
-
Previous Article
Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control
- DCDS-B Home
- This Issue
-
Next Article
Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation
Predator-prey interactions under fear effect and multiple foraging strategies
1. | Department of Mathematics, University of Kalyani, Nadia, West Bengal-741235, India |
2. | Department of Mathematics, Karimpur Pannadevi College, Nadia, West Bengal-741152, India |
We propose and analyze the effects of a generalist predator-driven fear effect on a prey population by considering a modified Leslie-Gower predator-prey model. We assume that the prey population suffers from reduced fecundity due to the fear of predators. We investigate the predator-prey dynamics by incorporating linear, Holling type Ⅱ and Holling type Ⅲ foraging strategies of the generalist predator. As a control strategy, we have considered density-dependent harvesting of the organisms in the system. We show that the systems with linear and Holling type Ⅲ foraging exhibit transcritical bifurcation, whereas the system with Holling type Ⅱ foraging has a much more complex dynamics with transcritical, saddle-node, and Hopf bifurcations. It is observed that the prey population in the system with Holling type Ⅲ foraging of the predator gets severely affected by the predation-driven fear effect in comparison with the same with linear and Holling type Ⅱ foraging rates of the predator. Our model simulation results show that an increase in the harvesting rate of the predator is a viable strategy in recovering the prey population.
References:
[1] |
M. A. Aziz-Alaoui and M. Daher 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. |
[2] |
G. Barabás and G. Meszéna,
When the exception becomes the rule: the disappearance of limiting similarity in the Lotka–Volterra model, J. Theoret. Biol., 258 (2009), 89-94.
doi: 10.1016/j.jtbi.2008.12.033. |
[3] |
S. Creel and D. Christianson,
Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.
doi: 10.1016/j.tree.2007.12.004. |
[4] |
W. Cresswell,
Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.
doi: 10.1007/s10336-010-0638-1. |
[5] |
Y.-J. Gong and J.-C. Huang,
Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 239-244.
doi: 10.1007/s10255-014-0279-x. |
[6] |
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. |
[7] |
R. P. Gupta, M. Banerjee and P. Chandra,
Bifurcation analysis and control of Leslie–Gower predator–prey model with Michaelis–Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.
doi: 10.1007/s12591-012-0142-6. |
[8] |
R. P. Gupta and P. Chandra,
Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.
doi: 10.1016/j.jmaa.2012.08.057. |
[9] |
S. Halder, J. Bhattacharyya and S. Pal,
Comparative studies on a predator–prey model subjected to fear and Allee effect with type Ⅰ and type Ⅱ foraging, J. Appl. Math. Comput., 62 (2020), 93-118.
doi: 10.1007/s12190-019-01275-w. |
[10] |
G. W. Harrison,
Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171.
doi: 10.1007/BF00279719. |
[11] |
C. S. Holling,
The components of predation as revealed by a study of small-mammal predation of the European pine sawfly1, The Canadian Entomologist, 91 (1959), 293-320.
|
[12] |
C. S. Holling,
Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.
doi: 10.4039/Ent91385-7. |
[13] |
V. Křivan,
On the Gause predator–prey model with a refuge: A fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.
doi: 10.1016/j.jtbi.2011.01.016. |
[14] |
P. H. Leslie and J. C. Gower,
The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.
doi: 10.1093/biomet/47.3-4.219. |
[15] |
M. Liu and K. Wang,
Dynamics of a Leslie–Gower Holling-type ii predator–prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213.
doi: 10.1016/j.na.2013.02.018. |
[16] |
K. J. MacLeod, C. J. Krebs, R. Boonstra and M. J. Sheriff,
Fear and lethality in snowshoe hares: The deadly effects of non-consumptive predation risk, Oikos, 127 (2018), 375-380.
doi: 10.1111/oik.04890. |
[17] |
P. Mishra, S. N. Raw and B. Tiwari,
Study of a Leslie–Gower predator-prey model with prey defense and mutual interference of predators, Chaos Solitons Fractals, 120 (2019), 1-16.
doi: 10.1016/j.chaos.2019.01.012. |
[18] |
A. Oaten and W. W. Murdoch,
Functional response and stability in predator-prey systems, The American Naturalist, 109 (1975), 289-298.
|
[19] |
L. Perko, Differential Equations and Dynamical Systems, vol. 7, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
[20] |
E. L. Preisser and D. I. Bolnick, The many faces of fear: Comparing the pathways and impacts of nonconsumptive predator effects on prey populations, PloS One, 3 (2008), e2465.
doi: 10.1371/journal.pone.0002465. |
[21] |
H. Seno,
A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model, J. Difference Equ. Appl., 13 (2007), 1155-1170.
doi: 10.1080/10236190701464996. |
[22] |
M. K. Singh, B. S. Bhadauria and B. K. Singh,
Bifurcation analysis of modified leslie-gower predator-prey model with double allee effect, Ain Shams Engineering Journal, 9 (2018), 1263-1277.
doi: 10.1016/j.asej.2016.07.007. |
[23] |
E. van Leeuwen, V. A. A. Jansen and P. W. Bright,
How population dynamics shape the functional response in a one-predator–two-prey system, Ecology, 88 (2007), 1571-1581.
doi: 10.1890/06-1335. |
[24] |
J. Wang, Y. Cai, S. Fu and W. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos, 29 (2019), 083109, 10 pp.
doi: 10.1063/1.5111121. |
[25] |
X. Wang, L. Zanette and X. Zou,
Modelling the fear effect in predator–prey interactions, J. Math. Biol., 73 (2016), 1179-1204.
doi: 10.1007/s00285-016-0989-1. |
[26] |
X. Wang and X. Zou,
Modeling the fear effect in predator–prey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79 (2017), 1325-1359.
doi: 10.1007/s11538-017-0287-0. |
[27] |
Y. Xia and S. Yuan,
Survival analysis of a stochastic predator–prey model with prey refuge and fear effect, J. Biol. Dyn., 14 (2020), 871-892.
doi: 10.1080/17513758.2020.1853832. |
[28] |
Z. Xiao, Z. Li et al., Stability analysis of a mutual interference predator-prey model with the fear effect, Journal of Applied Science and Engineering, 22 (2019), 205–211. |
[29] |
Z. Zhang, R. K. Upadhyay and J. Datta, Bifurcation analysis of a modified Leslie–Gower model with Holling type-Ⅳ functional response and nonlinear prey harvesting, Adv. Difference Equ., 2018 (2018), Paper No. 127, 21 pp.
doi: 10.1186/s13662-018-1581-3. |
[30] |
Z.-Z. Zhang and H.-Z. Yang,
Hopf bifurcation in a delayed predator-prey system with modified Leslie-Gower and Holling type Ⅲ schemes, Acta Automat. Sinica, 39 (2013), 610-616.
doi: 10.3724/SP.J.1004.2013.00610. |
[31] |
Y. Zhu and K. Wang,
Existence and global attractivity of positive periodic solutions for a predator–prey model with modified Leslie–Gower Holling-type Ⅱ schemes, J. Math. Anal. Appl., 384 (2011), 400-408.
doi: 10.1016/j.jmaa.2011.05.081. |
show all references
References:
[1] |
M. A. Aziz-Alaoui and M. Daher 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. |
[2] |
G. Barabás and G. Meszéna,
When the exception becomes the rule: the disappearance of limiting similarity in the Lotka–Volterra model, J. Theoret. Biol., 258 (2009), 89-94.
doi: 10.1016/j.jtbi.2008.12.033. |
[3] |
S. Creel and D. Christianson,
Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.
doi: 10.1016/j.tree.2007.12.004. |
[4] |
W. Cresswell,
Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.
doi: 10.1007/s10336-010-0638-1. |
[5] |
Y.-J. Gong and J.-C. Huang,
Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 239-244.
doi: 10.1007/s10255-014-0279-x. |
[6] |
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. |
[7] |
R. P. Gupta, M. Banerjee and P. Chandra,
Bifurcation analysis and control of Leslie–Gower predator–prey model with Michaelis–Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.
doi: 10.1007/s12591-012-0142-6. |
[8] |
R. P. Gupta and P. Chandra,
Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.
doi: 10.1016/j.jmaa.2012.08.057. |
[9] |
S. Halder, J. Bhattacharyya and S. Pal,
Comparative studies on a predator–prey model subjected to fear and Allee effect with type Ⅰ and type Ⅱ foraging, J. Appl. Math. Comput., 62 (2020), 93-118.
doi: 10.1007/s12190-019-01275-w. |
[10] |
G. W. Harrison,
Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171.
doi: 10.1007/BF00279719. |
[11] |
C. S. Holling,
The components of predation as revealed by a study of small-mammal predation of the European pine sawfly1, The Canadian Entomologist, 91 (1959), 293-320.
|
[12] |
C. S. Holling,
Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.
doi: 10.4039/Ent91385-7. |
[13] |
V. Křivan,
On the Gause predator–prey model with a refuge: A fresh look at the history, J. Theoret. Biol., 274 (2011), 67-73.
doi: 10.1016/j.jtbi.2011.01.016. |
[14] |
P. H. Leslie and J. C. Gower,
The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.
doi: 10.1093/biomet/47.3-4.219. |
[15] |
M. Liu and K. Wang,
Dynamics of a Leslie–Gower Holling-type ii predator–prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213.
doi: 10.1016/j.na.2013.02.018. |
[16] |
K. J. MacLeod, C. J. Krebs, R. Boonstra and M. J. Sheriff,
Fear and lethality in snowshoe hares: The deadly effects of non-consumptive predation risk, Oikos, 127 (2018), 375-380.
doi: 10.1111/oik.04890. |
[17] |
P. Mishra, S. N. Raw and B. Tiwari,
Study of a Leslie–Gower predator-prey model with prey defense and mutual interference of predators, Chaos Solitons Fractals, 120 (2019), 1-16.
doi: 10.1016/j.chaos.2019.01.012. |
[18] |
A. Oaten and W. W. Murdoch,
Functional response and stability in predator-prey systems, The American Naturalist, 109 (1975), 289-298.
|
[19] |
L. Perko, Differential Equations and Dynamical Systems, vol. 7, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
[20] |
E. L. Preisser and D. I. Bolnick, The many faces of fear: Comparing the pathways and impacts of nonconsumptive predator effects on prey populations, PloS One, 3 (2008), e2465.
doi: 10.1371/journal.pone.0002465. |
[21] |
H. Seno,
A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model, J. Difference Equ. Appl., 13 (2007), 1155-1170.
doi: 10.1080/10236190701464996. |
[22] |
M. K. Singh, B. S. Bhadauria and B. K. Singh,
Bifurcation analysis of modified leslie-gower predator-prey model with double allee effect, Ain Shams Engineering Journal, 9 (2018), 1263-1277.
doi: 10.1016/j.asej.2016.07.007. |
[23] |
E. van Leeuwen, V. A. A. Jansen and P. W. Bright,
How population dynamics shape the functional response in a one-predator–two-prey system, Ecology, 88 (2007), 1571-1581.
doi: 10.1890/06-1335. |
[24] |
J. Wang, Y. Cai, S. Fu and W. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos, 29 (2019), 083109, 10 pp.
doi: 10.1063/1.5111121. |
[25] |
X. Wang, L. Zanette and X. Zou,
Modelling the fear effect in predator–prey interactions, J. Math. Biol., 73 (2016), 1179-1204.
doi: 10.1007/s00285-016-0989-1. |
[26] |
X. Wang and X. Zou,
Modeling the fear effect in predator–prey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79 (2017), 1325-1359.
doi: 10.1007/s11538-017-0287-0. |
[27] |
Y. Xia and S. Yuan,
Survival analysis of a stochastic predator–prey model with prey refuge and fear effect, J. Biol. Dyn., 14 (2020), 871-892.
doi: 10.1080/17513758.2020.1853832. |
[28] |
Z. Xiao, Z. Li et al., Stability analysis of a mutual interference predator-prey model with the fear effect, Journal of Applied Science and Engineering, 22 (2019), 205–211. |
[29] |
Z. Zhang, R. K. Upadhyay and J. Datta, Bifurcation analysis of a modified Leslie–Gower model with Holling type-Ⅳ functional response and nonlinear prey harvesting, Adv. Difference Equ., 2018 (2018), Paper No. 127, 21 pp.
doi: 10.1186/s13662-018-1581-3. |
[30] |
Z.-Z. Zhang and H.-Z. Yang,
Hopf bifurcation in a delayed predator-prey system with modified Leslie-Gower and Holling type Ⅲ schemes, Acta Automat. Sinica, 39 (2013), 610-616.
doi: 10.3724/SP.J.1004.2013.00610. |
[31] |
Y. Zhu and K. Wang,
Existence and global attractivity of positive periodic solutions for a predator–prey model with modified Leslie–Gower Holling-type Ⅱ schemes, J. Math. Anal. Appl., 384 (2011), 400-408.
doi: 10.1016/j.jmaa.2011.05.081. |














Transformation | |||
Non-dimensional system | where |
Transformation | |||
Non-dimensional system | where |
(a) | ||
Original parameters | ||
Parameter | Description | Value |
$R_1$ | Intrinsic growth rate of prey | 0.03 |
$R_2$ | Intrinsic growth rate of predator | 0.03 |
$B$ | The level of fear | 4 |
$A_1$ | Consumption rate of predator | 0.5 |
$A_2$ | Intraspecific competition of predator | 0.5 |
$K$ | Carrying capacity of prey | 2 |
$\eta$ | Alternative prey density | 0.25 |
$B_1$ | Half saturation coefficient | 0.1 |
$H_1$ | Harvesting rate of prey | 0.01 |
$H_2$ | Harvesting rate of predator | 0.02 |
(b) | ||
Non-dimensional parameters | ||
Parameter | Value | |
$r$ | 1 | |
$\alpha$ | 0.03 | |
$\beta$ | 0.48 | |
$\eta_1$ | 0.125 | |
$b$ | 0.025 | |
$h_1$ | 0.333 | |
$h_2$ | 0.667 |
(a) | ||
Original parameters | ||
Parameter | Description | Value |
$R_1$ | Intrinsic growth rate of prey | 0.03 |
$R_2$ | Intrinsic growth rate of predator | 0.03 |
$B$ | The level of fear | 4 |
$A_1$ | Consumption rate of predator | 0.5 |
$A_2$ | Intraspecific competition of predator | 0.5 |
$K$ | Carrying capacity of prey | 2 |
$\eta$ | Alternative prey density | 0.25 |
$B_1$ | Half saturation coefficient | 0.1 |
$H_1$ | Harvesting rate of prey | 0.01 |
$H_2$ | Harvesting rate of predator | 0.02 |
(b) | ||
Non-dimensional parameters | ||
Parameter | Value | |
$r$ | 1 | |
$\alpha$ | 0.03 | |
$\beta$ | 0.48 | |
$\eta_1$ | 0.125 | |
$b$ | 0.025 | |
$h_1$ | 0.333 | |
$h_2$ | 0.667 |
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Bifurcation parameter | Linear | Holling type Ⅱ | Holling type Ⅲ | |||
Threshold | Bifurcation | Threshold | Bifurcation | Threshold | Bifurcation | |
$h_1$ ($h_2 < r$) |
$h_1^*=0.8971$ | TB | $h_1^{**}=0.1475$ $h_{1sn}^-=0.3685$ |
TB SNB |
$h_1^{\#}=0.79$ | TB |
$h_1$ ($h_2>r$) |
$h_1^*=1$ | TB | $h_1^{**}=1$ | TB | $h_1^{\#}=1$ | TB |
$h_2$ ($h_1 < 1$) |
$h_2^*=1$ | TB | $h_2^{**}=0.74$ $h_{2sn}=0.64$ |
TB SNB |
$h_2^{\#}=1$ | TB |
$\beta$ ($h_1 < 1$ & $h_2 < r$) |
$\beta^*=16.8$ | TB | $\beta^{**}=16$ $\beta_{sn}=18.51$ |
TB SNB |
$\beta^{\#}=1.5$ | TB |
$\eta_1$ ($h_1 < 1$ & $h_2 < r$) |
$\eta_1^*=0.825$ | TB | $\eta_1^{**}=0.2$ $\eta_{1sn}=0.441$ |
TB SNB |
$\eta_1^{\#}=0.3721$ | TB |
Bifurcation parameter | Linear | Holling type Ⅱ | Holling type Ⅲ | |||
Threshold | Bifurcation | Threshold | Bifurcation | Threshold | Bifurcation | |
$h_1$ ($h_2 < r$) |
$h_1^*=0.8971$ | TB | $h_1^{**}=0.1475$ $h_{1sn}^-=0.3685$ |
TB SNB |
$h_1^{\#}=0.79$ | TB |
$h_1$ ($h_2>r$) |
$h_1^*=1$ | TB | $h_1^{**}=1$ | TB | $h_1^{\#}=1$ | TB |
$h_2$ ($h_1 < 1$) |
$h_2^*=1$ | TB | $h_2^{**}=0.74$ $h_{2sn}=0.64$ |
TB SNB |
$h_2^{\#}=1$ | TB |
$\beta$ ($h_1 < 1$ & $h_2 < r$) |
$\beta^*=16.8$ | TB | $\beta^{**}=16$ $\beta_{sn}=18.51$ |
TB SNB |
$\beta^{\#}=1.5$ | TB |
$\eta_1$ ($h_1 < 1$ & $h_2 < r$) |
$\eta_1^*=0.825$ | TB | $\eta_1^{**}=0.2$ $\eta_{1sn}=0.441$ |
TB SNB |
$\eta_1^{\#}=0.3721$ | TB |
Parameters | Largest basin of recovery | Smallest basin of recovery |
Linear | Holling type Ⅱ | |
Linear | Holling type Ⅲ | |
Holling type Ⅲ | Holling type Ⅱ | |
Linear | Holling type Ⅱ | |
Linear | Holling type Ⅱ |
Parameters | Largest basin of recovery | Smallest basin of recovery |
Linear | Holling type Ⅱ | |
Linear | Holling type Ⅲ | |
Holling type Ⅲ | Holling type Ⅱ | |
Linear | Holling type Ⅱ | |
Linear | Holling type Ⅱ |
[1] |
Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233 |
[2] |
Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419 |
[3] |
Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923 |
[4] |
Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203 |
[5] |
Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 |
[6] |
Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 |
[7] |
John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 |
[8] |
Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 |
[9] |
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 |
[10] |
Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 |
[11] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[12] |
Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209 |
[13] |
Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 |
[14] |
Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 |
[15] |
Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71 |
[16] |
R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure and Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 |
[17] |
Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 |
[18] |
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 |
[19] |
Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 |
[20] |
Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022069 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]