# American Institute of Mathematical Sciences

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

 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.

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.
Phase diagram of system (2.2) with $s = b = 2,\; d = 0.5$ and $r = 0.2$
Saddle-node $(0,0)$ of system (2.11) with $s = 2,d = 0.5,b = 1,r = 1$
Phase diagram of system (2.2) with $s = 2,d = 0.5,b = 1,r = 1$
Phase diagram of system (1.3) with $s = r = 2,d = 0.1,b = 1$
Phase diagram of system (1.3) with $s = 3,d = 1.5,b = 1,r = 1$
Phase diagram of system (1.3) with $b = s = 2,d = 1.5,r = 2$
Phase diagram of system (1.3) with $s = 2,d = 0.625,r = 1,b = 0.8$
Phase diagram of system (1.3) with $s = 1,d = 0.5,b = 2.5,r = 1$
Phase diagram of system (1.3) with $s = 0.8,d = 0.5,r = 1,b = 1$
Phase diagram of system (1.3) with $s = 1.75625,d = 0.2,r = 2,b = 3$
Phase diagram of system (1.3) with $s = 1.5,d = 0.1,r = 2,b = 2$
Bifurcation sets and the corresponding phase portraits of system (4.6)
(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
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 $(K4):=\{b-1-bd<0, s-1-bd<0\}$
 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] Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130 [2] Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747 [3] Benjamin Leard, Catherine Lewis, Jorge Rebaza. Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting. Discrete and 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 and 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 and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [6] Tahani Mtar, Radhouane Fekih-Salem, Tewfik Sari. Mortality can produce limit cycles in density-dependent models with a predator-prey relationship. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022049 [7] Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022082 [8] 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 [9] Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1397-1420. doi: 10.3934/dcdsb.2021095 [10] Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321 [11] Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure and Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423 [12] Canan Çelik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 719-738. doi: 10.3934/dcdsb.2011.16.719 [13] Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117 [14] 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 and Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033 [15] Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209 [16] Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062 [17] Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130 [18] 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 and Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267 [19] 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 and Continuous Dynamical Systems - B, 2020, 25 (8) : 3049-3086. doi: 10.3934/dcdsb.2020051 [20] J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054

2020 Impact Factor: 1.327