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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 |
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.
References:
| [1] |
A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1995. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [7] |
A. Ducrot and J.-S. Guo,
Quenching behavior for a singular predator-prey model, Nonlinearity, 25 (2012), 2059-2073.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
J. Hainzl,
Multiparameter bifurcation of a predator-prey system, SIAM J. Math. Anal., 23 (1992), 150-180.
doi: 10.1137/0523008. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [7] |
A. Ducrot and J.-S. Guo,
Quenching behavior for a singular predator-prey model, Nonlinearity, 25 (2012), 2059-2073.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
J. Hainzl,
Multiparameter bifurcation of a predator-prey system, SIAM J. Math. Anal., 23 (1992), 150-180.
doi: 10.1137/0523008. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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