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doi: 10.3934/dcdsb.2021298
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A predator-prey model with cooperative hunting in the predator and group defense in the prey

1. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

School of Mathematics & Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China

3. 

Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China

4. 

School of Science, Jimei University, Xiamen 361021, China

*Corresponding author: Ben Niu

Received  July 2021 Revised  November 2021 Early access December 2021

In this paper we propose a predator-prey model with a non-differentiable functional response in which the prey exhibits group defense and the predator exhibits cooperative hunting. There is a separatrix curve dividing the phase portrait. The species with initial population above the separatrix result in extinction of prey in finite time, and the species with initial population below it can coexist, oscillate sustainably or leave the prey surviving only. Detailed bifurcation analysis is carried out to explore the effect of cooperative hunting in the predator and aggregation in the prey on the existence and stability of the coexistence state as well as the dynamics of system. The model undergoes transcritical bifurcation, Hopf bifurcation, homoclinic (heteroclinic) bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation, and through numerical simulations it is found that it possesses rich dynamics including bubble loop of limit cycles, and open ended branch of periodic orbits disappearing through a homoclinic cycle or a loop of heteroclinic orbits. Also, a continuous transition of different types of Hopf branches are investigated which forms a global picture of Hopf bifurcation in the model.

Citation: Yanfei Du, Ben Niu, Junjie Wei. A predator-prey model with cooperative hunting in the predator and group defense in the prey. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021298
References:
[1]

V. AjraldiM. Pittavino and E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal. RWA, 12 (2011), 2319-2338.  doi: 10.1016/j.nonrwa.2011.02.002.

[2]

V. AjraldiE. Venturino and B. Wade, Mimicking spatial effects in predator-prey models with group defense, Proc. Int. Conf. CMMSE, 1 (2009), 57-66. 

[3]

Q. An and W. H. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 487-510.  doi: 10.3934/dcdsb.2018183.

[4]

M. T. 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.

[5]

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.

[6]

C. Boesch, Cooperative hunting in wild chimpanzees, Anim. Behav., 48 (1994), 653-667.  doi: 10.1006/anbe.1994.1285.

[7]

P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal. RWA, 13 (2012), 1837-1843.  doi: 10.1016/j.nonrwa.2011.12.014.

[8]

R. Bshary, A. Hohner, K. Ait-el-Djoudi and H. Fricke, Interspecific communicative and coordinated hunting between groupers and giant moray eels in the Red Sea, PLoS Biol., 4 (2006). doi: 10.1371/journal.pbio.0040431.

[9]

I. M. Bulai and E. Venturino, Shape effects on herd behavior in ecological interacting population models, Math. Comput. Simulation, 141 (2017), 40-55.  doi: 10.1016/j.matcom.2017.04.009.

[10]

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.

[11]

F. Courchamp and D. W. Macdonald, Crucial importance of pack size in the African wild dog Lycaon pictus, Anim. Conserv., 4 (2001), 169-174.  doi: 10.1017/S1367943001001196.

[12]

A. DhoogeW. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Softw., 29 (2003), 141-164.  doi: 10.1145/779359.779362.

[13]

S. Djilali, Impact of prey herd shape on the predator-prey interaction, Chaos Solitons Fractals, 120 (2019), 139-148.  doi: 10.1016/j.chaos.2019.01.022.

[14] L. A. Dugatkin, Cooperation Among Animals: An Evolutionary Perspective, Oxford University Press, Oxford, 1997. 
[15]

H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508. 

[16]

D. P. Hector, Cooperative hunting and its relationship to foraging success and prey size in an avian predator, Ethology, 73 (1986), 247-257.  doi: 10.1111/j.1439-0310.1986.tb00915.x.

[17]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[18]

J. C. Holmes and W. M. Bethel, Modification of intermediate host behavior by parasites, Zoolog J. Linnean Soc., 51 (1972), 123-149. 

[19]

S. R. J. JangW. Zhang and V. Larriva, Cooperative hunting in a predator-prey system with Allee effects in the prey, Nat. Resour. Model, 31 (2018), 12194.  doi: 10.1111/nrm.12194.

[20]

N. D. Kazarinov and P. V. D. Driessche, A model predator-prey system with functional response, Math. Biosci., 39 (1978), 125-134.  doi: 10.1016/0025-5564(78)90031-7.

[21]

M. C. KhnkeI. Siekmann and H. Malchow, Taxis-driven pattern formation in a predator-prey model with group defense, Ecol. Complex., 43 (2020), 100848.  doi: 10.1016/j.ecocom.2020.100848.

[22]

B. W. Kooi and E. Venturino, Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey, Math. Biosci., 274 (2016), 58-72.  doi: 10.1016/j.mbs.2016.02.003.

[23]

Y. A. Kuznetsov, Element of Applied Bifurcation Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 1998.

[24]

M. W. Moffett, Foraging dynamics in the group-hunting myrmicine ant, pheidologeton diversus, J. Insect. Behav., 1 (1988), 309-331.  doi: 10.1007/BF01054528.

[25]

S. PalN. PalS. Samanta and J. Chattopadhyay, Effect of hunting cooperation and fear in a predator-prey model, Ecol. Complex, 39 (2019), 100770.  doi: 10.1016/j.ecocom.2019.100770.

[26]

J. A. Polking and D. Arnold, Ordinary Differential Equations Using MATLAB, Prentice-Hall, Englewood Cliffs, 2003.

[27]

S. N. RawP. MishraR. Kumar and S. Thakur, Complex behavior of prey-predator system exhibiting group defense: A mathematical modeling study, Chaos Soliton Fract., 100 (2017), 74-90.  doi: 10.1016/j.chaos.2017.05.010.

[28]

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

[29]

P. A. Schmidt and L. D. Mech, Wolf pack size and food acquisition, Am. Nat., 150 (1997), 513-517.  doi: 10.1086/286079.

[30]

D. SenS. Ghorai and M. Banerjee, Allee effect in prey versus hunting cooperation on predator-enhancement of stable coexistence, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950081.  doi: 10.1142/S0218127419500810.

[31]

D. SongC. Li and Y. Song, Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation functional response, Nonlinear Anal. RWA, 54 (2020), 103106.  doi: 10.1016/j.nonrwa.2020.103106.

[32]

Y. SongY. Peng and T. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differential Equations, 300 (2021), 597-624.  doi: 10.1016/j.jde.2021.08.010.

[33]

Y. Song, J. Shi and H. Wang, Spatiotemporal dynamics of a diffusive consumer-resource model with explicit spatial memory, Stud. Appl. Math., (2021), 1–23 doi: 10.1111/sapm.12443.

[34]

J. S. Tener, Muskoxen, , Queen's Printer, Ottawa, 1995.

[35]

G. W. Uetz, Foraging strategies of spiders, Trends. Ecol. Evol., 7 (1992), 155-159.  doi: 10.1016/0169-5347(92)90209-T.

[36]

E. Venturino and S. Petrovskii, Spatiotemporal behavior of a prey-predator system with a group defense for prey, Ecol. Complex, 14 (2013), 37-47.  doi: 10.1016/j.ecocom.2013.01.004.

[37]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003.

[38]

D. Wu and M. Zhao, Qualitative analysis for a diffusive predator-prey model with hunting cooperative, Physica A, 515 (2019), 299-309.  doi: 10.1016/j.physa.2018.09.176.

[39]

Z. Xu and Y. Song, Bifurcation analysis of a diffusive predator-prey system with a herd behavior and quadratic mortality, Math. Method Appl. Sci., 38 (2015), 2994-3006.  doi: 10.1002/mma.3275.

[40]

C. XuS. Yuan and T. Zhang, Global dynamics of a predator-prey model with defense mechanism for prey, Appl. Math. Lett., 62 (2016), 42-48.  doi: 10.1016/j.aml.2016.06.013.

[41]

S. YanD. JiaT. Zhang and S. Yuan, Pattern dynamics in a diffusive predator-prey model with hunting cooperations, Chaos Solitons Fract., 130 (2020), 109428.  doi: 10.1016/j.chaos.2019.109428.

[42]

H. Yin and X. Wen, Hopf bifurcation of a diffusive Gause-type predator-prey model induced by time fractional-order derivatives, Math. Method Appl. Sci., 41 (2018), 5178-5189.  doi: 10.1002/mma.5066.

[43]

S. YuanC. Xu and T. Zhang, Spatial dynamics in a predator-prey model with herd behavior, Chaos, 23 (2013), 033102.  doi: 10.1063/1.4812724.

show all references

References:
[1]

V. AjraldiM. Pittavino and E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal. RWA, 12 (2011), 2319-2338.  doi: 10.1016/j.nonrwa.2011.02.002.

[2]

V. AjraldiE. Venturino and B. Wade, Mimicking spatial effects in predator-prey models with group defense, Proc. Int. Conf. CMMSE, 1 (2009), 57-66. 

[3]

Q. An and W. H. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 487-510.  doi: 10.3934/dcdsb.2018183.

[4]

M. T. 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.

[5]

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.

[6]

C. Boesch, Cooperative hunting in wild chimpanzees, Anim. Behav., 48 (1994), 653-667.  doi: 10.1006/anbe.1994.1285.

[7]

P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal. RWA, 13 (2012), 1837-1843.  doi: 10.1016/j.nonrwa.2011.12.014.

[8]

R. Bshary, A. Hohner, K. Ait-el-Djoudi and H. Fricke, Interspecific communicative and coordinated hunting between groupers and giant moray eels in the Red Sea, PLoS Biol., 4 (2006). doi: 10.1371/journal.pbio.0040431.

[9]

I. M. Bulai and E. Venturino, Shape effects on herd behavior in ecological interacting population models, Math. Comput. Simulation, 141 (2017), 40-55.  doi: 10.1016/j.matcom.2017.04.009.

[10]

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.

[11]

F. Courchamp and D. W. Macdonald, Crucial importance of pack size in the African wild dog Lycaon pictus, Anim. Conserv., 4 (2001), 169-174.  doi: 10.1017/S1367943001001196.

[12]

A. DhoogeW. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Softw., 29 (2003), 141-164.  doi: 10.1145/779359.779362.

[13]

S. Djilali, Impact of prey herd shape on the predator-prey interaction, Chaos Solitons Fractals, 120 (2019), 139-148.  doi: 10.1016/j.chaos.2019.01.022.

[14] L. A. Dugatkin, Cooperation Among Animals: An Evolutionary Perspective, Oxford University Press, Oxford, 1997. 
[15]

H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508. 

[16]

D. P. Hector, Cooperative hunting and its relationship to foraging success and prey size in an avian predator, Ethology, 73 (1986), 247-257.  doi: 10.1111/j.1439-0310.1986.tb00915.x.

[17]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[18]

J. C. Holmes and W. M. Bethel, Modification of intermediate host behavior by parasites, Zoolog J. Linnean Soc., 51 (1972), 123-149. 

[19]

S. R. J. JangW. Zhang and V. Larriva, Cooperative hunting in a predator-prey system with Allee effects in the prey, Nat. Resour. Model, 31 (2018), 12194.  doi: 10.1111/nrm.12194.

[20]

N. D. Kazarinov and P. V. D. Driessche, A model predator-prey system with functional response, Math. Biosci., 39 (1978), 125-134.  doi: 10.1016/0025-5564(78)90031-7.

[21]

M. C. KhnkeI. Siekmann and H. Malchow, Taxis-driven pattern formation in a predator-prey model with group defense, Ecol. Complex., 43 (2020), 100848.  doi: 10.1016/j.ecocom.2020.100848.

[22]

B. W. Kooi and E. Venturino, Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey, Math. Biosci., 274 (2016), 58-72.  doi: 10.1016/j.mbs.2016.02.003.

[23]

Y. A. Kuznetsov, Element of Applied Bifurcation Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 1998.

[24]

M. W. Moffett, Foraging dynamics in the group-hunting myrmicine ant, pheidologeton diversus, J. Insect. Behav., 1 (1988), 309-331.  doi: 10.1007/BF01054528.

[25]

S. PalN. PalS. Samanta and J. Chattopadhyay, Effect of hunting cooperation and fear in a predator-prey model, Ecol. Complex, 39 (2019), 100770.  doi: 10.1016/j.ecocom.2019.100770.

[26]

J. A. Polking and D. Arnold, Ordinary Differential Equations Using MATLAB, Prentice-Hall, Englewood Cliffs, 2003.

[27]

S. N. RawP. MishraR. Kumar and S. Thakur, Complex behavior of prey-predator system exhibiting group defense: A mathematical modeling study, Chaos Soliton Fract., 100 (2017), 74-90.  doi: 10.1016/j.chaos.2017.05.010.

[28]

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

[29]

P. A. Schmidt and L. D. Mech, Wolf pack size and food acquisition, Am. Nat., 150 (1997), 513-517.  doi: 10.1086/286079.

[30]

D. SenS. Ghorai and M. Banerjee, Allee effect in prey versus hunting cooperation on predator-enhancement of stable coexistence, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950081.  doi: 10.1142/S0218127419500810.

[31]

D. SongC. Li and Y. Song, Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation functional response, Nonlinear Anal. RWA, 54 (2020), 103106.  doi: 10.1016/j.nonrwa.2020.103106.

[32]

Y. SongY. Peng and T. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differential Equations, 300 (2021), 597-624.  doi: 10.1016/j.jde.2021.08.010.

[33]

Y. Song, J. Shi and H. Wang, Spatiotemporal dynamics of a diffusive consumer-resource model with explicit spatial memory, Stud. Appl. Math., (2021), 1–23 doi: 10.1111/sapm.12443.

[34]

J. S. Tener, Muskoxen, , Queen's Printer, Ottawa, 1995.

[35]

G. W. Uetz, Foraging strategies of spiders, Trends. Ecol. Evol., 7 (1992), 155-159.  doi: 10.1016/0169-5347(92)90209-T.

[36]

E. Venturino and S. Petrovskii, Spatiotemporal behavior of a prey-predator system with a group defense for prey, Ecol. Complex, 14 (2013), 37-47.  doi: 10.1016/j.ecocom.2013.01.004.

[37]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003.

[38]

D. Wu and M. Zhao, Qualitative analysis for a diffusive predator-prey model with hunting cooperative, Physica A, 515 (2019), 299-309.  doi: 10.1016/j.physa.2018.09.176.

[39]

Z. Xu and Y. Song, Bifurcation analysis of a diffusive predator-prey system with a herd behavior and quadratic mortality, Math. Method Appl. Sci., 38 (2015), 2994-3006.  doi: 10.1002/mma.3275.

[40]

C. XuS. Yuan and T. Zhang, Global dynamics of a predator-prey model with defense mechanism for prey, Appl. Math. Lett., 62 (2016), 42-48.  doi: 10.1016/j.aml.2016.06.013.

[41]

S. YanD. JiaT. Zhang and S. Yuan, Pattern dynamics in a diffusive predator-prey model with hunting cooperations, Chaos Solitons Fract., 130 (2020), 109428.  doi: 10.1016/j.chaos.2019.109428.

[42]

H. Yin and X. Wen, Hopf bifurcation of a diffusive Gause-type predator-prey model induced by time fractional-order derivatives, Math. Method Appl. Sci., 41 (2018), 5178-5189.  doi: 10.1002/mma.5066.

[43]

S. YuanC. Xu and T. Zhang, Spatial dynamics in a predator-prey model with herd behavior, Chaos, 23 (2013), 033102.  doi: 10.1063/1.4812724.

Figure 1.  When $ (\frac{1}{\beta})^{\frac{1}{\alpha}}<K $, there is a unique interior equilibrium
Figure 2.  a) Bifurcation diagram of system (2) with $ r = 0.7,K = 4,\alpha = 0.5,\beta = 0.7 $. b) The period of the limit cycles. c) When $ c = 0.3>c_H $, $ E^* $ loses its stability, and there is a limit cycle. d) When $ c = c_{het} = 0.432 $, there is a loop of heteroclinic orbits
Figure 3.  If $ (\frac{1}{\beta})^{\frac{1}{\alpha}} >K $, the black curve is the left side of Eq. (8), the blue ones denote the right side of Eq. (8) with three different $ c $ values
Figure 4.  a) Bifurcation diagram of system (2) with $ r = 2,K = 4,\alpha = 0.5,\beta = 0.35 $. b) When $ c = c_{hom} = 1.662 $, there is a homoclinic cycle
Figure 5.  a) The diagram of $ u- $nullclines with $ \alpha_1<\alpha_2 $. b) The diagram of $ v- $nullclines with $ \alpha_1<\alpha_2 $
Figure 6.  The diagrams of $ u- $nullclines and $ v- $nullclines for different values of $ \alpha $ for a) case (Ⅰ-ⅰa); b) case (Ⅰ-ⅰb); c) case (Ⅰ-ⅰc) and d) case (Ⅰ-ⅱ)
Figure 7.  Choose $ r = 2, K = 2, c = 2, \beta = 0.7 $ for case (I-ia). a) The bifurcation diagram of $ \alpha-u-v $. b) When $ \alpha = \alpha_{het} = 0.688 $, there is a loop of heteroclinic orbits
Figure 8.  Choose $ r = 2, K = 1.2, c = 2, \beta = 0.7 $ for case (I-ia). a) The bifurcation diagram of $ \alpha-u-v $. b) When $ \alpha = 0.41 $, there is a homoclinic cycle
Figure 9.  Choose $ r = 2, K = 1.2, c = 1.5, \beta = 0.7 $ for case (I-ib). a) The bifurcation diagram of $ \alpha-u-v $. b) When $ \alpha = 0.26 $, there is a homoclinic cycle
Figure 10.  Choose $ r = 1.7, K = 7, c = 0.3, \beta = 0.7 $ for case (I-ic). a) The bifurcation diagram of $ \alpha-u-v $. b) When $ \alpha = 0.86 $, there is a loop of heteroclinic orbits
Figure 11.  The bifurcation diagram for case (I-ii) choosing $ K = 4,c = 0.3,\beta = 0.7 $ and a) $ r = 0.7 $; b) $ r = 1.033 $; c) $ r = 1.05 $ d) $ r = 1.4 $
Figure 12.  a) The diagram of $ u- $nullclines and $ v- $nullclines for different $ \alpha $ ($ \alpha_1<\alpha_2 $) when $ K>1,\beta>1 $ (case Ⅱ). b) Bifurcation diagram of $ \alpha-u-v $ with $ r = 1.7, K = 2.1, c = 0.7, \beta = 1.1 $. c) Choosing $ r = 1.7, K = 2.1, c = 0.7, \beta = 1.1 $, there is a loop of heteroclinic orbits when $ \alpha = 0.735 $
Figure 13.  a) The bifurcation diagram for case (Ⅲ-ⅰ) with $ r = 1, K = 0.8,c = 3, \beta = 1.3 $. b) When $ \alpha = 0.63 $, there is a loop of heteroclinic orbits
Figure 14.  a) The diagram of $ u- $nullclines and $ v- $nullclines for different values of $ \alpha $ when $ K<1,\beta>1, \frac{1}{\beta}>K $ (case Ⅲ-ⅱ). b) The bifurcation diagram of $ \alpha-u-v $ with $ r = 1, K = 0.5,c = 3, \beta = 1.2 $. c) Choosing $ r = 1, K = 0.5,c = 3, \beta = 1.2 $, there is a homoclinic cycle when $ \alpha = 0.3899 $
Figure 15.  a) The diagram of $ u- $nullclines and $ v- $nullclines for different values of $ \alpha $ when $ K<1,\beta<1 $ (case (Ⅳ-ⅰ)). b) Bifurcation diagram of $ \alpha-u-v $ with $ r = 2, K = 0.8, c = 2, \beta = 0.9 $. c) Choose $ r = 2, K = 0.8, c = 2, \beta = 0.9 $, there is a homoclinic cycle when $ \alpha = 0.4065 $
Figure 16.  The bifurcation set near Bogdanov-Takens bifurcation point BT with $ r = 2, K = 4, \beta = 0.35 $
Table 1.  Main results for system (2) taking $ c $ as bifurcation parameter
$ Case $ $ \begin{array}{l} Interior \;equilibria \end{array} $ $ \begin{array}{l} Type \; of \;bifurcation \end{array} $ $ \begin{array}{l} Shape\; of\; Hopf \;branch \end{array} $ $ \begin{array}{l} Periodic \;orbits\; disappear \;through \end{array} $ $ Figure $
$ (\frac{1}{\beta})^{\frac{1}{\alpha}}<K $ $ one $ $ \begin{array}{l} Hopf \end{array} $ $ \begin{array}{l} open \; ended \end{array} $ $ \begin{array}{l} a \; loop \; of \;heteroclinic \;orbits \end{array} $ Figure 1
Figure 2
$ (\frac{1}{\beta})^{\frac{1}{\alpha}}>K $ $ \begin{array}{ll} c>c_{sn}\\ c=c_{sn} \\ c<c_{sn}\end{array} $ two
one
none
$ \begin{array}{l} saddle-node \;Hopf \end{array} $ $ \begin{array}{l} open \; ended \end{array} $ $ \begin{array}{l} a\; homoclinic \;cycle \end{array} $ Figure 4
$ Case $ $ \begin{array}{l} Interior \;equilibria \end{array} $ $ \begin{array}{l} Type \; of \;bifurcation \end{array} $ $ \begin{array}{l} Shape\; of\; Hopf \;branch \end{array} $ $ \begin{array}{l} Periodic \;orbits\; disappear \;through \end{array} $ $ Figure $
$ (\frac{1}{\beta})^{\frac{1}{\alpha}}<K $ $ one $ $ \begin{array}{l} Hopf \end{array} $ $ \begin{array}{l} open \; ended \end{array} $ $ \begin{array}{l} a \; loop \; of \;heteroclinic \;orbits \end{array} $ Figure 1
Figure 2
$ (\frac{1}{\beta})^{\frac{1}{\alpha}}>K $ $ \begin{array}{ll} c>c_{sn}\\ c=c_{sn} \\ c<c_{sn}\end{array} $ two
one
none
$ \begin{array}{l} saddle-node \;Hopf \end{array} $ $ \begin{array}{l} open \; ended \end{array} $ $ \begin{array}{l} a\; homoclinic \;cycle \end{array} $ Figure 4
Table 2.  Main results for system (2) taking $ \alpha $ as bifurcation parameter when $ K\neq 1 $ and $ \beta\neq 1 $
$ Case$ $Condition$ $\begin{array}{l} Interior \;equilibria \end{array}$ $\begin{array}{l} Type\; of \;Bifurcation \end{array}$ $\begin{array}{l} Shape\; of\; Hopf\; branch \end{array}$ $\begin{array}{l} Periodic \;orbits \; disappear \; through \end{array}$ Figure
$ Case \; (I-ia)$ $\begin{array}{l} K>1, \beta<1\\c>c_2 \end{array}$ $\begin{array}{ll} 0<\alpha<\alpha_T \;\;E_1^*, E_2^* \\ \alpha>\alpha_T \;\;\;\;\;\;\;\;E_1^*\end{array}$ $\begin{array}{l} Hopf \; transcritical \end{array}$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a\; homoclinic \;cycle\; or\; a \;loop\; of\; heteroclinic \;orbits \end{array}$ Figure 6 a)
Figure 7
Figure 8
$Case \; (I-ib)$ $\begin{array}{l} K>1, \beta<1\\c_1<c<c_2, \\K<2 \end{array}$ $\begin{array}{ll} 0<\alpha<\alpha_{sn} \;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_{sn} \;\;\;\;\;\;\;\;none \end{array}$ $\begin{array}{l} saddle-node \; Hopf \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$ Figure 6 b)
Figure 9
$Case \; (I-ic)$ $\begin{array}{l} K>1, \beta<1\\c_1<c<c_2, \\K>2 \end{array}$ $\begin{array}{ll} 0<\alpha<\alpha_T\;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_T\;\;\;\;\;\;\;\;E_1^* \end{array}$ $\begin{array}{l} Hopf \; transcritical \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a \;loop \; of \; heteroclinic \;orbits \end{array}$ Figure 6 c)
Figure 10
$Case \; (I-ii)$ $\begin{array}{l} K>1, \beta<1\\c<c_1 \end{array}$ $\begin{array}{ll} \alpha<\alpha_{sn} \;\;\;\;\;none\\ \alpha_{sn}<\alpha<\alpha_T \;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_T \;\;\;\;E_1^* \end{array}$ $\begin{array}{l} saddle-node \; Hopf \; transcritical \end{array}$ $\begin{array}{l} a \;bubble; \;or open\; ended(two \;branches)\end{array}$ $\begin{array}{l} an \; equilibrium; \;or a\; homoclinic \;cycle \; or \; a \;loop \; of \; heteroclinic \; orbits \end{array}$ Figure 6 d)
Figure 11
$Case \;(II)$ $K>1, \beta>1$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none\\\alpha>\alpha_s \;\;\;\;\;E_1^* \end{array}$ $Hopf$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} \; a \;loop \; of \; heteroclinic \;orbits \end{array}$ Figure 12
$Case \;(III-i)$ $\begin{array}{l} K<1, \beta>1\\\frac{1}{\beta}<K \end{array}$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none\\\alpha>\alpha_s\;\;\;\;\;E_1^* \end{array}$ $Hopf$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a \;loop \; of \; heteroclinic \;orbits \end{array}$ Figure 13
$ Case \; (III-ii)$ $\begin{array}{l} K<1, \beta>1\\\frac{1}{\beta}>K \end{array}$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none \\ \alpha_s<\alpha<\alpha_T \;\;\;\;E_1^* \\ \alpha_T<\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_{sn} \;\;\;\;\;none \end{array}$ $\begin{array}{l} saddle-node \;\\ Hopf \; \\transcritical \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$ Figure 14
$Case \; (IV-i)$ $\begin{array}{l} K<1, \beta<1\\c>c_1 \end{array}$ $\begin{array}{ll} \alpha>\alpha_{sn} \;\;\;\;\;none\\ 0<\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \end{array}$ $\begin{array}{l} saddle-node \; \\ Hopf \; \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$ Figure 15
$ Case \; (IV-ii)$ $\begin{array}{l} K<1, \beta<1\\c<c_1 \end{array}$ $none$ $none$ $none$ $none$ $none$
$ Case$ $Condition$ $\begin{array}{l} Interior \;equilibria \end{array}$ $\begin{array}{l} Type\; of \;Bifurcation \end{array}$ $\begin{array}{l} Shape\; of\; Hopf\; branch \end{array}$ $\begin{array}{l} Periodic \;orbits \; disappear \; through \end{array}$ Figure
$ Case \; (I-ia)$ $\begin{array}{l} K>1, \beta<1\\c>c_2 \end{array}$ $\begin{array}{ll} 0<\alpha<\alpha_T \;\;E_1^*, E_2^* \\ \alpha>\alpha_T \;\;\;\;\;\;\;\;E_1^*\end{array}$ $\begin{array}{l} Hopf \; transcritical \end{array}$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a\; homoclinic \;cycle\; or\; a \;loop\; of\; heteroclinic \;orbits \end{array}$ Figure 6 a)
Figure 7
Figure 8
$Case \; (I-ib)$ $\begin{array}{l} K>1, \beta<1\\c_1<c<c_2, \\K<2 \end{array}$ $\begin{array}{ll} 0<\alpha<\alpha_{sn} \;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_{sn} \;\;\;\;\;\;\;\;none \end{array}$ $\begin{array}{l} saddle-node \; Hopf \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$ Figure 6 b)
Figure 9
$Case \; (I-ic)$ $\begin{array}{l} K>1, \beta<1\\c_1<c<c_2, \\K>2 \end{array}$ $\begin{array}{ll} 0<\alpha<\alpha_T\;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_T\;\;\;\;\;\;\;\;E_1^* \end{array}$ $\begin{array}{l} Hopf \; transcritical \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a \;loop \; of \; heteroclinic \;orbits \end{array}$ Figure 6 c)
Figure 10
$Case \; (I-ii)$ $\begin{array}{l} K>1, \beta<1\\c<c_1 \end{array}$ $\begin{array}{ll} \alpha<\alpha_{sn} \;\;\;\;\;none\\ \alpha_{sn}<\alpha<\alpha_T \;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_T \;\;\;\;E_1^* \end{array}$ $\begin{array}{l} saddle-node \; Hopf \; transcritical \end{array}$ $\begin{array}{l} a \;bubble; \;or open\; ended(two \;branches)\end{array}$ $\begin{array}{l} an \; equilibrium; \;or a\; homoclinic \;cycle \; or \; a \;loop \; of \; heteroclinic \; orbits \end{array}$ Figure 6 d)
Figure 11
$Case \;(II)$ $K>1, \beta>1$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none\\\alpha>\alpha_s \;\;\;\;\;E_1^* \end{array}$ $Hopf$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} \; a \;loop \; of \; heteroclinic \;orbits \end{array}$ Figure 12
$Case \;(III-i)$ $\begin{array}{l} K<1, \beta>1\\\frac{1}{\beta}<K \end{array}$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none\\\alpha>\alpha_s\;\;\;\;\;E_1^* \end{array}$ $Hopf$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a \;loop \; of \; heteroclinic \;orbits \end{array}$ Figure 13
$ Case \; (III-ii)$ $\begin{array}{l} K<1, \beta>1\\\frac{1}{\beta}>K \end{array}$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none \\ \alpha_s<\alpha<\alpha_T \;\;\;\;E_1^* \\ \alpha_T<\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \\ \alpha>\alpha_{sn} \;\;\;\;\;none \end{array}$ $\begin{array}{l} saddle-node \;\\ Hopf \; \\transcritical \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$ Figure 14
$Case \; (IV-i)$ $\begin{array}{l} K<1, \beta<1\\c>c_1 \end{array}$ $\begin{array}{ll} \alpha>\alpha_{sn} \;\;\;\;\;none\\ 0<\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \end{array}$ $\begin{array}{l} saddle-node \; \\ Hopf \; \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$ Figure 15
$ Case \; (IV-ii)$ $\begin{array}{l} K<1, \beta<1\\c<c_1 \end{array}$ $none$ $none$ $none$ $none$ $none$
Table 3.  Main results for system (2) taking $ \alpha $ as bifurcation parameter when $ K = 1 $ or $ \beta = 1 $
$ Case$ $Condition$ $\begin{array}{l} Interior \;equilibria \end{array}$ $\begin{array}{l} Type\; of\\Bifurcation \end{array}$ $\begin{array}{l} Shape\; of\; Hopf\;branch \end{array}$ $\begin{array}{l} Periodic \;orbits \; disappear \;through \end{array}$
$Case \;(V)$ $\begin{array}{l} K=1, \beta>1 \end{array}$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none\\\alpha>\alpha_s\;\;\;\;\;E_1^* \end{array}$ $Hopf$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a \;loop\;of\; heteroclinic \;orbits \end{array}$
$Case \; (VI-i)$ $\begin{array}{l} K=1, \beta<1\\c>c_1 \end{array}$ $\begin{array}{ll} \alpha>\alpha_{sn} \;\;\;\;\;none\\ 0 <\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \end{array}$ $\begin{array}{l} saddle-node\; \\ Hopf \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$
$Case \; (VI-ii)$ $\begin{array}{l} K=1, \beta<1\\c<c_1 \end{array}$ $none$ $none$ $none$ $none$
$ Case \; (VII)$ $\begin{array}{l} K>1, \beta=1 \end{array}$ $\begin{array}{ll} \alpha>0\;\;\;\;\;E_1^* \end{array}$ $\begin{array}{l} Hopf \end{array}$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \; or \; a \;loop \; of \; heteroclinic \;orbits \end{array}$
$Case \; ( VIII)$ $\begin{array}{l} K<1, \beta=1 \end{array}$ $\begin{array}{ll} \alpha>\alpha_{sn} \;\;\;\;\;none\\ 0 <\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \end{array}$ $\begin{array}{l} saddle-node\;Hopf \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$
$ Case$ $Condition$ $\begin{array}{l} Interior \;equilibria \end{array}$ $\begin{array}{l} Type\; of\\Bifurcation \end{array}$ $\begin{array}{l} Shape\; of\; Hopf\;branch \end{array}$ $\begin{array}{l} Periodic \;orbits \; disappear \;through \end{array}$
$Case \;(V)$ $\begin{array}{l} K=1, \beta>1 \end{array}$ $\begin{array}{ll} \alpha<\alpha_s\;\;\;\;\;none\\\alpha>\alpha_s\;\;\;\;\;E_1^* \end{array}$ $Hopf$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a \;loop\;of\; heteroclinic \;orbits \end{array}$
$Case \; (VI-i)$ $\begin{array}{l} K=1, \beta<1\\c>c_1 \end{array}$ $\begin{array}{ll} \alpha>\alpha_{sn} \;\;\;\;\;none\\ 0 <\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \end{array}$ $\begin{array}{l} saddle-node\; \\ Hopf \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$
$Case \; (VI-ii)$ $\begin{array}{l} K=1, \beta<1\\c<c_1 \end{array}$ $none$ $none$ $none$ $none$
$ Case \; (VII)$ $\begin{array}{l} K>1, \beta=1 \end{array}$ $\begin{array}{ll} \alpha>0\;\;\;\;\;E_1^* \end{array}$ $\begin{array}{l} Hopf \end{array}$ $\begin{array}{l} open\; ended\end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \; or \; a \;loop \; of \; heteroclinic \;orbits \end{array}$
$Case \; ( VIII)$ $\begin{array}{l} K<1, \beta=1 \end{array}$ $\begin{array}{ll} \alpha>\alpha_{sn} \;\;\;\;\;none\\ 0 <\alpha<\alpha_{sn}\;\;\;\;\;E_1^*, E_2^* \end{array}$ $\begin{array}{l} saddle-node\;Hopf \end{array}$ $\begin{array}{l} open\; ended \end{array}$ $\begin{array}{l} a\; homoclinic \;cycle \end{array}$
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