
-
Previous Article
On fractional Leibniz rule for Dirichlet Laplacian in exterior domain
- DCDS Home
- This Issue
-
Next Article
Pathological center foliation with dimension greater than one
Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey
1. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
2. | School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China |
It is well known that the Leslie-Gower prey-predator model (without Allee effect) has a unique globally asymptotically stable positive equilibrium point, thus there is no Hopf bifurcation branching from positive equilibrium point. In this paper we study the Leslie-Gower prey-predator model with strong Allee effect in prey, and perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the steady-state bifurcation of PDE model. Moreover, by the center manifold theory and the normal form method, the direction and stability of Hopf bifurcation solutions are established. Finally, some numerical simulations are presented. Apparently, Allee effect changes the topology structure of the original Leslie-Gower model.
References:
[1] |
W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949. Google Scholar |
[2] |
M. A. Aziz-Alaoui and M. D. 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. |
[3] |
Y. L. Cai, C. D. Zhao, W. M. Wang and J. F. Wang,
Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Appl. Math. Lett., 39 (2015), 2092-2106.
doi: 10.1016/j.apm.2014.09.038. |
[4] |
F. Courchamp, T. Clutton-Brock and B. Grenfell,
Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405-410.
doi: 10.1016/S0169-5347(99)01683-3. |
[5] |
R. H. Cui, J. P. Shi and B. Y. Wu,
Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.
doi: 10.1016/j.jde.2013.08.015. |
[6] |
L. L. Du and M. X. Wang,
Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485.
doi: 10.1016/j.jmaa.2010.02.002. |
[7] |
Y. H. Du and S. B. Hsu,
A diffusive predator-prey model in heterogeneous environment, J. Diff. Equat., 203 (2004), 331-364.
doi: 10.1016/j.jde.2004.05.010. |
[8] |
E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma and J. D. Flores,
Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model, 35 (2011), 366-381.
doi: 10.1016/j.apm.2010.07.001. |
[9] |
B. Hassard, N. Kazarinoff and Y. H. Wan,
Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. |
[10] |
A. Korobeinikov,
A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.
doi: 10.1016/S0893-9659(01)80029-X. |
[11] |
P. H. Leslie,
Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245.
doi: 10.1093/biomet/35.3-4.213. |
[12] |
P. H. Leslie,
A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31.
doi: 10.1093/biomet/45.1-2.16. |
[13] |
S. B. Li, J. H. Wu and H. Nie,
Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model, Comput. Math. Appl., 70 (2015), 3043-3056.
doi: 10.1016/j.camwa.2015.10.017. |
[14] |
Y. Li,
Hopf bifurcations in general systems of Brusselator type, Nonlinear Anal.: Real World Appl., 28 (2016), 32-47.
doi: 10.1016/j.nonrwa.2015.09.004. |
[15] |
Y. Li and M. X. Wang,
Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal.: Real World Appl., 14 (2013), 1806-1816.
doi: 10.1016/j.nonrwa.2012.11.012. |
[16] |
N. Min and X. M. Wang,
Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689.
doi: 10.1016/j.camwa.2016.07.028. |
[17] |
N. Min and X. M. Wang,
Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1721-1737.
doi: 10.3934/dcdsb.2018073. |
[18] |
W. J. Ni and M. X. Wang,
Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.
doi: 10.3934/dcdsb.2017172. |
[19] |
W. J. Ni and M. X. Wang,
Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.
doi: 10.1016/j.jde.2016.06.022. |
[20] |
W. M. Ni,
Diffusion, cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
|
[21] |
P. Y. H. Pang and M. X. Wang,
Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh, 133 (2003), 919-942.
doi: 10.1017/S0308210500002742. |
[22] |
P. Y. H. Pang and M. X. Wang,
Strategy and stationary pattern in a three-species predator-prey model, J. Diff. Equat., 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[23] |
E. C. Pielou,
Mathematical Ecology, John Wiley & Sons, New York, RI, 1977. |
[24] |
Y. W. Qi and Y. Zhu,
Global stability of Lesile-type predator-prey model, Meth. Appl. Anal., 23 (2016), 259-268.
doi: 10.4310/MAA.2016.v23.n3.a3. |
[25] |
P. A. Stephens and W. J. Sutherland,
Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401-405.
doi: 10.1016/S0169-5347(99)01684-5. |
[26] |
J. F. Wang, J. P. Shi and J. J. W,
Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[27] |
J. F. Wang, J. J. Wei and J. P. Shi,
Global bifurcation analysis and pattern formation inhomogeneous diffusive predator-prey systems, J. Diff. Equat., 260 (2016), 3495-3523.
doi: 10.1016/j.jde.2015.10.036. |
[28] |
M. X. Wang,
Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196 (2004), 172-192.
doi: 10.1016/j.physd.2004.05.007. |
[29] |
M. X. Wang and Q. Y. Zhang,
Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Cont. Dyn. Syst. A, 38 (2018), 2591-2607.
doi: 10.3934/dcds.2018109. |
[30] |
Y. X. Wang and W. T. Li,
Spatial patterns of the Holling-Tanner predator-prey model with nonlinear diffusion effects, Appl. Anal., 92 (2013), 2168-2181.
doi: 10.1080/00036811.2012.724402. |
[31] |
S. Wiggins,
Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second edition. Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003. |
[32] |
F. Q. Yi, J. J. Wei and J. P. Shi,
Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal.: Real World Appl., 9 (2008), 1038-1051.
doi: 10.1016/j.nonrwa.2007.02.005. |
[33] |
F. Q. Yi, J. J. Wei and J. P. Shi,
Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.
doi: 10.1016/j.jde.2008.10.024. |
show all references
References:
[1] |
W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949. Google Scholar |
[2] |
M. A. Aziz-Alaoui and M. D. 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. |
[3] |
Y. L. Cai, C. D. Zhao, W. M. Wang and J. F. Wang,
Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Appl. Math. Lett., 39 (2015), 2092-2106.
doi: 10.1016/j.apm.2014.09.038. |
[4] |
F. Courchamp, T. Clutton-Brock and B. Grenfell,
Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405-410.
doi: 10.1016/S0169-5347(99)01683-3. |
[5] |
R. H. Cui, J. P. Shi and B. Y. Wu,
Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.
doi: 10.1016/j.jde.2013.08.015. |
[6] |
L. L. Du and M. X. Wang,
Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl., 366 (2010), 473-485.
doi: 10.1016/j.jmaa.2010.02.002. |
[7] |
Y. H. Du and S. B. Hsu,
A diffusive predator-prey model in heterogeneous environment, J. Diff. Equat., 203 (2004), 331-364.
doi: 10.1016/j.jde.2004.05.010. |
[8] |
E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma and J. D. Flores,
Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model, 35 (2011), 366-381.
doi: 10.1016/j.apm.2010.07.001. |
[9] |
B. Hassard, N. Kazarinoff and Y. H. Wan,
Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. |
[10] |
A. Korobeinikov,
A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.
doi: 10.1016/S0893-9659(01)80029-X. |
[11] |
P. H. Leslie,
Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213-245.
doi: 10.1093/biomet/35.3-4.213. |
[12] |
P. H. Leslie,
A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31.
doi: 10.1093/biomet/45.1-2.16. |
[13] |
S. B. Li, J. H. Wu and H. Nie,
Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model, Comput. Math. Appl., 70 (2015), 3043-3056.
doi: 10.1016/j.camwa.2015.10.017. |
[14] |
Y. Li,
Hopf bifurcations in general systems of Brusselator type, Nonlinear Anal.: Real World Appl., 28 (2016), 32-47.
doi: 10.1016/j.nonrwa.2015.09.004. |
[15] |
Y. Li and M. X. Wang,
Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal.: Real World Appl., 14 (2013), 1806-1816.
doi: 10.1016/j.nonrwa.2012.11.012. |
[16] |
N. Min and X. M. Wang,
Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689.
doi: 10.1016/j.camwa.2016.07.028. |
[17] |
N. Min and X. M. Wang,
Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1721-1737.
doi: 10.3934/dcdsb.2018073. |
[18] |
W. J. Ni and M. X. Wang,
Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.
doi: 10.3934/dcdsb.2017172. |
[19] |
W. J. Ni and M. X. Wang,
Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.
doi: 10.1016/j.jde.2016.06.022. |
[20] |
W. M. Ni,
Diffusion, cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
|
[21] |
P. Y. H. Pang and M. X. Wang,
Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh, 133 (2003), 919-942.
doi: 10.1017/S0308210500002742. |
[22] |
P. Y. H. Pang and M. X. Wang,
Strategy and stationary pattern in a three-species predator-prey model, J. Diff. Equat., 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[23] |
E. C. Pielou,
Mathematical Ecology, John Wiley & Sons, New York, RI, 1977. |
[24] |
Y. W. Qi and Y. Zhu,
Global stability of Lesile-type predator-prey model, Meth. Appl. Anal., 23 (2016), 259-268.
doi: 10.4310/MAA.2016.v23.n3.a3. |
[25] |
P. A. Stephens and W. J. Sutherland,
Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401-405.
doi: 10.1016/S0169-5347(99)01684-5. |
[26] |
J. F. Wang, J. P. Shi and J. J. W,
Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004. |
[27] |
J. F. Wang, J. J. Wei and J. P. Shi,
Global bifurcation analysis and pattern formation inhomogeneous diffusive predator-prey systems, J. Diff. Equat., 260 (2016), 3495-3523.
doi: 10.1016/j.jde.2015.10.036. |
[28] |
M. X. Wang,
Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196 (2004), 172-192.
doi: 10.1016/j.physd.2004.05.007. |
[29] |
M. X. Wang and Q. Y. Zhang,
Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Cont. Dyn. Syst. A, 38 (2018), 2591-2607.
doi: 10.3934/dcds.2018109. |
[30] |
Y. X. Wang and W. T. Li,
Spatial patterns of the Holling-Tanner predator-prey model with nonlinear diffusion effects, Appl. Anal., 92 (2013), 2168-2181.
doi: 10.1080/00036811.2012.724402. |
[31] |
S. Wiggins,
Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second edition. Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003. |
[32] |
F. Q. Yi, J. J. Wei and J. P. Shi,
Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal.: Real World Appl., 9 (2008), 1038-1051.
doi: 10.1016/j.nonrwa.2007.02.005. |
[33] |
F. Q. Yi, J. J. Wei and J. P. Shi,
Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.
doi: 10.1016/j.jde.2008.10.024. |



![]() One Hopf bifurcation value |
![]() Two Hopf bifurcation values |
Null | |
![]() One Hopf bifurcation value |
Null | Null | |
![]() One Hopf bifurcation value |
![]() Two Hopf bifurcation values |
Null | |
![]() One Hopf bifurcation value |
Null | Null | |
![]() |
![]() |
Null | |
![]() |
Null | Null | |
![]() |
![]() |
Null | |
![]() |
Null | Null | |
![]() bifurcation values |
![]() bifurcation values |
Null | |
![]() bifurcation values |
![]() bifurcation values |
![]() bifurcation values |
|
![]() bifurcation values |
![]() bifurcation values |
Null | |
![]() bifurcation values |
![]() bifurcation values |
![]() bifurcation values |
|
![]() bifurcation values |
![]() bifurcation values |
![]() bifurcation values |
|
![]() bifurcation values |
![]() bifurcation values |
Null | |
![]() bifurcation values |
![]() bifurcation values |
![]() bifurcation values |
|
![]() bifurcation values |
![]() bifurcation values |
Null | |
1 | 0.03 | 0.1 | 22.44329 | 1 | 0.1 | 1 |
2 | 0.05 | 0.1 | 11.46339 | 1 | 0.1 | 1 |
3 | 0.06 | 0.1 | 9.485507 | 1 | 0.1 | 1 |
4 | 0.06 | 0.1 | 6.305220 | 1 | 0.1 | 1 |
1 | 0.03 | 0.1 | 22.44329 | 1 | 0.1 | 1 |
2 | 0.05 | 0.1 | 11.46339 | 1 | 0.1 | 1 |
3 | 0.06 | 0.1 | 9.485507 | 1 | 0.1 | 1 |
4 | 0.06 | 0.1 | 6.305220 | 1 | 0.1 | 1 |
$b$ | $\mu$ | $\beta$ | $d_1$ | $d_2$ | $l$ | |
1 | 0.25 | 0.292 | 0.972 | 0.5 | 3 | 0.531 |
2 | 0.062 | 2.431 | 8.667 | 0.5 | 2 | 1.283 |
3 | 0.25 | 1 | 0.667 | 1 | 1 | 1 |
4 | 0.062 | 1 | 10 | 1 | 1 | 2 |
$b$ | $\mu$ | $\beta$ | $d_1$ | $d_2$ | $l$ | |
1 | 0.25 | 0.292 | 0.972 | 0.5 | 3 | 0.531 |
2 | 0.062 | 2.431 | 8.667 | 0.5 | 2 | 1.283 |
3 | 0.25 | 1 | 0.667 | 1 | 1 | 1 |
4 | 0.062 | 1 | 10 | 1 | 1 | 2 |
[1] |
Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148 |
[2] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[3] |
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 |
[4] |
Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020468 |
[5] |
Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263 |
[6] |
Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020342 |
[7] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021013 |
[8] |
Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020356 |
[9] |
Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 |
[10] |
Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 |
[11] |
Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 |
[12] |
Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 |
[13] |
Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053 |
[14] |
Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328 |
[15] |
Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021007 |
[16] |
Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020281 |
[17] |
Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3657-3682. doi: 10.3934/dcds.2020051 |
[18] |
Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021005 |
[19] |
Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 |
[20] |
Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]