American Institute of Mathematical Sciences

Advanced
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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.   Google Scholar

[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.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.   Google Scholar

[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.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Phase diagram of system (2.2) with $ s = b = 2,\; d = 0.5 $ and $ r = 0.2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Saddle-node $ (0,0) $ of system (2.11) with $ s = 2,d = 0.5,b = 1,r = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Phase diagram of system (2.2) with $ s = 2,d = 0.5,b = 1,r = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Phase diagram of system (1.3) with $ s = r = 2,d = 0.1,b = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Phase diagram of system (1.3) with $ s = 3,d = 1.5,b = 1,r = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Phase diagram of system (1.3) with $ b = s = 2,d = 1.5,r = 2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Phase diagram of system (1.3) with $ s = 2,d = 0.625,r = 1,b = 0.8 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Phase diagram of system (1.3) with $ s = 1,d = 0.5,b = 2.5,r = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Phase diagram of system (1.3) with $ s = 0.8,d = 0.5,r = 1,b = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Phase diagram of system (1.3) with $ s = 1.75625,d = 0.2,r = 2,b = 3 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Phase diagram of system (1.3) with $ s = 1.5,d = 0.1,r = 2,b = 2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  Bifurcation sets and the corresponding phase portraits of system (4.6)
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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 $(K4):=\{b-1-bd<0, s-1-bd<0\}$
Table Options

Download as excel

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 $(K4):=\{b-1-bd<0, s-1-bd<0\}$
[1]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747
[2]

Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130
[3]

Benjamin Leard, Catherine Lewis, Jorge Rebaza. Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 303-315. doi: 10.3934/dcdss.2008.1.303
[4]

Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283
[5]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
[6]

Inkyung Ahn, Wonlyul Ko, Kimun Ryu. Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey. Conference Publications, 2013, 2013 (special) : 11-19. doi: 10.3934/proc.2013.2013.11
[7]

Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321
[8]

Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure & Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423
[9]

Canan Çelik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 719-738. doi: 10.3934/dcdsb.2011.16.719
[10]

Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117
[11]

Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020033
[12]

Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209
[13]

Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062
[14]

Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130
[15]

Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267
[16]

Jian Zu, Miaolei Li, Yuexi Gu, Shuting Fu. Modelling the evolutionary dynamics of host resistance-related traits in a susceptible-infected community with density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3049-3086. doi: 10.3934/dcdsb.2020051
[17]

J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054
[18]

Hanwu Liu, Lin Wang, Fengqin Zhang, Qiuying Li, Huakun Zhou. Dynamics of a predator-prey model with state-dependent carrying capacity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4739-4753. doi: 10.3934/dcdsb.2019028
[19]

Meng Fan, Qian Wang. Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 563-574. doi: 10.3934/dcdsb.2004.4.563
[20]

Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (158)
  • HTML views (284)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences