doi: 10.3934/dcdsb.2020040

Recent developments on a singular predator-prey model

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.

Citation: Yu-Shuo Chen, Jong-Shenq Guo, Masahiko Shimojo. Recent developments on a singular predator-prey model. Discrete & Continuous Dynamical Systems - B, 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

[3]

Y.-Y. ChenJ.-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. CourchampM. 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. DucrotJ.-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. WeinbergerM. 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. ChenJ.-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. CourchampM. 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. DucrotJ.-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. WeinbergerM. 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

[1]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

[2]

Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067

[3]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083

[4]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129

[5]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

[6]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[7]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405

[8]

Sungrim Seirin Lee. Dependence of propagation speed on invader species: The effect of the predatory commensalism in two-prey, one-predator system with diffusion. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 797-825. doi: 10.3934/dcdsb.2009.12.797

[9]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213

[10]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[11]

Sebastian Acosta. A control approach to recover the wave speed (conformal factor) from one measurement. Inverse Problems & Imaging, 2015, 9 (2) : 301-315. doi: 10.3934/ipi.2015.9.301

[12]

Chase Mathison. Thermoacoustic Tomography with circular integrating detectors and variable wave speed. Inverse Problems & Imaging, 2020, 14 (4) : 665-682. doi: 10.3934/ipi.2020030

[13]

Hans Weinberger. On sufficient conditions for a linearly determinate spreading speed. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2267-2280. doi: 10.3934/dcdsb.2012.17.2267

[14]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

[15]

Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883

[16]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1483-1508. doi: 10.3934/cpaa.2019071

[17]

Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133

[18]

Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591

[19]

Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199

[20]

Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355

2019 Impact Factor: 1.27

Article outline

[Back to Top]