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April  2021, 26(4): 1811-1825. doi: 10.3934/dcdsb.2020040

Recent developments on a singular predator-prey model

Yu-Shuo Chen 1, , Jong-Shenq Guo 1,, and  Masahiko Shimojo 2,

1. 

Department of Mathematics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan
2. 

Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan

* Corresponding author: Jong-Sheng Guo

Received  May 2019 Published  February 2020

Fund Project: This work is partially supported by the Ministry of Science and Technology of Taiwan under the grants 106-2811-M-032-008 and 105-2115-M-032-003-MY3

This work is concerned with the dynamical behaviors of a singular predator-prey model. We first review some well-known results obtained recently. Then we give some new results on the spreading speed of the predator, the existence vs non-existence of traveling waves connecting the predator-free state to the co-existence state, and the existence vs non-existence of spatially periodic traveling waves to this singular predator-prey system.

Keywords: Predator-prey system, spreading speed, traveling wave, wave speed, minimal speed.
Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 92D25, 92D40.
Citation: Yu-Shuo Chen, Jong-Shenq Guo, Masahiko Shimojo. Recent developments on a singular predator-prey model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1811-1825. doi: 10.3934/dcdsb.2020040
References:
[1]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1995.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math., 446, Springer, Berlin, 1975, 5-49. doi: 10.1007/BFb0070595.  Google Scholar

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Y.-Y. Chen, J.-S. Guo and C.-H. Yao, Traveling wave solutions for a continuous and discrete diffusive predator-prey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071.  Google Scholar

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F. Courchamp, M. Langlais and G. Sugihara, Controls of rabbits to protect birds from cat predation, Biol. Conservations, 89 (1999), 219-225.  doi: 10.1016/S0006-3207(98)00131-1.  Google Scholar

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F. Courchamp and G. Sugihara, Modelling the biological control of an alien predator to protect island species from extinction, Ecological Appl., 9 (1999), 112-123.   Google Scholar

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Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[7]

A. Ducrot and J.-S. Guo, Quenching behavior for a singular predator-prey model, Nonlinearity, 25 (2012), 2059-2073.   Google Scholar

[8]

A. Ducrot, J.-S. Guo and M. Shimojo, Behaviors of solutions for a singular prey-predator model and its shadow system, J. Dynam. Differential Equations, 30 (2018), 1063-1079.  doi: 10.1007/s10884-017-9587-1.  Google Scholar

[9]

A. Ducrot and M. Langlais, A singular reaction-diffusion system modelling prey-predator interactions: Invasion and co-extinction waves, J. Differential Equations, 253 (2012), 502-532.  doi: 10.1016/j.jde.2012.04.005.  Google Scholar

[10]

A. Ducrot and M. Langlais, Global weak solution for a singular two component reaction-diffusion system, Bull. Lond. Math. Soc., 46 (2014), 1-13.  doi: 10.1112/blms/bdt058.  Google Scholar

[11]

S. Gaucel and M. Langlais, Some remarks on a singular reaction-diffusion arising in predator-prey modelling, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 61-72.  doi: 10.3934/dcdsb.2007.8.61.  Google Scholar

[12]

J.-S. Guo and M. Shimojo, Spatio-temporal oscillation for a singular predator-prey model, J. Math. Anal. Appl., 459 (2018), 1-9.  doi: 10.1016/j.jmaa.2017.10.080.  Google Scholar

[13]

J. Hainzl, Multiparameter bifurcation of a predator-prey system, SIAM J. Math. Anal., 23 (1992), 150-180.  doi: 10.1137/0523008.  Google Scholar

[14]

B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.  Google Scholar

[15]

S.-B. Hsu and T.-W. Hwang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[16]

S.-B. Hsu and T.-W. Hwang, Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.   Google Scholar

[17]

G. Lin, Spreading speeds of a Lotka-Volterra predator-prey system: The role of the predator, Nonlinear Anal., 74 (2011), 2448-2461.  doi: 10.1016/j.na.2010.11.046.  Google Scholar

[18]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[19]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[20]

S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.  doi: 10.1016/j.jmaa.2013.05.031.  Google Scholar

[21]

S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

[22]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[23]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[24]

W. Zuo and J. Shi, Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay, Comm. Pure. Appl. Anal., 17 (2018), 1179-1200.  doi: 10.3934/cpaa.2018057.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1995.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math., 446, Springer, Berlin, 1975, 5-49. doi: 10.1007/BFb0070595.  Google Scholar

[3]

Y.-Y. Chen, J.-S. Guo and C.-H. Yao, Traveling wave solutions for a continuous and discrete diffusive predator-prey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071.  Google Scholar

[4]

F. Courchamp, M. Langlais and G. Sugihara, Controls of rabbits to protect birds from cat predation, Biol. Conservations, 89 (1999), 219-225.  doi: 10.1016/S0006-3207(98)00131-1.  Google Scholar

[5]

F. Courchamp and G. Sugihara, Modelling the biological control of an alien predator to protect island species from extinction, Ecological Appl., 9 (1999), 112-123.   Google Scholar

[6]

Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[7]

A. Ducrot and J.-S. Guo, Quenching behavior for a singular predator-prey model, Nonlinearity, 25 (2012), 2059-2073.   Google Scholar

[8]

A. Ducrot, J.-S. Guo and M. Shimojo, Behaviors of solutions for a singular prey-predator model and its shadow system, J. Dynam. Differential Equations, 30 (2018), 1063-1079.  doi: 10.1007/s10884-017-9587-1.  Google Scholar

[9]

A. Ducrot and M. Langlais, A singular reaction-diffusion system modelling prey-predator interactions: Invasion and co-extinction waves, J. Differential Equations, 253 (2012), 502-532.  doi: 10.1016/j.jde.2012.04.005.  Google Scholar

[10]

A. Ducrot and M. Langlais, Global weak solution for a singular two component reaction-diffusion system, Bull. Lond. Math. Soc., 46 (2014), 1-13.  doi: 10.1112/blms/bdt058.  Google Scholar

[11]

S. Gaucel and M. Langlais, Some remarks on a singular reaction-diffusion arising in predator-prey modelling, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 61-72.  doi: 10.3934/dcdsb.2007.8.61.  Google Scholar

[12]

J.-S. Guo and M. Shimojo, Spatio-temporal oscillation for a singular predator-prey model, J. Math. Anal. Appl., 459 (2018), 1-9.  doi: 10.1016/j.jmaa.2017.10.080.  Google Scholar

[13]

J. Hainzl, Multiparameter bifurcation of a predator-prey system, SIAM J. Math. Anal., 23 (1992), 150-180.  doi: 10.1137/0523008.  Google Scholar

[14]

B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.  Google Scholar

[15]

S.-B. Hsu and T.-W. Hwang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[16]

S.-B. Hsu and T.-W. Hwang, Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.   Google Scholar

[17]

G. Lin, Spreading speeds of a Lotka-Volterra predator-prey system: The role of the predator, Nonlinear Anal., 74 (2011), 2448-2461.  doi: 10.1016/j.na.2010.11.046.  Google Scholar

[18]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[19]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[20]

S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.  doi: 10.1016/j.jmaa.2013.05.031.  Google Scholar

[21]

S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

[22]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[23]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[24]

W. Zuo and J. Shi, Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay, Comm. Pure. Appl. Anal., 17 (2018), 1179-1200.  doi: 10.3934/cpaa.2018057.  Google Scholar

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