., (2011), Art. ID 473949, 14 pp. doi: 10.1155/2011/473949.

[18]

Y. Zhu and K. Wang, Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, J. Math. Anal. Appl., 384 (2011), 400-408. doi: 10.1016/j.jmaa.2011.05.081.

[1]

Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065
[2]

Yong Yao, Lingling Liu. Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021252
[3]

C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289
[4]

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
[5]

Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236
[6]

Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203
[7]

Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations and Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115
[8]

Hongwei Yin, Xiaoyong Xiao, Xiaoqing Wen. Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2121-2151. doi: 10.3934/dcdsb.2018228
[9]

Jun Zhou. Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1127-1145. doi: 10.3934/cpaa.2015.14.1127
[10]

Rong Zou, Shangjiang Guo. Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4189-4210. doi: 10.3934/dcdsb.2020093
[11]

Baifeng Zhang, Guohong Zhang, Xiaoli Wang. Threshold dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021260
[12]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2189-2219. doi: 10.3934/dcdsb.2021129
[13]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
[14]

Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172
[15]

Safia Slimani, Paul Raynaud de Fitte, Islam Boussaada. Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5003-5039. doi: 10.3934/dcdsb.2019042
[16]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[17]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162
[18]

Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621
[19]

Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268
[20]

Xiaoying Wang, Xingfu Zou. Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Mathematical Biosciences & Engineering, 2018, 15 (3) : 775-805. doi: 10.3934/mbe.2018035

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy