Modeling the p53-Mdm2 Dynamics Triggered by DNA Damage

Abstract

In this paper, a p53-Mdm2 network model with Michaelis–Menten function involved in DNA damage repair is studied by using the dynamical system approach. Detailed bifurcations of the model including saddle-node bifurcation, Hopf bifurcation of codimension 3, and cusp-type Bogdanov–Takens bifurcation of codimension 3 are investigated. Meanwhile, the coexistence of three limit cycles and the coexistence of a homoclinic cycle and a limit cycle are also obtained. To our best knowledge, it is the first time that the coexistence of an isola and a cusp of limit cycles is observed for the p53 model, which indicates complex transitions of different oscillating dynamics. Furthermore, we find that oscillation could persist whether the damage is nonexistent, small or large, and multiple oscillations can be involved in the DNA repair process. In addition, the dynamics induced by the interaction between p53 and Mdm2 unveil the relationship between the positive feedback brought by p53 itself and the negative feedback brought by Mdm2, which triggers the digital oscillation modes for DNA damage. Of particular interest is that multiple oscillations and the threshold value of DNA damage are found to reveal the intrinsic mechanism of DNA damage repair.

Appendices

Appendix A: Analysis on the Existence of Positive Equilibria

In this appendix, we give a detailed analysis on the existence of positive equilibria. We first define the following equation and let

$$\begin{aligned} \begin{aligned} F_0(x)&=A_7x^7+A_6x^6+A_5x^5+A_4x^4+A_3x^3+A_2x^2+A_1x+A_0,\\ F_1(x)&:=F'(x)=7A_7x^6+6A_6x^5+5A_5x^4+4A_4x^3+3A_3x^2+2A_2x+A_1,\\ F_2(x)&:=F_1'(x)=42A_7x^5+30A_6x^4+20A_5x^3+12A_4x^2+6A_3x+2A_2,\\ F_3(x)&:=F_2'(x)=210A_7x^4+120A_6x^3+60A_5x^2+24A_4x+6A_3,\\ F_4(x)&:=F_3'(x)=840A_7x^3+360A_6x^2+120A_5x+24A_4,\\ F_5(x)&:=F_4'(x)=120(21A_7x^2+6A_6x+A_5), \end{aligned} \end{aligned}$$

(42)

where

$$\begin{aligned} \begin{aligned}&A_7=-d_1 d_2<0,{} & {} A_6=d_2 (\alpha -d_1k_3)-r\beta ,\\&A_5=d_2(\alpha k_3-d_1k_1),{} & {} A_4=-r\beta k_1 -d_1 d_2 k_3 k_1<0,\\&A_3=-d_1 d_2 k_2<0,{} & {} A_2=d_2 k_2(\alpha -d_1k_3),\\&A_1=d_2 k_2(\alpha k_3-d_1k_1),{} & {} A_0=-k_3d_1 d_2 k_1 k_2<0. \end{aligned} \end{aligned}$$

From equation $F_5(x)=0$, we have $\Delta =12(3A_6^2-7A_5A_7)$. Obviously, the number of positive roots of equation $F_5(x)=0$ will affect the number of positive equilibrium. The sign of $\Delta$ determines the existence of the root of equation $F_5(x)=0$. Next, we judge the existence of positive equilibrium from the three cases of $\Delta > 0$, $\Delta = 0$ and $\Delta < 0$.

Case 1. $\Delta < 0$. Since $A_7<0$, $\Delta < 0$ hold, which implies that $A_5<0$. Further, we get $A_1<0$ and $F_5(x)<0$, which implies function $F_4(x)$ is monotonically decreasing. Since $F_4(0)=A_4<0$, then $F_3(x)$ is monotonically decreasing. Similarly, $F_2(x)$ is monotonically decreasing for $F_3(0)<0$, then $F_2(0)=A_2$. Here, the sign of $A_2$ largely determines the number of positive roots of equation $F_0(x)=0$. We give the existence of positive roots of equation $F_0(x)=0$ in different cases through the following tables.

Remark 5

In this case, if equation $F_0(x)=0$ has only one positive root $x^*$, then system (2) has a unique positive equilibrium $E^*(x^*,y^*)$, where $x^*=x_{22}>x_1$. Moreover, $F_1(x^*)=F_1(x_{22})=0$ and $F_2(x^*)=F_2(x_{22})< 0$. From Theorem 1 we can obtain that $E^*$ is a higher-order singular point, and saddle-node bifurcation will occur at this equilibrium. On the other hand, if $F_0(x)=0$ has two positive roots $x_1^*<x_2^*$, and we can deduce that $F_1(x_1^*)>0$ and $F_1(x_2^*)<0$. Further, Theorem 1 shows that the corresponding equilibrium $E_1$ is a saddle and $E_2$ is a node or focus. Then Hopf bifurcation will occur at equilibrium $E_2$.

Table 1 The existence of positive roots of equation $F_0(x)=0$ when $\Delta <0$

Case 2. $\Delta = 0$. When $\Delta = 0$, equation $F_5(x)=0$ has one root $x^*=-\frac{120}{7A_7}A_6$. However, at this time, the existence of the positive root of equation $F_0(x)=0$ is the same as that of the positive root obtained in case 1, which we will not repeat here.

Case 3. $\Delta > 0$. The two roots of equation $F_5(x)=0$ are $x_{1,2}=\frac{-6A_6\pm \sqrt{\Delta }}{42A_7}$. Obviously, the signs of $A_5$ and $A_6$ will affect the number of positive roots of equation $F_5(x)=0$. Let’s discuss from the following three situations. (i) If $A_5>0$, then we get $x_1<0$ and $x_2>0$. (ii) If $A_5<0$ and $A_6<0$, $x_{1,2}<0$ since $x_1+x_2=-\frac{6A_6}{21A_7}<0$ and $x_1x_2=\frac{A_5}{21A_7}>0$. (iii) If $A_5<0$ and $A_6>0$, $x_{1,2}>0$ since $x_1+x_2=-\frac{6A_6}{21A_7}>0$ and $x_1x_2=\frac{A_5}{21A_7}>0$. Obviously, case (ii) is similar to that when $\Delta <0$. Therefore, we analyze the other two cases here.

Table 2 The existence of positive roots of equation $F_0(x)=0$ when $\Delta >0$

Case (3i) If $A_5>0$, then there is a positive number $x_{2}=\frac{-6A_6+\sqrt{\Delta }}{42A_7}$ such that $F_5(x_2)=0$. Function $F_4(x)$ is monotonically increasing on interval $(0,x_2)$ and monotonically decreasing on interval $(x_2,+\infty )$. Moreover, the sign of $F_4(x_2)$ will affect the subsequent analysis. If $F_4(x_2)\le 0$, we have $F_3(x)$ is monotonically decreasing for all $x>0$ since $F_4(0)<0$. Further, $F_2(x)$ is monotonically decreasing for all $x>0$ for $F_3(0)<0$. Then, we give a table as shown in Table 1 to illustrate the existence of positive equilibria.

If $F_4(x_2)>0$, there are two positive numbers $x_{21}$ and $x_{22}$, $x_{21}<x_2<x_{22}$, and $F_4(x_{21})=F_4(x_{22})=0$. Then $F_3(x)$ is monotonically increasing on $(x_{21},x_{22})$ and monotonically decreasing on $(0,x_{21})\cup (x_{22},+\infty )$. The sign of $F_3(x_{22})$ will affect the subsequent analysis since $F_3(0)<0$. If $F_3(x_{22})\le 0$, $F_2(x)$ is monotonically increasing for all $x>0$, then the existence of positive roots of equation $F_0(x)=0$ can refer to the results given in Table 2. Next, we mainly give the analysis process when $F_3(x_{22})>0$. If condition $F_3(x_{22})>0$ holds, it means that there exist $x_{31}$ and $x_{32}$ such that $x_{31}<x_{22}<x_{32}$ and $F_3(x_{31})=F_3(x_{32})=0$. Further, we obtain that $F_2(x)$ is monotonically increasing on $(x_{31},x_{32})$ and monotonically decreasing on $(0,x_{31})\cup (x_{32},+\infty )$. Then, we give some tables as shown in Table 1 to illustrate the existence of positive equilibria.

Table 3 The existence of positive roots of equation $F_0(x)=0$ when $\Delta >0$

Note that, system (2) have four positive equilibria $E^*_i(x^*_i,y^*_i)(i=1,2,3,4)$ if there are four positive roots $x^*_i(i=1,2,3,4)$. In general, we assume that $x^*_1<x^*_2<x^*_3<x^*_4$. We obtain that $F_1(x^*_{1,3})>0$, $F_1(x^*_{2,4})<0$, thus $E^*_1$ and $E^*_3$ are saddle points and Hopf bifurcation will occur at equilibrium $E^*_2$ or $E^*_4$.

Table 4 The existence of positive roots of equation $F_0(x)=0$ when $\Delta >0$

Case (3ii) Conditions $A_5<0$ and $A_6>0$ imply that $A_1<0$ and $A_2>0$ hold. In this case, equation $F_5(x)=0$ has two positive roots $x_{1,2}$, which implies that $F_4(x)$ is monotonically decreasing in $(0,x_1)\cup (x_2,+\infty )$ and monotonically increasing in $(x_1,x_2)$. Similarly, the sign of $F_4(x_2)$ will affect the subsequence analysis since $F_4(0)<0$. If $F_4(x_2)\le 0$, then $F_3(x)$ is monotonically decreasing for all $x>0$ as $F_4(0)<0$. Further, $F_2(x)$ is monotonically decreasing for all $x>0$ as $F_3(0)<0$. Since $F_2(0)=A_2>0$, there exists $x_{21}$ such that $F_2(x_{21})=0$, then we obtain that $F_1(x)$ is monotonically decreasing in $(\tilde{x},+\infty )$ and monotonically increasing in $(0,\tilde{x})$. If $F_1(\tilde{x})\le 0$, there is no positive equilibrium. If $F_1(\tilde{x})>0$, there exist $x_{31}$ and $x_{32}$ such that $F_1(x_{3i})=0$. Then $F_0(x)$ is monotonically decreasing in $(0,x^*_1)\cup (x^*_2,+\infty )$ and monotonically increasing in $(x_{31},x_{32})$. Further, the number of positive roots depends on the sign of $F(x_{32})$, and we can get that there are at most two positive equilibria.

If $F_4(x_2)>0$, there exist $x_{21}$ and $x_{22}$ such that $F_1(x_{2i})=0, i=1, 2$. This result shows that $F_3(x)$ is monotonically decreasing in $(0,x_{21})\cup (x_{22},+\infty )$ and monotonically increasing in $(x_{21},x_{22})$. And the sign of $F_3(x_{22})$ will affect the subsequence analysis since $F_3(0)<0$. On the one hand, if $F_3(x_{22})\le 0$, then $F_2(x)$ is monotonically increasing for all $x>0$, and through the same analysis, we can get that there are at most two positive equilibria. On the other hand, if $F_3(x_{22})>0$, there exist $x_{31}$ and $x_{32}$ such that $F_3(x_{3i})=0, i=1, 2$. And $F_2(x)$ is monotonically decreasing in $(0,x_{31})\cup (x_{32},+\infty )$ and monotonically increasing in $(x_{31},x_{32})$. Next, we give a table as shown in Table 4 to illustrate the existence of positive equilibria.

Table 5 The existence of positive roots of equation $F_0(x)=0$ when $\Delta >0$

Appendix B: The Specific Expression of Parameters in (26)

The specific expression of parameters $q_{20}$, $q_{11}$, $q_{02}$, $p_{20}$, $p_{11}$ and $p_{02}$ in Eq. (26) is given as follows:

$$\begin{aligned}{} & {} q_{20}=-\frac{1}{W_1^4 x_1^2 (k_1+x_1^2)^4 (k_3+x_1)^6}\\ {}{} & {} \times [2 d_2 (T_1 x_1 (k_3+x_1)-a_{10} k_3)^3 (x_1 (k_3+x_1) (d_2 T_1 (k_1^2 (3 k_3^2 (W_1^2 x_1^2-3)+6 k_3 x_1 (W_1^2 x_1^2-1) \\{} & {} +x_1^2 (3 W_1^2 x_1^2+4))+k_1 x_1^2 (3 k_3^2 (2 W_1^2 x_1^2-7)+4 k_3 x_1 (3 W_1^2 x_1^2-8)+6 W_1^2 x_1^4-9 x_1^2)\\{} & {} +x_1^4 (k_3^2 (3 W_1^2 x_1^2+1)+6 k_3 W_1^2 x_1^3+3 W_1^2 x_1^4))-8 W_1 x_1 (k_3+x_1) (k_1^2-3 k_1 x_1 (k_3+x_1) \\ {}{} & {} +k_3 x_1^3))-a_{10} d_2 k_3 (k_1^2 (3 k_3^2 (W_1^2 x_1^2-3)+6 k_3 x_1 (W_1^2 x_1^2-1)+x_1^2 (3 W_1^2 x_1^2+4)) \\ {}{} & {} +k_1 x_1^2 (3 k_3^2 (2 W_1^2 x_1^2-7) +4 k_3 x_1 (3 W_1^2 x_1^2-8)+6 W_1^2 x_1^4-9 x_1^2)+x_1^4 (k_3^2 (3 W_1^2 x_1^2+1) \\ {}{} & {} +6 k_3 W_1^2 x_1^3+3 W_1^2 x_1^4)))];\\{} & {} q_{11}=\frac{d_2 W_3^3 (k_3+x_1)^5}{r^8 W_1^4 x_1^7 (k_1+x_1^2)^5}[d_2 W_3 (k_3+x_1) (\alpha x_1 (-6 k_1^2 k_3^2-2 k_1^2 k_3 x_1+3 x_1^6 (2 W_1^2 (2 k_1 \\ {}{} & {} +k_3^2)+3)+x_1^4 (6 k_1 W_1^2 (k_1+2 k_3^2)+11 (k_3^2-3 k_1))+k_1 x_1^2 (6 k_1 (k_3^2 W_1^2+1)-43 k_3^2) \\ {}{} & {} +6 k_3 x_1^5 (4 k_1 W_1^2+3) +4 k_1 k_3 x_1^3 (3 k_1 W_1^2-20)+12 k_3 W_1^2 x_1^7+6 W_1^2 x_1^8)\\ {}{} & {} +W_2 (k_1+x_1^2)^3 (k_3+x_1) \times (6 k_1 k_3-k_1 x_1+7 k_3 x_1^2)) -W_1 x_1 (8 k_1^4 k_3 r^2 W_2 x_1 \\ {}{} & {} +k_1^3 (k_3^3 W_3+3 k_3^2 W_3 x_1+x_1^3 (32 k_3 r^2 W_2+W_3)+3 k_3 W_3 x_1^2)+k_1^2 x_1^2 (3 k_3^3 W_3+9 k_3^2 W_3 x_1 \\ {}{} & {} +3 x_1^3 (16 k_3 r^2 W_2+W_3)+16 \alpha r^2 (k_3+x_1)+9 k_3 W_3 x_1^2) +k_1 x_1^4 (3 k_3^3 W_3+9 k_3^2 W_3 x_1 \\ {}{} & {} +32 k_3 r^2 W_2 x_1^3-8 \alpha r^2 (9 k_3+11 x_1)+9 k_3 W_3 x_1^2+3 W_3 x_1^3)+x_1^6 (k_3^3 W_3+3 k_3^2 W_3 x_1 \\ {}{} & {} +8 k_3 r^2 W_2 x_1^3+8 \alpha r^2 (5 k_3+3 x_1)+3 k_3 W_3 x_1^2+W_3 x_1^3))];\\{} & {} q_{02}=\frac{W_3^3 (k_3+x_1)^5}{2 r^8 W_1^4 x_1^8 (k_1+x_1^2)^6}[-12 \alpha d_2^2 W_3 x_1 (k_3+x_1)^2 (\alpha x_1 (k_1^2-8 k_1 x_1^2+3 x_1^4) (k_3+x_1) \\ {}{} & {} +\alpha W_1^2 x_1^3 (k_1+x_1^2)^2 (k_3+x_1)+2 k_3 W_2 (k_1+x_1^2)^4)+32 \alpha d_2 r^2 W_1 x_1^3 (k_1^4 k_3 W_2 \\ {}{} & {} +4 k_1^3 k_3 W_2 x_1^2+k_1^2 x_1 (2 k_3 (\alpha +3 W_2 x_1^3)+\alpha x_1)+2 k_1 x_1^3 (-3 \alpha k_3+2 k_3 W_2 x_1^3-4 \alpha x_1) \\ {}{} & {} +x_1^5 (4 \alpha k_3+k_3 W_2 x_1^3+3 \alpha x_1))+W_3 (k_1+x_1^2)^2 (k_3+x_1) (4 \alpha d_2 W_1 x_1^2 (k_1+x_1^2) \\ {}{} & {} (k_3+x_1)^2-(d_2 W_2 (k_1+x_1^2)^2 (k_3+x_1)+2 \alpha d_2 k_3 x_1)^2)],\\{} & {} p_{20}=\frac{2 (2 d_2 W_3 (k_3+x_1)^2 (k_1-k_3 x_1)+r^2 W_1 x_1 (k_3 x_1^2-k_1 (k_3+2 x_1)))}{r^4 W_1^3 x_1^2 (k_1+x_1^2)^5 (k_3+x_1)^2}\times \\ {}{} & {} [d_2 W_3 (k_3+x_1)^2 (k_1^2 (k_3 (W_1^2 x_1^2-3)+W_1^2 x_1^3+x_1)+k_1 x_1^2 (2 k_3 W_1^2 x_1^2-7 k_3 \\ {}{} & {} +2 W_1^2 x_1^3-3 x_1)+W_1^2 x_1^6 (k_3+x_1))-2 r^2 W_1 x_1^3 (k_1^2-3 k_1 x_1 (k_3+x_1)+k_3 x_1^3)];\\ p_{11}{} & {} =\frac{1}{2 r^4 W_1^3 x_1^4 (k_1+x_1^2)^6 (k_3+x_1)^2}[2 r^2 W_1 (x_1^2+k_1) (k_3 x_1^3-3 k_1 (k_3+x_1) x_1+k_1^2) \\ {}{} & {} (2 k_3 r^2 W_1 \alpha x_1^3+2 r^2 W_1 W_2 (k_3+x_1) (x_1^2+k_1)^2 x_1^2-W_3 (k_3+x_1)^2 (d_2 W_2 (k_3+x_1) (x_1^2+k_1)^2 \\ {}{} & {} +2 d_2 k_3 x_1 \alpha )) x_1^3-4 r^2 W_1 (W_1 x_1 (k_1 (k_3+2 x_1)-k_3 x_1^2) r^2+d_2 W_3 (k_3+x_1)^2 (k_3 x_1-k_1)) \\ {}{} & {} ((k_3 W_2 x_1^3+3 \alpha x_1+4 k_3 \alpha ) x_1^5+2 k_1 (2 k_3 W_2 x_1^3-4 \alpha x_1-3 k_3 \alpha ) x_1^3+4 k_1^3 k_3 W_2 x_1^2+k_1^2 (x_1 \alpha \\ {}{} & {} +2 k_3 (3 W_2 x_1^3+\alpha )) x_1+k_1^4 k_3 W_2) x_1^3+2 d_2 W_3 (k_3+x_1)^2 (W_1 x_1 (k_1 (k_3+2 x_1)-k_3 x_1^2) r^2 \\ {}{} & {} +d_2 W_3 (k_3+x_1)^2 (k_3 x_1-k_1)) (2 k_3 W_2 (x_1^2+k_1)^4 +W_1^2 x_1^3 (k_3+x_1) \alpha (x_1^2+k_1)^2 \\ {}{} & {} +x_1 (k_3+x_1) (3 x_1^4-8 k_1 x_1^2+k_1^2) \alpha ) x_1+d_2 W_3 (k_3+x_1)^2 (x_1^2+k_1)(-W_1^2 x_1^2 (k_3+x_1) \\ {}{} & {} (x_1^2+k_1)^2+4 k_1 k_3 (x_1^2+k_1)-k_1 (k_3+x_1) (k_1-3 x_1^2))(2 k_3 r^2 W_1 \alpha x_1^3+2 r^2 W_1 W_2 \\ {}{} & {} (k_3+x_1) (x_1^2+k_1)^2 x_1^2-W_3 (k_3+x_1)^2 (d_2 W_2 (k_3+x_1) (x_1^2+k_1)^2+2 d_2 k_3 x_1 \alpha )) \\ {}{} & {} +W_3 (k_3+x_1)^2 (2 r^2 W_1^2 W_2 x_1^2 (k_3+x_1)^2 (x_1^2+k_1)^6+2 k_3 r^2 W_1^2 x_1^3 (k_3+x_1) (x_1^2+k_1)^4 \\ {}{} & {} -W_1 W_3 (k_3+x_1)^3(W_2 (k_3+x_1) (x_1^2+k_1)^2+2 d_2 k_3 x_1) (x_1^2+k_1)^4 \\ {}{} & {} -2 d_2 r^2 W_1 x_1^3 (k_3 x_1^3-3 k_1 (k_3+x_1) x_1+k_1^2) (W_2 x_1^5+k_3 W_2 x_1^4+2 k_1 W_2 (k_3+x_1) x_1^2 \\ {}{} & {} +2 k_3 \alpha x_1+k_1^2 W_2 (k_3+x_1)) (x_1^2+k_1)-d_2 W_3 (k_3+x_1)^2 (-W_1^2 x_1^2 (k_3+x_1) (x_1^2+k_1)^2 \\ {}{} & {} +4 k_1 k_3 (x_1^2+k_1)-k_1 (k_3+x_1) (k_1-3 x_1^2)) (d_2 W_2 (k_3+x_1) (x_1^2+k_1)^2+2 d_2 k_3 x_1 \alpha ) \\{} & {} (x_1^2+k_1)-2 d_2^2 W_3 x_1 (k_3+x_1)^2 (k_1-k_3 x_1) (2 k_3 W_2 (x_1^2+k_1)^4+W_1^2 x_1^3 (k_3+x_1) \alpha (x_1^2+k_1)^2 \\ {}{} & {} +x_1 (k_3+x_1) (3 x_1^4-8 k_1 x_1^2+k_1^2) \alpha )+4 d_2 r^2 W_1 x_1^3 (k_1-k_3 x_1) ((k_3 W_2 x_1^3+3 \alpha x_1 \\ {}{} & {} +4 k_3 \alpha ) x_1^5+2 k_1 (2 k_3 W_2 x_1^3-4 \alpha x_1-3 k_3 \alpha ) x_1^3+4 k_1^3 k_3 W_2 x_1^2\\{} & {} +k_1^2 (x_1 \alpha +2 k_3 (3 W_2 x_1^3+\alpha )) x_1+k_1^4 k_3 W_2))];\\ p_{02}{} & {} =\frac{1}{r^4 W_1^3 x_1^5 (k_1+x_1^2)^4 (k_3+x_1)^3}[d_2 W_3 (k_3+x_1)^3 (2 k_3 W_2 (k_1+x_1^2)^2-W_4 x_1 (k_1+x_1^2)^2 \\ {}{} & {} (k_3+x_1)+\alpha W_1^2 x_1^3 (k_3+x_1)) (d_2 W_3 (k_3+x_1)^2 (-W_2 (k_1+x_1^2)^2 (k_3+x_1)-2 \alpha k_3 x_1) \\ {}{} & {} +2 r^2 W_1 x_1^2 (W_2 (k_1+x_1^2)^2 (k_3+x_1)+\alpha k_3 x_1))+W_3 (k_3+x_1)^3(-2 r^2 W_1 x_1^2 (d_2 W_2 (k_1+x_1^2)^2 \\ {}{} & {} (k_3+x_1)+2 \alpha d_2 k_3 x_1) ((k_1+x_1^2)^2 (-k_3 W_2+k_3 W_4 x_1+W_4 x_1^2)-\alpha k_3 x_1)-d_2 W_3 (k_3+x_1)^2 \\{} & {} (d_2 W_2 (k_1+x_1^2)^2 (k_3+x_1)+2 \alpha d_2 k_3 x_1) (2 k_3 W_2 (k_1+x_1^2)^2-W_4 x_1 (k_1+x_1^2)^2 (k_3+x_1) \\ {}{} & {} +\alpha W_1^2 x_1^3 (k_3+x_1))+2 \alpha d_2 W_1 W_3 x_1^3 (k_3+x_1)^3 (x_1 (3 k_3+2 x_1)-k_1)+12 \alpha k_1 r^2 W_1^2 x_1^4 \\ {}{} & {} (k_3+x_1)^2-8 \alpha r^2 W_1^2 x_1^4 (k_1+x_1^2) (k_3+x_1)^2-2 \alpha k_3 r^2 W_1^2 x_1^4 (k_1+x_1^2) (k_3+x_1)) \\ {}{} & {} +2 r^2 W_1 x_1^2 (k_3+x_1) ((k_1+x_1^2)^2 (-k_3 W_2+k_3 W_4 x_1+W_4 x_1^2)-\alpha k_3 x_1) \\{} & {} (d_2 W_3 (k_3+x_1)^2 (-W_2 (k_1+x_1^2)^2 (k_3+x_1)-2 \alpha k_3 x_1)\\ {}{} & {} +2 r^2 W_1 x_1^2 (W_2 (k_1+x_1^2)^2 (k_3+x_1)+\alpha k_3 x_1))],\\ W_2{} & {} =\frac{2 \alpha x_1}{(k_1+x_1^2)^2}-\frac{4 \alpha k_1 x_1}{(k_1+x_1^2)^3}, \quad W_3=\frac{r^2 T_1 x_1^2}{(k_3+x_1)^2}-\frac{a_{10} k_3 r^2 x_1}{(k_3+x_1)^3},\ \\W_4{} & {} =\frac{\alpha (2 k_1-3 x_1^2)}{(k_1+x_1^2)^3}-\frac{3 \alpha (k_1^2-3 k_1 x_1^2)}{(k_1+x_1^2)^4}. \end{aligned}$$

Appendix C: The Second Lyapunov Coefficient

The second Lyapunov coefficient is

$$\begin{aligned}{} & {} \sigma _2=40 a_{01}^7 a_{20} b_{20}^3-20 a_{01}^6 a_{10} b_{20}^2 (a_{11} b_{20}-6 a_{20}^2)+60 a_{01}^5 a_{10}^2 a_{20} b_{20} (2 a_{20}^2-3 a_{11} b_{20}) \\{} & {} +20 a_{01}^4 a_{10}^3 (3 a_{11}^2 b_{20}^2-15 a_{11} a_{20}^2 b_{20}+2 a_{20}^4)-20 a_{01}^3 a_{10}^4 a_{11} (7 a_{20}^3-12 a_{11} a_{20}b_{20}) \\{} & {} +60 a_{01}^2 a_{10}^5 a_{11}^2 (3 a_{20}^2-a_{11} b_{20})+\omega ^6 (30 a_{01}^3 a_{11} a_{40}-9 a_{01}^3 (6 a_{20} a_{31}+a_{21} a_{30}) \\{} & {} +12 a_{01}^2 a_{10} a_{11} a_{31}+18 a_{01}^2 a_{10} a_{21}^2+3 a_{01}^2 a_{11} (a_{11} a_{30}+7 a_{20} a_{21})-25 a_{01} a_{10} a_{11}^2 a_{21} \\{} & {} -5 a_{01} a_{11}^3 a_{20}+5 a_{10} a_{11}^4)+\omega ^4 [36 a_{01}^5 a_{20} b_{40}+3 a_{01}^5 (9 a_{30} b_{30}+40 a_{40} b_{20}) \\{} & {} -18 a_{01}^4 a_{10} a_{11} b_{40}+156 a_{01}^4 a_{10} a_{20} a_{40}+3 a_{01}^4 a_{10} (-6 a_{21} b_{30}+9 a_{30}^2-50 a_{31} b_{20}) \\{} & {} +3 a_{01}^4 (5 a_{11} a_{20} b_{30}+9 a_{11} a_{30} b_{20}+20 a_{20}^2 a_{30}-14 a_{20} a_{21} b_{20})-138 a_{01}^3 a_{10}^2 a_{11} a_{40} \\{} & {} -3 a_{01}^3 a_{10}^2 (62 a_{20} a_{31}+15 a_{21} a_{30})-a_{01}^3 a_{10} (81 a_{11} a_{20} a_{30}+a_{11} (3 a_{11} b_{30}-43 a_{21} b_{20}) \\{} & {} +118 a_{20}^2 a_{21})+9 a_{01}^3 a_{11}^2 a_{20} b_{20}-20 a_{01}^3 a_{11} a_{20}^3+168 a_{01}^2 a_{10}^3 a_{11} a_{31}+18 a_{01}^2 a_{10}^3 a_{21}^2 \\{} & {} +2 a_{01}^2 a_{10}^2 a_{11} (15 a_{11} a_{30}+109 a_{20} a_{21})-10 a_{01}^2 a_{10} a_{11}^3 b_{20}+70 a_{01}^2 a_{10} a_{11}^2 a_{20}^2 \\{} & {} -104 a_{01} a_{10}^3 a_{11}^2 a_{21}-80 a_{01} a_{10}^2 a_{11}^3 a_{20}+30 a_{10}^3 a_{11}^4]+\omega ^2 [6 a_{01}^6 b_{20} (13 a_{20} b_{30} +10 a_{30} b_{20})\\{} & {} +a_{01}^5 a_{10} (-b_{20} (39 a_{11} b_{30}+40 a_{21} b_{20})+78 a_{20}^2 b_{30}+198 a_{20} a_{30} b_{20}) +18 a_{01}^5 a_{11} a_{20} b_{20}^2 \\{} & {} +40 a_{01}^5 a_{20}^3 b_{20}+a_{01}^4 a_{10}^2 (-117 a_{11} a_{20} b_{30}-159 a_{11} a_{30} b_{20}+138 a_{20}^2 a_{30})+a_{01}^4 a_{10} a_{11}^2 b_{20}^2\\{} & {} -158 a_{01}^4 a_{10}^2 a_{20} a_{21} b_{20}-102 a_{01}^4 a_{10} a_{11} a_{20}^2 b_{20}+40 a_{01}^4 a_{10} a_{20}^4-237a_{01}^3 a_{10}^3 a_{11} a_{20} a_{30}\\{} & {} +a_{11} (39 a_{11} b_{30}+119 a_{21} b_{20})-118 a_{20}^2 a_{21})+113 a_{01}^3 a_{10}^2 a_{11}^2 a_{20} b_{20}-160 a_{01}^3 a_{10}^2 a_{11} a_{20}^3\\{} & {} +a_{01}^2 a_{10}^4 a_{11} (99 a_{11} a_{30}+197 a_{20} a_{21})-46 a_{01}^2 a_{10}^3 a_{11}^3 b_{20}+250 a_{01}^2 a_{10}^3 a_{11}^2a_{20}^2\\{} & {} -79 a_{01} a_{10}^5 a_{11}^2 a_{21}-175 a_{01} a_{10}^4 a_{11}^3 a_{20}+45 a_{10}^5 a_{11}^4]-100 a_{01} a_{10}^6 a_{11}^3 a_{20}+20 a_{10}^7 a_{11}^4. \end{aligned}$$