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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

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