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August  2018, 23(6): 2121-2151. doi: 10.3934/dcdsb.2018228

Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response

1. 

School of Sciences, Nanchang University Nanchang 330031, China

2. 

Numerical Simulation and High-Performance Computing Laboratory, Nanchang University, Nanchang 330031, China

* Corresponding author: Xiaoqing Wen

Received  June 2016 Revised  March 2017 Published  July 2018

Fund Project: Hongwei Yin is supported by NSF of China (61563033 and 11461044) and NSF of Jiangxi Province (20161BAB201010). Xiaoqing Wen is supported by NSF of China (11563005)

In this paper, a Lévy-diffusion Leslie-Gower predator-prey model with a nonmonotonic functional response is studied. We show the existence, uniqueness and attractiveness of the globally positive solution to this model. Moreover, to its corresponding steady-state model, we obtain the stability of the semi-trivial solutions, the existence and nonexistence of coexistence states by the method of topological degree, the uniqueness and stability of coexistence state, and the multiplicity and stability of coexistence states by Grandall-Rabinowitz bifurcation theorem. In addition, to get these results, we study the property of the Lévy diffusion operator, and give out the comparison principle of the generalized parabolic Lévy-diffusion differential equation, as well as the existence and stability of the solution for the steady-state Logistic equation with Lévy diffusion. Furthermore, we obtain the comparison principle of the steady-state Lévy-diffusion equation. As far as we know, these results are new in the ecological model.

Citation: Hongwei Yin, Xiaoyong Xiao, Xiaoqing Wen. Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2121-2151. doi: 10.3934/dcdsb.2018228
References:
[1]

P. AguirreE. González-Olivares and E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10 (2009), 1401-1416. doi: 10.1016/j.nonrwa.2008.01.022. Google Scholar

[2]

N. Ali and M. Jazar, Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses, Journal of Applied Mathematics and Computing, 43 (2013), 271-293. doi: 10.1007/s12190-013-0663-3. Google Scholar

[3]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404. Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[5]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, Journal of Mathematical Analysis and Applications, 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7. Google Scholar

[6]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin Des Sciences Mathematiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[7]

S. FuL. Zhang and P. Hu, Global behavior of solutions in a Lotka-Volterra predator-prey model with prey-stage structure, Nonlinear Analysis: Real World Applications, 14 (2013), 2027-2045. doi: 10.1016/j.nonrwa.2013.02.007. Google Scholar

[8]

D. HnaienF. Kellil and R. Lassoued, Asymptotic behavior of global solutions of an anomalous diffusion system, Journal of Mathematical Analysis and Applications, 421 (2014), 1519-1530. doi: 10.1016/j.jmaa.2014.07.083. Google Scholar

[9]

N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. Brunnschweiler, T. K. Doyle, J. D. R. Houghton and Others, Environmental context explains Lévy and brownian movement patterns of marine predators, Nature, 465 (2010), 1066–1069. doi: 10.1038/nature09116. Google Scholar

[10]

Y.-J. KimO. Kwon and F. Li, Global asymptotic stability and the ideal free distribution in a starvation driven diffusion, Journal of Mathematical Biology, 68 (2014), 1341-1370. doi: 10.1007/s00285-013-0674-6. Google Scholar

[11]

E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, Journal of Mathematical Analysis and Applications, 411 (2014), 107-118. doi: 10.1016/j.jmaa.2013.09.039. Google Scholar

[12]

Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Solitons & Fractals, 34 (2007), 606-620. doi: 10.1016/j.chaos.2006.03.068. Google Scholar

[13]

J. Liu, H. Zhou and L. Zhang, Cross-diffusion induced turing patterns in a sex-structured predator-prey model, International Journal of Biomathematics, 5 (2012), 1250016, 23PP. doi: 10.1142/S179352451100157X. Google Scholar

[14]

Y. F. LvR. Yuan and Y. Z. Pei, Turing pattern formation in a three species model with generalist predator and cross-diffusion, Nonlinear Analysis-Theory Methods & Applications, 85 (2013), 214-232. doi: 10.1016/j.na.2013.03.001. Google Scholar

[15]

R. PengM. Wang and G. Yang, Stationary patterns of the Holling-Tanner prey-predator model with diffusion and cross-diffusion, Applied Mathematics and Computation, 196 (2008), 570-577. doi: 10.1016/j.amc.2007.06.019. Google Scholar

[16]

R. Peng and M. Wang, Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Applied Mathematics Letters, 20 (2007), 664-670. doi: 10.1016/j.aml.2006.08.020. Google Scholar

[17]

S. Secchi, Ground state solutions for nonlinear fractional schrödinger equations in $\mathbb R^n$, Journal of Mathematical Physics, 54 (2013), 031501, 17PP. doi: 10.1063/1.4793990. Google Scholar

[18]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, Journal of Mathematical Analysis and Applications, 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar

[19]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137. Google Scholar

[20]

X. Shang and J. Zhang, Ground states for fractional schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. doi: 10.1088/0951-7715/27/2/187. Google Scholar

[21]

D. W. SimsE. J. SouthallN. E. HumphriesG. C. HaysC. J. A. BradshawJ. W. PitchfordA. JamesM. Z. AhmedA. S. BrierleyM. A. HindellD. MorrittM. K. MusylD. RightonE. L. C. ShepardV. J. WearmouthR. P. WilsonM. J. Witt and J. D. Metcalfe, Scaling laws of marine predator search behaviour, Nature, 451 (2008), 1098-1102. doi: 10.1038/nature06518. Google Scholar

[22]

Y. Song and X. Zou, Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point, Computers & Mathematics with Applications, 67 (2014), 1978-1997. doi: 10.1016/j.camwa.2014.04.015. Google Scholar

[23]

V. Volterra, Variazioni E fluttuazioni del numero D'individui in specie conviventi, Mem Acad Lincei Roma, 2 (1926), 31-113. Google Scholar

[24]

M. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D: Nonlinear Phenomena, 196 (2004), 172-192. doi: 10.1016/j.physd.2004.05.007. Google Scholar

[25]

D. Xiao and S. Ruan, Codimension two bifurcations in a predator-prey system with group defense, International Journal of Bifurcation and Chaos, 11 (2001), 2123-2131. doi: 10.1142/S021812740100336X. Google Scholar

[26]

D. Xiao and H. Zhu, Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response, Siam Journal On Applied Mathematics, 66 (2006), 802-819. doi: 10.1137/050623449. Google Scholar

[27]

W. Yang, Global asymptotical stability and persistent property for a diffusive predator-prey system with modified Leslie-Gower functional response, Nonlinear Analysis: Real World Applications, 14 (2013), 1323-1330. doi: 10.1016/j.nonrwa.2012.09.020. Google Scholar

[28]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, 2nd edition, Science Press, Beijing, 2011.Google Scholar

[29]

H. Yin, X. Xiao and X. Wen, Turing patterns in a predator-prey system with self-diffusion, Abstract and Applied Analysis, 2013 (2013), Art. ID 891738, 10 pp. Google Scholar

[30]

H. YinJ. ZhouX. Xiao and X. Wen, Analysis of a diffusive Leslie-Gower predator-prey model with nonmonotonic functional response, Chaos, Solitons & Fractals, 65 (2014), 51-61. doi: 10.1016/j.chaos.2014.04.010. Google Scholar

[31]

H. YinX. XiaoX. Wen and K. Liu, Pattern analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional response and diffusion, Computers & Mathematics with Applications, 67 (2014), 1607-1621. doi: 10.1016/j.camwa.2014.02.016. Google Scholar

[32]

J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Journal of Mathematical Analysis and Applications, 389 (2012), 1380-1393. doi: 10.1016/j.jmaa.2012.01.013. Google Scholar

[33]

J. Zhou, Positive steady state solutions of a Leslie-Gower predator-prey model with Holling type Ⅱ functional response and density-dependent diffusion, Nonlinear Analysis: Theory, Methods & Applications, 82 (2013), 47-65. doi: 10.1016/j.na.2012.12.014. Google Scholar

[34]

H. ZhuG. S. K. Wolkowicz and S. A. Campbell, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal On Applied Mathematics, 63 (2002), 636-682. doi: 10.1137/S0036139901397285. Google Scholar

[35]

B. ZimmermannH. SandP. WabakkenO. Liberg and H. P. Andreassen, Predator-dependent functional response in wolves: From food limitation to surplus killing, Journal of Animal Ecology, 84 (2015), 102-112. doi: 10.1111/1365-2656.12280. Google Scholar

show all references

References:
[1]

P. AguirreE. González-Olivares and E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10 (2009), 1401-1416. doi: 10.1016/j.nonrwa.2008.01.022. Google Scholar

[2]

N. Ali and M. Jazar, Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses, Journal of Applied Mathematics and Computing, 43 (2013), 271-293. doi: 10.1007/s12190-013-0663-3. Google Scholar

[3]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404. Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[5]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, Journal of Mathematical Analysis and Applications, 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7. Google Scholar

[6]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin Des Sciences Mathematiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[7]

S. FuL. Zhang and P. Hu, Global behavior of solutions in a Lotka-Volterra predator-prey model with prey-stage structure, Nonlinear Analysis: Real World Applications, 14 (2013), 2027-2045. doi: 10.1016/j.nonrwa.2013.02.007. Google Scholar

[8]

D. HnaienF. Kellil and R. Lassoued, Asymptotic behavior of global solutions of an anomalous diffusion system, Journal of Mathematical Analysis and Applications, 421 (2014), 1519-1530. doi: 10.1016/j.jmaa.2014.07.083. Google Scholar

[9]

N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. Brunnschweiler, T. K. Doyle, J. D. R. Houghton and Others, Environmental context explains Lévy and brownian movement patterns of marine predators, Nature, 465 (2010), 1066–1069. doi: 10.1038/nature09116. Google Scholar

[10]

Y.-J. KimO. Kwon and F. Li, Global asymptotic stability and the ideal free distribution in a starvation driven diffusion, Journal of Mathematical Biology, 68 (2014), 1341-1370. doi: 10.1007/s00285-013-0674-6. Google Scholar

[11]

E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, Journal of Mathematical Analysis and Applications, 411 (2014), 107-118. doi: 10.1016/j.jmaa.2013.09.039. Google Scholar

[12]

Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Solitons & Fractals, 34 (2007), 606-620. doi: 10.1016/j.chaos.2006.03.068. Google Scholar

[13]

J. Liu, H. Zhou and L. Zhang, Cross-diffusion induced turing patterns in a sex-structured predator-prey model, International Journal of Biomathematics, 5 (2012), 1250016, 23PP. doi: 10.1142/S179352451100157X. Google Scholar

[14]

Y. F. LvR. Yuan and Y. Z. Pei, Turing pattern formation in a three species model with generalist predator and cross-diffusion, Nonlinear Analysis-Theory Methods & Applications, 85 (2013), 214-232. doi: 10.1016/j.na.2013.03.001. Google Scholar

[15]

R. PengM. Wang and G. Yang, Stationary patterns of the Holling-Tanner prey-predator model with diffusion and cross-diffusion, Applied Mathematics and Computation, 196 (2008), 570-577. doi: 10.1016/j.amc.2007.06.019. Google Scholar

[16]

R. Peng and M. Wang, Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Applied Mathematics Letters, 20 (2007), 664-670. doi: 10.1016/j.aml.2006.08.020. Google Scholar

[17]

S. Secchi, Ground state solutions for nonlinear fractional schrödinger equations in $\mathbb R^n$, Journal of Mathematical Physics, 54 (2013), 031501, 17PP. doi: 10.1063/1.4793990. Google Scholar

[18]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, Journal of Mathematical Analysis and Applications, 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar

[19]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137. Google Scholar

[20]

X. Shang and J. Zhang, Ground states for fractional schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. doi: 10.1088/0951-7715/27/2/187. Google Scholar

[21]

D. W. SimsE. J. SouthallN. E. HumphriesG. C. HaysC. J. A. BradshawJ. W. PitchfordA. JamesM. Z. AhmedA. S. BrierleyM. A. HindellD. MorrittM. K. MusylD. RightonE. L. C. ShepardV. J. WearmouthR. P. WilsonM. J. Witt and J. D. Metcalfe, Scaling laws of marine predator search behaviour, Nature, 451 (2008), 1098-1102. doi: 10.1038/nature06518. Google Scholar

[22]

Y. Song and X. Zou, Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point, Computers & Mathematics with Applications, 67 (2014), 1978-1997. doi: 10.1016/j.camwa.2014.04.015. Google Scholar

[23]

V. Volterra, Variazioni E fluttuazioni del numero D'individui in specie conviventi, Mem Acad Lincei Roma, 2 (1926), 31-113. Google Scholar

[24]

M. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D: Nonlinear Phenomena, 196 (2004), 172-192. doi: 10.1016/j.physd.2004.05.007. Google Scholar

[25]

D. Xiao and S. Ruan, Codimension two bifurcations in a predator-prey system with group defense, International Journal of Bifurcation and Chaos, 11 (2001), 2123-2131. doi: 10.1142/S021812740100336X. Google Scholar

[26]

D. Xiao and H. Zhu, Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response, Siam Journal On Applied Mathematics, 66 (2006), 802-819. doi: 10.1137/050623449. Google Scholar

[27]

W. Yang, Global asymptotical stability and persistent property for a diffusive predator-prey system with modified Leslie-Gower functional response, Nonlinear Analysis: Real World Applications, 14 (2013), 1323-1330. doi: 10.1016/j.nonrwa.2012.09.020. Google Scholar

[28]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, 2nd edition, Science Press, Beijing, 2011.Google Scholar

[29]

H. Yin, X. Xiao and X. Wen, Turing patterns in a predator-prey system with self-diffusion, Abstract and Applied Analysis, 2013 (2013), Art. ID 891738, 10 pp. Google Scholar

[30]

H. YinJ. ZhouX. Xiao and X. Wen, Analysis of a diffusive Leslie-Gower predator-prey model with nonmonotonic functional response, Chaos, Solitons & Fractals, 65 (2014), 51-61. doi: 10.1016/j.chaos.2014.04.010. Google Scholar

[31]

H. YinX. XiaoX. Wen and K. Liu, Pattern analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional response and diffusion, Computers & Mathematics with Applications, 67 (2014), 1607-1621. doi: 10.1016/j.camwa.2014.02.016. Google Scholar

[32]

J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Journal of Mathematical Analysis and Applications, 389 (2012), 1380-1393. doi: 10.1016/j.jmaa.2012.01.013. Google Scholar

[33]

J. Zhou, Positive steady state solutions of a Leslie-Gower predator-prey model with Holling type Ⅱ functional response and density-dependent diffusion, Nonlinear Analysis: Theory, Methods & Applications, 82 (2013), 47-65. doi: 10.1016/j.na.2012.12.014. Google Scholar

[34]

H. ZhuG. S. K. Wolkowicz and S. A. Campbell, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal On Applied Mathematics, 63 (2002), 636-682. doi: 10.1137/S0036139901397285. Google Scholar

[35]

B. ZimmermannH. SandP. WabakkenO. Liberg and H. P. Andreassen, Predator-dependent functional response in wolves: From food limitation to surplus killing, Journal of Animal Ecology, 84 (2015), 102-112. doi: 10.1111/1365-2656.12280. Google Scholar

Figure 1.  Response functions
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