Article Contents
Article Contents

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

• * Corresponding author: Shigui Ruan

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.

Mathematics Subject Classification: 92D25, 34D45, 34C23.

 Citation:

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

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

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

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

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

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

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

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

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

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

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

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

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

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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