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doi: 10.3934/dcdsb.2021252
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Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting

1. 

School of Mathematics and Physics, Wuhan Institute of Technology, Wuhan, Hubei 430205, China

2. 

School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China, Institute for Artificial Intelligence, Southwest Petroleum University, Chengdu, Sichuan 610500, China

* Corresponding author: Lingling Liu

Received  March 2021 Revised  July 2021 Early access October 2021

Fund Project: The first author is supported by NSFC #12101470, WIT #K2021077. The second author is supported by NSFC #11771308, #11871041, SWPU #2017CXTD02, #18TD0013, #2019CXTD08

In this paper, we study the dynamics of a Leslie-Gower predator-prey system with hunting cooperation among predator population and constant-rate harvesting for prey population. It is shown that there are a weak focus of multiplicity up to three and a cusp of codimension at most two for various parameter values, and the system exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension two and a Hopf bifurcation as the bifurcation parameters vary. The results developed in this article reveal far more complex dynamics compared to the Leslie-Gower system and show how the prey harvesting and the hunting cooperation affect the dynamics of the system. In particular, there exist some critical values of prey harvesting and hunting cooperation such that the predator and prey populations are at risk of extinction if the intensities of harvesting and hunting cooperation are greater than these critical values. Moreover, numerical simulations are presented to illustrate our theoretical results.

Citation: Yong Yao, Lingling Liu. Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021252
References:
[1]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.  doi: 10.1016/S0022-5193(89)80211-5.  Google Scholar

[2]

N. Bacaër, A Short History of Mathematical Population Dynamics, Springer Verlag, New York, 2011. doi: 10.1007/978-0-85729-115-8.  Google Scholar

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[4]

L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72 (2010), 94-121.  doi: 10.1007/s11538-009-9439-1.  Google Scholar

[5]

J. Carr, Applications of Center Manifold Theory, Springer, New York, 1981.  Google Scholar

[6]

X. Chen and W. Zhang, Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232 (2009), 565-581.  doi: 10.1016/j.cam.2009.06.029.  Google Scholar

[7]

C. CosnerD. L. DeAngelisJ. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65-75.  doi: 10.1006/tpbi.1999.1414.  Google Scholar

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.  Google Scholar

[9]

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.  Google Scholar

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[11]

R. P. GuptaM. 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.  Google Scholar

[12]

R. P. GuptaP. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 423-443.  doi: 10.3934/dcdsb.2015.20.423.  Google Scholar

[13]

M. P. Hassell and G. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137.  doi: 10.1038/2231133a0.  Google Scholar

[14]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar

[15]

D. Hu and H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal. Real World Appl., 33 (2017), 58-82.  doi: 10.1016/j.nonrwa.2016.05.010.  Google Scholar

[16]

J. Huang, Y. Gong and J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350164, 24 pp. doi: 10.1142/S0218127413501642.  Google Scholar

[17]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101.  Google Scholar

[18]

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

[19]

R. KimunK. Wonlyul and H. Mainul, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dynam., 94 (2018), 1639-1656.   Google Scholar

[20]

L. Kong and C. Zhu, Bogdanov-Takens bifurcations of codimensions 2 and 3 in a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting, Math. Meth. Appl. Sci., 40 (2017), 6715-6731.  doi: 10.1002/mma.4484.  Google Scholar

[21]

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.  Google Scholar

[22]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1995. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[23]

K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Anal. Real World Appl., 12 (2011), 1961-1973.  doi: 10.1016/j.nonrwa.2010.12.012.  Google Scholar

[24]

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.  Google Scholar

[25]

A. Lotka, Elements of Physical Biology, Williams and Williams, Baltimore, 1925. Google Scholar

[26] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 1973.   Google Scholar
[27]

R. M. MayJ. R. BeddingtonC. W. ClarkS. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277.  doi: 10.1126/science.205.4403.267.  Google Scholar

[28]

S. PalN. PalS. Samanta and J. Chattopadhyay, Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model, Math. Biosci. Eng., 16 (2019), 5146-5179.  doi: 10.3934/mbe.2019258.  Google Scholar

[29]

E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.  Google Scholar

[30]

D. Scheel and C. Packer, Group hunting behavior of lions: A search for cooperation, Anim. Behav., 41 (1991), 697-709.  doi: 10.1016/S0003-3472(05)80907-8.  Google Scholar

[31]

G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type II model, Ecology, 82 (2001), 3083-3092.   Google Scholar

[32]

M. Teixeira Alves and F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13-22.  doi: 10.1016/j.jtbi.2017.02.002.  Google Scholar

[33] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, New Jersey, 2003.   Google Scholar
[34]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.  Google Scholar

[35]

D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.  doi: 10.1137/S0036139903428719.  Google Scholar

[36]

D. XiaoW. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.  doi: 10.1016/j.jmaa.2005.11.048.  Google Scholar

[37]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Ins. Commun., 21 (1999), 493-506.   Google Scholar

[38]

Y. Yao, Dynamics of a prey-predator system with foraging facilitation in predators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050009, 24 pp. doi: 10.1142/S0218127420500091.  Google Scholar

[39]

P. Ye and D. Wu, Dynamics of a prey-predator system with foraging facilitation in predators, Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system, Chinese J. Phys., 68 (2020), 49-64.  doi: 10.1016/j.cjph.2020.07.021.  Google Scholar

[40]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[41]

C. Zhu and K. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.  Google Scholar

show all references

References:
[1]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.  doi: 10.1016/S0022-5193(89)80211-5.  Google Scholar

[2]

N. Bacaër, A Short History of Mathematical Population Dynamics, Springer Verlag, New York, 2011. doi: 10.1007/978-0-85729-115-8.  Google Scholar

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[4]

L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72 (2010), 94-121.  doi: 10.1007/s11538-009-9439-1.  Google Scholar

[5]

J. Carr, Applications of Center Manifold Theory, Springer, New York, 1981.  Google Scholar

[6]

X. Chen and W. Zhang, Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232 (2009), 565-581.  doi: 10.1016/j.cam.2009.06.029.  Google Scholar

[7]

C. CosnerD. L. DeAngelisJ. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65-75.  doi: 10.1006/tpbi.1999.1414.  Google Scholar

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.  Google Scholar

[9]

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.  Google Scholar

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[11]

R. P. GuptaM. 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.  Google Scholar

[12]

R. P. GuptaP. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 423-443.  doi: 10.3934/dcdsb.2015.20.423.  Google Scholar

[13]

M. P. Hassell and G. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137.  doi: 10.1038/2231133a0.  Google Scholar

[14]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar

[15]

D. Hu and H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal. Real World Appl., 33 (2017), 58-82.  doi: 10.1016/j.nonrwa.2016.05.010.  Google Scholar

[16]

J. Huang, Y. Gong and J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350164, 24 pp. doi: 10.1142/S0218127413501642.  Google Scholar

[17]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101.  Google Scholar

[18]

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

[19]

R. KimunK. Wonlyul and H. Mainul, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dynam., 94 (2018), 1639-1656.   Google Scholar

[20]

L. Kong and C. Zhu, Bogdanov-Takens bifurcations of codimensions 2 and 3 in a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting, Math. Meth. Appl. Sci., 40 (2017), 6715-6731.  doi: 10.1002/mma.4484.  Google Scholar

[21]

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.  Google Scholar

[22]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1995. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[23]

K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Anal. Real World Appl., 12 (2011), 1961-1973.  doi: 10.1016/j.nonrwa.2010.12.012.  Google Scholar

[24]

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.  Google Scholar

[25]

A. Lotka, Elements of Physical Biology, Williams and Williams, Baltimore, 1925. Google Scholar

[26] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 1973.   Google Scholar
[27]

R. M. MayJ. R. BeddingtonC. W. ClarkS. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277.  doi: 10.1126/science.205.4403.267.  Google Scholar

[28]

S. PalN. PalS. Samanta and J. Chattopadhyay, Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model, Math. Biosci. Eng., 16 (2019), 5146-5179.  doi: 10.3934/mbe.2019258.  Google Scholar

[29]

E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.  Google Scholar

[30]

D. Scheel and C. Packer, Group hunting behavior of lions: A search for cooperation, Anim. Behav., 41 (1991), 697-709.  doi: 10.1016/S0003-3472(05)80907-8.  Google Scholar

[31]

G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type II model, Ecology, 82 (2001), 3083-3092.   Google Scholar

[32]

M. Teixeira Alves and F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13-22.  doi: 10.1016/j.jtbi.2017.02.002.  Google Scholar

[33] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, New Jersey, 2003.   Google Scholar
[34]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.  Google Scholar

[35]

D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.  doi: 10.1137/S0036139903428719.  Google Scholar

[36]

D. XiaoW. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.  doi: 10.1016/j.jmaa.2005.11.048.  Google Scholar

[37]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Ins. Commun., 21 (1999), 493-506.   Google Scholar

[38]

Y. Yao, Dynamics of a prey-predator system with foraging facilitation in predators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050009, 24 pp. doi: 10.1142/S0218127420500091.  Google Scholar

[39]

P. Ye and D. Wu, Dynamics of a prey-predator system with foraging facilitation in predators, Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system, Chinese J. Phys., 68 (2020), 49-64.  doi: 10.1016/j.cjph.2020.07.021.  Google Scholar

[40]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[41]

C. Zhu and K. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.  Google Scholar

Figure 1.  Saddle-nodes of system (5) when $ \alpha = \beta = \gamma = 1 $. (A) Saddle-nodes $ E_{1*} $ when $ h = 0.25 $. (B) Saddle-nodes $ E_{2*} $ when $ h = 0.1126 $
Figure 2.  Codimension 2 cusp when $ \beta = 1 $, $ \gamma = 0.3 $, $ \alpha = 0.82645 $ and $ h = 0.1144 $
Figure 3.  The Bogdanov-Takens bifurcation diagram and corresponding phase portraits of system (5) when $ \beta = 1 $ and $ \gamma = 0.3 $. (A) Bifurcation diagram. (B) No equilibrium when $ (\alpha, h) = (1.2, 0.1108)\in\mathcal{I}_1 $. (C) An unstable focus $ E_{22} $ and a saddle $ E_{21} $ when $ (\alpha, h) = (1.5, 0.108)\in\mathcal{I}_2 $. (D) An unstable limit cycle around a stable focus $ E_{22} $ and a saddle $ E_{21} $ when $ (\alpha, h) = (1.5, 0.1078)\in\mathcal{I}_3 $. (E) A homoclinic orbit when $ (\alpha, h) = (1.2, 0.11057)\in\mathcal{HL} $. (F) A stable focus $ E_{22} $ and a saddle $ E_{21} $ when $ (\alpha, h) = (1.5, 0.107)\in\mathcal{I}_4 $
Figure 4.  Limit cycles induced by Hopf bifurcation at $ E_{22} $ in system (5). (A) An unstable limit cycle around $ E_{22} $ with $ \beta = 1.8 $, $ \gamma = 0.3 $, $ \alpha = 10.49382716 $ and $ h = 0.038 $. (B) Two limit cycles with $ \beta = 8 $, $ \gamma = 0.1 $, $ \alpha = 86.65647719836 $ and $ h = 0.0013277768 $. (C) The orbit from $ P_2 $ spirals outward. (D) The orbit from $ P_3 $ spirals inward
Table 1.  Dynamical behaviors near $ E_{2*} $
Parameters Equilibria and properties Closed orbits and homoclinic orbits
$ (\beta, \gamma) $ $ (h, \alpha) $
$(0,+\infty)$ $\times[\frac{1}{2}, \frac{2}{3})$ or $(0,\frac{2\gamma}{1-2\gamma})$ $\times(0, \frac{1}{2})$ $\mathcal{I}_{1}$ No equilibria No
$ \mathcal{SN}^+\cup\mathcal{SN}^- $ $ E_{2*} $(saddle node) No
$ \mathcal{I}_{2} $ $E_{21}$(saddle)
$E_{22}$(unstable focus or node)
No
$ \mathcal{H} $ $E_{21}$(saddle)
$E_{22}$(unstable weak focus)
No
$ \mathcal{I}_3 $ $E_{21}$(saddle)
$E_{22}$(stable focus)
A unstable limit cycle
$ \mathcal{HL} $ $E_{21}$(saddle)
$E_{22}$(stable focus)
A homoclinic orbit
$ \mathcal{I}_{4} $ $E_{21}$(saddle)
$E_{22}$(stable focus or node)
No
$ (h_*, \alpha_*) $ $ E_{2*} $(cusp) No
Parameters Equilibria and properties Closed orbits and homoclinic orbits
$ (\beta, \gamma) $ $ (h, \alpha) $
$(0,+\infty)$ $\times[\frac{1}{2}, \frac{2}{3})$ or $(0,\frac{2\gamma}{1-2\gamma})$ $\times(0, \frac{1}{2})$ $\mathcal{I}_{1}$ No equilibria No
$ \mathcal{SN}^+\cup\mathcal{SN}^- $ $ E_{2*} $(saddle node) No
$ \mathcal{I}_{2} $ $E_{21}$(saddle)
$E_{22}$(unstable focus or node)
No
$ \mathcal{H} $ $E_{21}$(saddle)
$E_{22}$(unstable weak focus)
No
$ \mathcal{I}_3 $ $E_{21}$(saddle)
$E_{22}$(stable focus)
A unstable limit cycle
$ \mathcal{HL} $ $E_{21}$(saddle)
$E_{22}$(stable focus)
A homoclinic orbit
$ \mathcal{I}_{4} $ $E_{21}$(saddle)
$E_{22}$(stable focus or node)
No
$ (h_*, \alpha_*) $ $ E_{2*} $(cusp) No
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