American Institute of Mathematical Sciences

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April  2021, 26(4): 1967-1990. doi: 10.3934/dcdsb.2020041

Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response

Xin Jiang 1, , Zhikun She 1, and  Shigui Ruan 2,,

1. 

School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
2. 

Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA

* Corresponding author: Shigui Ruan

Received  July 2019 Revised  October 2019 Published  April 2021 Early access  February 2020

Fund Project: Research of the first author was supported by China Scholarship Council (201806020127) and the Academic Excellence Foundation of BUAA for Ph.D. Students. Research of the second author was supported by Beijing Natural Science Foundation (Z180005) and National Natural Science Foundation of China (11422111)

In this paper, we study the global dynamics of a density-dependent predator-prey system with ratio-dependent functional response. The main features and challenges are that the origin of this model is a degenerate equilibrium of higher order and there are multiple positive equilibria. Firstly, local qualitative behavior of the system around the origin is explicitly described. Then, based on the dynamics around the origin and other equilibria, global qualitative analysis of the model is carried out. Finally, the existence of Bogdanov-Takens bifurcation (cusp case) of codimension two is analyzed. This shows that the system undergoes various bifurcation phenomena, including saddle-node bifurcation, Hopf bifurcation, and homoclinic bifurcation along with different topological sectors near the degenerate origin. Numerical simulations are presented to illustrate the theoretical results.

Keywords: Predator-prey system, density-dependent mortality, ratio-dependent, global dynamics, Bogdanov-Takens bifurcation.
Mathematics Subject Classification: 92D25, 34D45, 34C23.
Citation: Xin Jiang, Zhikun She, Shigui Ruan. Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1967-1990. doi: 10.3934/dcdsb.2020041
References:
[1]

H. R. Akcakaya, Population cycles of mammals: Evidence for a ratio-dependent predation hypothesis, Ecol. Monogr., 62 (1992), 119-142.  doi: 10.2307/2937172.

[2]

H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004.  doi: 10.2307/1939362.

[3]

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

[4]

A. D. Bazykin, Structural and Dynamic Stability of Model Predator-Prey Systems, Int. Inst. Appl. Syst. Analysis, Laxenburg, 1976.

[5]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 11, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798725.

[6]

F. Berezovskaya, G. Karev and R. Arditi, Parametric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43 (2001), 221-246.  doi: 10.1007/s002850000078.

[7]

R. Bogdonov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Mathe. Soviet., 1 (1981), 373-388. 

[8]

R. Bogdonov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta Mathe. Soviet., 1 (1981), 389-421. 

[9] S. N. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.
[10]

A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholson's blowflies as an example, Ecology, 73 (1992), 1552-1563. 

[11]

J. Hainzl, Stability and Hopf bifurcaiton in a predator-prey system with several parameters, SIAM J. Appl. Math., 48 (1998), 170-190.  doi: 10.1137/0148008.

[12]

J. Hainzl, Multiparameter bifurcation of a predator-prey system, SIAM J. Math. Anal., 23 (1992), 150-180.  doi: 10.1137/0523008.

[13]

J. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.

[14]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.  doi: 10.1007/s002850100079.

[15]

J. Huang, S. Ruan and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differential Equations, 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024.

[16]

J. Huang, Y. 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.

[17]

X. Jiang, Z. She, Z. Feng and X. Zheng, Structural stability of a density dependent predator-prey system with ratio-dependent functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 21pp. doi: 10.1142/S0218127417502224.

[18]

C. Jost, O. Arino and R. Arditi, About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61 (1999), 19-32.  doi: 10.1006/bulm.1998.0072.

[19]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependence predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105.

[20]

B. Li and Y. Kuang, Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J. Appl. Math., 67 (2007), 1453-1464.  doi: 10.1137/060662460.

[21]

P. Parrilo and S. Lall, Semidefinite programming relaxations and algebraic optimization in control, European J. Control, 9 (2003), 307-321.  doi: 10.3166/ejc.9.307-321.

[22]

S. Ruan, Y. Tang and W. Zhang, Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response, J. Math. Biol., 57 (2008), 223-241.  doi: 10.1007/s00285-007-0153-z.

[23]

S. Ruan, Y. Tang and W. Zhang, Versal unfoldings of predator-prey systems with ratio-dependent functional response, J. Differential Equations, 249 (2010), 1410-1435.  doi: 10.1016/j.jde.2010.06.015.

[24]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896.

[25]

Z. She, H. Li, B. Xue, Z. Zheng and B. Xia, Discovering polynomial Lyapunov functions for continuous dynamical systems, J. Symbolic Comput., 58 (2013), 41-63.  doi: 10.1016/j.jsc.2013.06.003.

[26]

F. Takens, Forced oscillations and bifurcations, in Applications of Global Analysis, I, Comm. Math. Inst. Rijksuniv. Utrecht, 3, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, 1-59. doi: 10.1201/9781420034288-1.

[27]

Y. Tang and W. Zhang, Heteroclinic bifurcation in a ratio-dependent predator-prey system, J. Math. Biol., 50 (2005), 699-712.  doi: 10.1007/s00285-004-0307-1.

[28]

S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, Texts in Applied Mathematics, 2, Spring-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

[29]

D. Xiao and S. Ruan, Bogdoanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in Differential Equations with Applications to Biology, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999,493-506.

[30]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.  doi: 10.1007/s002850100097.

[31]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101, American Mathematical Society, Providence, RI, 1992.

[32]

X. Zheng, Z. She, Q. Liang and M. Li, Inner approximations of domains of attraction for a class of switched systems by computing Lyapunov-like functions, Internat. J. Robust Nonlinear Control, 28 (2018), 2191-2208.  doi: 10.1002/rnc.4010.

show all references

References:
[1]

H. R. Akcakaya, Population cycles of mammals: Evidence for a ratio-dependent predation hypothesis, Ecol. Monogr., 62 (1992), 119-142.  doi: 10.2307/2937172.

[2]

H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004.  doi: 10.2307/1939362.

[3]

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

[4]

A. D. Bazykin, Structural and Dynamic Stability of Model Predator-Prey Systems, Int. Inst. Appl. Syst. Analysis, Laxenburg, 1976.

[5]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 11, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798725.

[6]

F. Berezovskaya, G. Karev and R. Arditi, Parametric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43 (2001), 221-246.  doi: 10.1007/s002850000078.

[7]

R. Bogdonov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Mathe. Soviet., 1 (1981), 373-388. 

[8]

R. Bogdonov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta Mathe. Soviet., 1 (1981), 389-421. 

[9] S. N. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.
[10]

A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholson's blowflies as an example, Ecology, 73 (1992), 1552-1563. 

[11]

J. Hainzl, Stability and Hopf bifurcaiton in a predator-prey system with several parameters, SIAM J. Appl. Math., 48 (1998), 170-190.  doi: 10.1137/0148008.

[12]

J. Hainzl, Multiparameter bifurcation of a predator-prey system, SIAM J. Math. Anal., 23 (1992), 150-180.  doi: 10.1137/0523008.

[13]

J. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.

[14]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42 (2001), 489-506.  doi: 10.1007/s002850100079.

[15]

J. Huang, S. Ruan and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differential Equations, 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024.

[16]

J. Huang, Y. 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.

[17]

X. Jiang, Z. She, Z. Feng and X. Zheng, Structural stability of a density dependent predator-prey system with ratio-dependent functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 21pp. doi: 10.1142/S0218127417502224.

[18]

C. Jost, O. Arino and R. Arditi, About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61 (1999), 19-32.  doi: 10.1006/bulm.1998.0072.

[19]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependence predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105.

[20]

B. Li and Y. Kuang, Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J. Appl. Math., 67 (2007), 1453-1464.  doi: 10.1137/060662460.

[21]

P. Parrilo and S. Lall, Semidefinite programming relaxations and algebraic optimization in control, European J. Control, 9 (2003), 307-321.  doi: 10.3166/ejc.9.307-321.

[22]

S. Ruan, Y. Tang and W. Zhang, Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response, J. Math. Biol., 57 (2008), 223-241.  doi: 10.1007/s00285-007-0153-z.

[23]

S. Ruan, Y. Tang and W. Zhang, Versal unfoldings of predator-prey systems with ratio-dependent functional response, J. Differential Equations, 249 (2010), 1410-1435.  doi: 10.1016/j.jde.2010.06.015.

[24]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896.

[25]

Z. She, H. Li, B. Xue, Z. Zheng and B. Xia, Discovering polynomial Lyapunov functions for continuous dynamical systems, J. Symbolic Comput., 58 (2013), 41-63.  doi: 10.1016/j.jsc.2013.06.003.

[26]

F. Takens, Forced oscillations and bifurcations, in Applications of Global Analysis, I, Comm. Math. Inst. Rijksuniv. Utrecht, 3, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, 1-59. doi: 10.1201/9781420034288-1.

[27]

Y. Tang and W. Zhang, Heteroclinic bifurcation in a ratio-dependent predator-prey system, J. Math. Biol., 50 (2005), 699-712.  doi: 10.1007/s00285-004-0307-1.

[28]

S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, Texts in Applied Mathematics, 2, Spring-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

[29]

D. Xiao and S. Ruan, Bogdoanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in Differential Equations with Applications to Biology, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999,493-506.

[30]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.  doi: 10.1007/s002850100097.

[31]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101, American Mathematical Society, Providence, RI, 1992.

[32]

X. Zheng, Z. She, Q. Liang and M. Li, Inner approximations of domains of attraction for a class of switched systems by computing Lyapunov-like functions, Internat. J. Robust Nonlinear Control, 28 (2018), 2191-2208.  doi: 10.1002/rnc.4010.

Figure 1.  Phase diagram of system (2.2) with $ s = b = 2,\; d = 0.5 $ and $ r = 0.2 $
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Figure 2.  Saddle-node $ (0,0) $ of system (2.11) with $ s = 2,d = 0.5,b = 1,r = 1 $
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Figure 3.  Phase diagram of system (2.2) with $ s = 2,d = 0.5,b = 1,r = 1 $
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Figure 4.  Phase diagram of system (1.3) with $ s = r = 2,d = 0.1,b = 1 $
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Figure 5.  Phase diagram of system (1.3) with $ s = 3,d = 1.5,b = 1,r = 1 $
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Figure 6.  Phase diagram of system (1.3) with $ b = s = 2,d = 1.5,r = 2 $
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Figure 7.  Phase diagram of system (1.3) with $ s = 2,d = 0.625,r = 1,b = 0.8 $
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Figure 8.  Phase diagram of system (1.3) with $ s = 1,d = 0.5,b = 2.5,r = 1 $
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Figure 9.  Phase diagram of system (1.3) with $ s = 0.8,d = 0.5,r = 1,b = 1 $
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Figure 10.  Phase diagram of system (1.3) with $ s = 1.75625,d = 0.2,r = 2,b = 3 $
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Figure 11.  Phase diagram of system (1.3) with $ s = 1.5,d = 0.1,r = 2,b = 2 $
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Figure 12.  Bifurcation sets and the corresponding phase portraits of system (4.6)
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Figure 13.  (Ⅰ): When $ u_1 = -0.1 $ and $ u_2 = 0.515 $ lie in the region Ⅰ, there exists no positive equilibrium; (Ⅱ): When $ u_1 = -0.1 $ and $ u_2 = 0.55 $ lie in the region Ⅱ, there exist a saddle point and an unstable focus; (Ⅲ): When $ u_1 = -0.1 $ and $ u_2 = 0.605 $ lie in the region Ⅲ, there exist a saddle point, a stable focus and an unstable limit cycle; (Ⅳ): When $ u_1 = -0.1 $ and $ u_2 = 0.8 $ lie in the region Ⅳ, there exist a saddle point and an stable focus
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Table 1.  The global dymamics of system (1.3)
Condition $ 1 $ Condition $ 2 $ Global Results Hopf bifurcation
$ (H0) $ $ d<1 $ $ (K1) $ Theorem 3.3 Does not exist
$ (K2) $ Theorem 3.3 Does not exist
$ (K3) $ Theorem 3.3 Does not exist
$ (K4) $ $ \emptyset $ Does not exist
$ d>1 $ $ (K1) $ $ \emptyset $ Does not exist
$ (K2) $ Theorem 3.4 Does not exist
$ (K3) $ $ \emptyset $ Does not exist
$ (K4) $ Theorem 3.5 Does not exist
$ (H1) $ $ (K1) $ $ \emptyset $ Does not exist
$ (K2) $ Theorem 3.6 Remark 1
$ (K3) $ Theorem 3.7 Does not exist
$ (K4) $ Theorem 3.8 Does not exist
$ (H2)\wedge (H6) $ $ (K1) $ $ \emptyset $ Does not exist
$ (K2) $ Theorem 3.9 Remark 3
$ (K3) $ $ \emptyset $ Does not exist
$ (K4) $ $ \emptyset $ Does not exist
$ (H3)\wedge (H5) $ $ (K1) $ Theorem 3.10 Does not exist
$ (K2) $ Theorem 3.10 Does not exist
$ (K3) $ $ \emptyset $ Does not exist
$ (K4) $ $ \emptyset $ Does not exist
$ (H4) $ $ (K1) $ Theorem 3.11 Remark 5
$ (K2) $ Theorem 3.11 Remark 5
$ (K3) $ Theorem 3.11 Does not exist
$ (K4) $ $ \emptyset $ Does not exist
Here $(H0):=\overline{(H1)\vee (H2)\vee (H3)\vee (H4)}$,
$(K1):=\{b-1-bd\geq0, s-1-bd\geq0\}$, $(K2):=\{b-1-bd<0, s-1-bd\geq 0\}$,
$(K3):=\{b-1-bd\geq0,s-1-bd<0\}$ and $(K4):=\{b-1-bd<0, s-1-bd<0\}$
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Condition $ 1 $ Condition $ 2 $ Global Results Hopf bifurcation
$ (H0) $ $ d<1 $ $ (K1) $ Theorem 3.3 Does not exist
$ (K2) $ Theorem 3.3 Does not exist
$ (K3) $ Theorem 3.3 Does not exist
$ (K4) $ $ \emptyset $ Does not exist
$ d>1 $ $ (K1) $ $ \emptyset $ Does not exist
$ (K2) $ Theorem 3.4 Does not exist
$ (K3) $ $ \emptyset $ Does not exist
$ (K4) $ Theorem 3.5 Does not exist
$ (H1) $ $ (K1) $ $ \emptyset $ Does not exist
$ (K2) $ Theorem 3.6 Remark 1
$ (K3) $ Theorem 3.7 Does not exist
$ (K4) $ Theorem 3.8 Does not exist
$ (H2)\wedge (H6) $ $ (K1) $ $ \emptyset $ Does not exist
$ (K2) $ Theorem 3.9 Remark 3
$ (K3) $ $ \emptyset $ Does not exist
$ (K4) $ $ \emptyset $ Does not exist
$ (H3)\wedge (H5) $ $ (K1) $ Theorem 3.10 Does not exist
$ (K2) $ Theorem 3.10 Does not exist
$ (K3) $ $ \emptyset $ Does not exist
$ (K4) $ $ \emptyset $ Does not exist
$ (H4) $ $ (K1) $ Theorem 3.11 Remark 5
$ (K2) $ Theorem 3.11 Remark 5
$ (K3) $ Theorem 3.11 Does not exist
$ (K4) $ $ \emptyset $ Does not exist
Here $(H0):=\overline{(H1)\vee (H2)\vee (H3)\vee (H4)}$,
$(K1):=\{b-1-bd\geq0, s-1-bd\geq0\}$, $(K2):=\{b-1-bd<0, s-1-bd\geq 0\}$,
$(K3):=\{b-1-bd\geq0,s-1-bd<0\}$ and $(K4):=\{b-1-bd<0, s-1-bd<0\}$
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