Bifurcations driven by generalist and specialist predation: mathematical interpretation of Fennoscandia phenomenon

Abstract

In this paper, we revisit a predator–prey model with specialist and generalist predators proposed by Hanski et al. (J Anim Ecol 60:353–367, 1991) , where the density of generalist predators is assumed to be a constant. It is shown that the model admits a nilpotent cusp of codimension 4 or a nilpotent focus of codimension 3 for different parameter values. As the parameters vary, the model can undergo cusp type (or focus type) degenerate Bogdanov–Takens bifurcations of codimension 4 (or 3). Our results indicate that generalist predation can induce more complex dynamical behaviors and bifurcation phenomena, such as three small-amplitude limit cycles enclosing one equilibrium, one or two large-amplitude limit cycles enclosing one or three equilibria, three limit cycles appearing in a Hopf bifurcation of codimension 3 and dying in a homoclinic bifurcation of codimension 3. In addition, we show that generalist predation stabilizes the limit cycle driven by specialist predators to a stable equilibrium, which clearly explains the famous Fennoscandia phenomenon.

Appendices

Appendix A: The proof of Theorem 1

In this Appendix, we first show that $E_*$ is a cusp of codimension at most 4 in Theorem 1. We need to show that

$$\begin{aligned} \begin{array}{ll} \textrm{V}(M,\,M_1,\,M_2)\cap \Omega _{24}=\emptyset , \end{array} \end{aligned}$$

(A1)

where $\Omega _{24}$ is shown in (3.21), and the other notations are shown in Theorem 1.

We denote “$\textrm{V}(f_1,\,f_2,...,f_n)$” as the set of common zeros of $f_1,\,f_2,...,f_n$, “$\textrm{res}(f_1,\,f_2,\,x)$” as the resultant of $f_1$ and $f_2$ with respect to x, “$\textrm{prem}(f_1,\,f_2,\,x)$” as the pseudo-remainder of $f_1$ divided by $f_2$ with respect to x, and “$\textrm{lcoeff}(f_1,\,x)$” as the leading coefficient of $f_1$ with respect to x.

Step 1. Simplify the algebraic variety $\textrm{V}(M, M_1, M_2)\cap \Omega _{24}$.

By eliminating variables in the order $\eta \prec \alpha$, we have

$$\begin{aligned} \textrm{res}(M,\,M_1,\,\eta )&= 1024 \alpha ^2 u_*^{20} (\alpha +u_*)^8 (\alpha +u_*^2)^4r_{11},\nonumber \\ \textrm{res}(M,\,M_2,\,\eta )&= -262144 \alpha ^3 u_*^{34} (3 u_*-1) (\alpha +u_*)^{13} (\alpha +u_*^2)^7 r_{12},\nonumber \\ \textrm{res}(r_{11},\,r_{12},\,\alpha )&= 3676258543978604182634496000 (u_*-3){}^8 (u_*-1){}^7 u_*^{175} \nonumber \\&\times (3 u_*-1){}^9 r_{21}r_{22}, \end{aligned}$$

(A2)

where $r_{11}$ and $r_{12}$ are polynomials of $(u_*, \alpha )$, $r_{21}$ and $r_{22}$ are 17- and 54-order polynomials of $u_*$, and we omit their complicated expressions.

Notice that, $\textrm{lcoeff}(M, \eta )=\alpha (\alpha +3 u_*)> 0$ and $\textrm{lcoeff}(r_{11}, \alpha )=27 u_*^4+86 u_*^3-112 u_*^2-8 u_*-8< 0$ in $\Omega _{24}$. Similarly, from Theorem 1 in Chen and Zhang (2009) we can get that

$$\begin{aligned} \begin{array}{ll} \textrm{V}(M,\,M_1,\,M_2)\cap \Omega _{24}=\textrm{V}(M,\,M_1,\,M_2,\,r_{11},\,r_{12},\,r_{21}r_{22})\cap \Omega _{24}. \end{array} \end{aligned}$$

(A3)

Step 2. Simplify the algebraic variety $\textrm{V}(r_{11},\,r_{12},\,r_{21}r_{22})\cap \Omega _{24}$.

First, by using the Maple command $``\textrm{realroot}''$, we know that there exist one root $u_{*0}\in I_0$ for $\textrm{realroot}(r_{21}, 1/10^{10})$, and four roots $u_{*i}\in I_i$ (i=1,...,4) for $\textrm{realroot}(r_{22}, 1/10^{10})$ in $\Omega _{24}$, such that $r_{21}\mid _{u_{*}=u_{*0}}=r_{22}\mid _{u_{*}=u_{*i}}=0$ (i=1,...,4), where

$$\begin{aligned} \begin{array}{ll} &{}u_{*0}\doteq 0.005234,\ u_{*1}\doteq 0.017194,\ u_{*2}\doteq 0.022588,\\ &{} u_{*3}\doteq 0.070624,\ u_{*4}\doteq 0.2989611526, \end{array} \end{aligned}$$

(A4)

and

$$\begin{aligned} \begin{array}{ll} I_0=[\frac{1581734929692734191289}{302231454903657293676544},\frac{395433732423183547829}{75557863725914323419136}],\\[1.0ex] I_1=[\frac{20786838110638649144145}{1208925819614629174706176},\frac{41573676221277298288317}{2417851639229258349412352}],\\[1.0ex] I_2=[\frac{101727566390447}{4503599627370496},\frac{406910265561815}{18014398509481984}],\\[1.0ex] I_3=[\frac{5336187978172332291869}{75557863725914323419136},\frac{10672375956344664583765}{151115727451828646838272}],\\[1.0ex] I_4=[\frac{2629690108339}{8796093022208},\frac{5259380216687}{17592186044416}]. \end{array} \end{aligned}$$

(A5)

Second, by using pseudo-division we can get nine pseudo-remainders

$$\begin{aligned} \begin{array}{ll} &{}w_1=\textrm{prem}(r_{12},\,r_{11},\,\alpha ),\quad w_2=\textrm{prem} (r_{11},\,w_1,\,\alpha ),\\ {} &{} w_i=\textrm{prem}(w_{i-2},\,w_{i-1},\,\alpha ), \ (i=3,...,9), \end{array} \end{aligned}$$

(A6)

where $w_i$ (i=1,...,9) are (10-i)-order functions of $\alpha$, and the coefficients are functions of $u_*$.

Notice that, $\textrm{lcoeff}(r_{11},\, \alpha )\ne 0$, and by using Sturm’s theorem we can get that $\textrm{lcoeff}(w_i,\, \alpha )\ne 0$, (i=1,...,9) when $u_{*}\in \cup _{j=0}^{4}I_j$.

Combining above analysis, we can get that in $\cup _{j=0}^{4}I_j$, $\textrm{V}(r_{12},\,r_{11})\ne \emptyset$ and

$$\begin{aligned} \begin{array}{ll} \textrm{V}(r_{12},\,r_{11})=\textrm{V}(r_{11},\,w_1) =\textrm{V}(w_1,\,w_2)=...=\textrm{V}(w_8,\,w_9)\subseteq \textrm{V}(w_9). \end{array} \end{aligned}$$

(A7)

From (A6), we know that $w_9$ is a linear function of $\alpha$ which has a unique root, and from $w_9=0$ we can get that $\alpha =\alpha _0(u_*)$. By using Sturm’s theorem again, we can show that $\alpha _0(u_*)$ is a well defined monotone function and has no roots in $I_i$ (i=0,...,4). Therefore, we can obtain that

$$\begin{aligned} \begin{array}{ll} \alpha _0(u_{*0})\doteq 0.002379,\ \alpha _0(u_{*1})\doteq 0.022203,\ \alpha _0(u_{*2})\doteq 0.015347,\\ \alpha _0(u_{*3})\doteq 0.048550,\ \alpha _0(u_{*4})\doteq -0.199313, \end{array} \end{aligned}$$

(A8)

which show that when $u_*=u_{*4}$, we have $\alpha _0(u_*)<0$, that is $\textrm{V}(w_9,\,r_{22})\cap \Omega _{24}=\emptyset$. So that, we only need to consider the cases $u_*=u_{*i}$ (i=0,...,3).

Notice that, $(u_*, \alpha _0(u_*))\in \Omega _{21}$. In the following, we first consider the monotonicity of $\alpha _1$ and $\alpha _2$ respect to $u_*$ where $u_*\in (0, \frac{3-\sqrt{6}}{12} ]$. Through simple calculation, we can get that

$$\begin{aligned} \begin{array}{ll} \frac{d\alpha _1}{du_*}=\frac{36 u_*-1-96 u_*^3-72 u_*^2+(1-16 u_*^2-16 u_*) \sqrt{48 u_*^2-24 u_*+1} }{2 (2 u_*+1){}^2 \sqrt{48 u_*^2-24 u_*+1}}\triangleq \frac{\alpha _{11}+\alpha _{12}\sqrt{48 u_*^2-24 u_*+1} }{2 (2 u_*+1){}^2 \sqrt{48 u_*^2-24 u_*+1}}, \end{array} \end{aligned}$$

moreover, we have $\alpha _{11}^{2}-\alpha _{12}^{2}(48 u_*^2-24 u_*+1)<0\ (=0,\ \textrm{or}\ >0)$ if $0<u_*<0.035607\ (u_*=0.035607,\ \textrm{or}\ 0.035607<u_*\le \frac{3-\sqrt{6}}{12})$, that is $\textrm{sign}\frac{d\alpha _1}{du_*}=\textrm{sign}\alpha _{12}\ (\frac{\textrm{sign}\alpha _{11}+\textrm{sign}\alpha _{12}}{2},\ \textrm{or}\ \textrm{sign}\alpha _{11}) \textrm{if} 0<u_*<0.035607\ (u_*=0.035607,\ \textrm{or}\ 0.035607<u_*\le \frac{3-\sqrt{6}}{12})$. Further, we have $\alpha _{11}>0$ when $0.035607\le u_*\le \frac{3-\sqrt{6}}{12}$, and $\alpha _{12}>0$ when $0<u_*\le 0.035607$. Therefore, we can get that $\frac{d\alpha _1}{du_*}>0$ when $u_*\in (0, \frac{3-\sqrt{6}}{12} ]$. Similarly, we can show that $\frac{d\alpha _2}{du_*}>0\ (=0,\ <0)$ when $0<u_*<0.035607\ (u_*=0.035607,\ \textrm{or}\ 0.035607<u_*\le \frac{3-\sqrt{6}}{12})$.

Therefore, from the above analysis, we know that $\alpha _1$ and $\alpha _2$ are monotone when $u_*\in \cup _{j=0}^{2}I_j$, and we have

$$\begin{aligned} \begin{array}{ll} \alpha _1(u_{*0})\doteq 0.000058,\ \alpha _2(u_{*0})\doteq 0.004905,\ \alpha _1(u_{*1})\doteq 0.000722,\\ \alpha _2(u_{*1})\doteq 0.013614,\ \alpha _1(u_{*2})\doteq 0.001348,\ \alpha _2(u_{*2})\doteq 0.016358. \end{array} \end{aligned}$$

(A9)

Combining (A4), (A8) and (A9), when $u_*=u_{*i}$ (i=0, 2, 4), we have $(u_{*i}, \alpha _0(u_{*i}))\notin \Omega _{21}$, that is $\textrm{V}(w_9,\,r_{21}r_{22})\cap \Omega _{24}=\emptyset$, i.e., $\textrm{V}(r_{11},\,r_{12},\,r_{21}r_{22})\cap \Omega _{24}=\emptyset$. When $u_*=u_{*i}$ (i=1, 3), we have $(u_{*i}, \alpha _0(u_{*i}))\in \Omega _{21}$, that is $\textrm{V}(w_9,\,r_{21}r_{22})\cap \Omega _{24}=\textrm{V}(w_9,\,r_{22})\cap \Omega _{24}=\{(u_*, \alpha ):\ u_*=u_{*i}\ \textrm{and}\ \alpha =\alpha _0(u_{*i}),\ i=1,\ 3\}\ne \emptyset$, then we can get that

$$\begin{aligned} \textrm{V}(r_{11},\,r_{12},\,r_{21}r_{22})\cap \Omega _{24}&=\ \textrm{V}(r_{11},\,r_{12},\,r_{22})\cap \Omega _{24}\nonumber \\&=\ \cup _{i=1, 3} \{(u_*, \alpha ):\ u_*=u_{*i}\ \textrm{and}\ \alpha =\alpha _0(u_{*i})\}, \end{aligned}$$

(A10)

where $u_{*i}$ and $\alpha _0(u_{*i})$ are shown in (A4) and (A8), respectively.

Summarizing the above analysis and from (A3) we can get that

$$\begin{aligned}&\textrm{V}(M,\,M_1,\,M_2,\,r_{11},\,r_{12},\,r_{21}r_{22})\cap \Omega _{24}\nonumber \\&\quad =\textrm{V}(M,\,M_1,\,M_2,\,r_{11},\,r_{12},\,r_{22})\cap \Omega _{24}\nonumber \\&\quad =\textrm{V}(M,\,M_1,\,M_2)\cap \big (\cup _{i=1, 3} \big \{(u_*, \alpha ):\ (u_*,\,\alpha )=(u_{*i},\,\alpha _0(u_{*i}))\big \}\big )\cap \Omega _{24}\nonumber \\&\quad \triangleq \cup _{i=1, 3} \mathrm{V_i}, \end{aligned}$$

(A11)

where

$$\begin{aligned} \begin{array}{ll} \mathrm{V_i}=\textrm{V}(M,\,M_1,\,M_2)\cap \big \{(u_*, \alpha ):\ (u_*,\,\alpha )= (u_{*i},\,\alpha _0(u_{*i}))\big \}\cap \Omega _{24}. \end{array} \end{aligned}$$

(A12)

Step 3. Prove that $\textrm{V}_1=\emptyset$  in (A11).

First, we simplify the algebraic variety $\textrm{V}(M,\,M_1)$ in (A11). By using pseudo-division again, we can get one pseudo-remainder

$$\begin{aligned} \begin{array}{ll} w_{10}=\textrm{prem}(M_1,\,M,\,\eta ), \end{array} \end{aligned}$$

(A13)

where $w_{10}$ is 1-order function of $\eta$, and the coefficients are functions of $(u_*,\,\alpha )$. Then we can get that

$$\begin{aligned} \begin{array}{ll} \textrm{V}(M_1,\,M)=\textrm{V}(M,\,w_{10})\subseteq \textrm{V}(w_{10}). \end{array} \end{aligned}$$

(A14)

From (A13), we know that $w_{10}$ is a linear function of $\eta$ which has a unique root, and from $w_{10}=0$ we can get that $\eta =\eta _{00}(u_*,\,\alpha )$, where

$$\begin{aligned} \begin{array}{ll} &{}\eta _{00}(u_*,\,\alpha )\\ &{}\quad =[u_*^2 (-\alpha ^{10} (27 u_*^3-130 u_*^2+64 u_*+68)-\alpha ^9 (231 u_*^3-1358 u_*^2+882 u_*\\ &{}\qquad +276) u_*-2 \alpha ^8 (252 u_*^3-2567 u_*^2+1931 u_*+180) u_*^2+2 \alpha ^7 (438 u_*^3+3625 u_*^2\\ &{}\qquad -3981 u_*+128) u_*^3+2 \alpha ^6 (2385 u_*^3-2701 u_*^2-2579 u_*+618) u_*^4+2 \alpha ^5 \\ &{}\qquad \times (1311 u_*^3-14171 u_*^2+5013 u_*+402) u_*^5-6 \alpha ^4 (1956 u_*^3+3817 u_*^2-3453 u_*\\ &{}\qquad +132) u_*^6-6 \alpha ^3 (2406 u_*^3-1897 u_*^2-1827 u_*+144) u_*^7+9 \alpha ^2 (777 u_*^2+2032 u_*\\ &{}\qquad -18) u_*^9+27 \alpha (547 u_*+80) u_*^{11}+3240 u_*^{13})]\big /[\alpha ^{10} (27 u_*^3-238 u_*^2+584 u_*\\ &{}\qquad -396)+\alpha ^9 (123 u_*^3-1438 u_*^2+3850 u_*-2396) u_*-2 \alpha ^8 (48 u_*^3+1063 u_*^2\\ &{}\qquad -4771 u_*+3060) u_*^2+2 \alpha ^7 (-654 u_*^3+2495 u_*^2+2993 u_*-3384) u_*^3-2 \alpha ^6\\ &{}\qquad \times (477 u_*^3-8865 u_*^2+8669 u_*-1022) u_*^4+2 \alpha ^5 (2565 u_*^3+2883 u_*^2-16321 u_*\\ &{}\qquad +7254) u_*^5+18 \alpha ^4 (360 u_*^3-1805 u_*^2-295 u_*+852) u_*^6-54 \alpha ^3 (162 u_*^3\\ &{}\qquad +553 u_*^2-489 u_*-104) u_*^7-27 \alpha ^2 (507 u_*^2-540 u_*-622) u_*^9+81 \alpha (67 u_*\\ &{}\qquad +240) u_*^{11}+9720 u_*^{13}], \end{array}\nonumber \\ \end{aligned}$$

(A15)

and notice that $\big (u_{*1},\,\alpha _0(u_{*1}),\,\eta _{00}\big (u_{*1},\,\alpha _0(u_{*1})\big )\big )\in \Omega _{24}$.

Denote $\eta _{00}(u_*,\,\alpha )\triangleq \frac{\eta _{001}}{\eta _{002}}$, where $\eta _{001}$ and $\eta _{002}$ are polynomials of $(u_*,\,\alpha )$, and we know that $(u_{*1},\,\alpha _0(u_{*1}))\in I_1\times [\frac{2220321857376}{10^{14}}, \frac{2220332345838}{10^{14}}]\triangleq [u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$, where $I_1$ is shown in (A5).

We first prove that $\eta _{002}\ne 0$ in the region $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$ by the following 2 substeps.

Step 3.1. Prove that $\eta _{002}$  has no critical point in the interior of the region.

The corresponding first-order partial derivatives of $\eta _{002}$ as $\frac{\partial \eta _{002}}{\partial u_*}$ and $\frac{\partial \eta _{002}}{\partial \alpha }$, where we omit the detailed expressions. By using Sturm’s theorem, we can get that $\textrm{res}\big (\frac{\partial \eta _{002}}{\partial u_*},\, \frac{\partial \eta _{002}}{\partial \alpha },\, \alpha \big )$ has no root in $[u_{00}, u_{01}]$, and we have $\textrm{lcoeff}(\frac{\partial \eta _{002}}{\partial u_*},\,\alpha )=81{u_*}^{2}-476u_*+584 \ne 0$. Then, we can get that

$$\begin{aligned} \begin{aligned} \textrm{V}\Big (\frac{\partial \eta _{002}}{\partial u_*},\, \frac{\partial \eta _{002}}{\partial \alpha }\Big )=\textrm{V} \Big (\frac{\partial \eta _{002}}{\partial u_*},\, \frac{\partial \eta _{002}}{\partial \alpha },\, \textrm{res}\big (\frac{\partial \eta _{002}}{\partial u_*},\, \frac{\partial \eta _{002}}{\partial \alpha },\, \alpha \big )\Big )=\emptyset \end{aligned} \end{aligned}$$

in $[u_{00}, u_{01}]$, which shows that $\eta _{002}$ has no critical point in the interior of $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$, moreover, the maximum and minimum of $\eta _{002}$ can only be achieved on the boundary of this region.

Step 3.2. Prove that  $\eta _{002}$  is monotone and rootless at the boundary of the region.

By using Sturm’s theorem again, we can get that

$$\begin{aligned} \begin{aligned} \eta _{002}\mid _{u_*=u_{00}},\ \ \eta _{002}\mid _{u_*=u_{01}},\ \ \frac{d (\eta _{002}\mid _{u_*=u_{00}})}{d\alpha },\ \ \frac{d ( \eta _{002}\mid _{u_*=u_{01}})}{d\alpha } \end{aligned} \end{aligned}$$

have no roots in $[\alpha _{00}, \alpha _{01}]$, and

$$\begin{aligned} \begin{aligned} \eta _{002}\mid _{\alpha =\alpha _{00}},\ \ \eta _{002}\mid _{\alpha =\alpha _{01}},\ \ \frac{d (\eta _{002}\mid _{\alpha =\alpha _{00}})}{du_*},\ \ \frac{d ( \eta _{002}\mid _{\alpha =\alpha _{01}})}{du_*} \end{aligned} \end{aligned}$$

have no roots in $[u_{00}, u_{01}]$.

Hence, by calculating the values of four vertices of rectangular field $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$, we can get that $\eta _{002}\in [-1.719412\times 10^{-16}, -1.676935\times 10^{-16}]$ in $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$, which shows that $\eta _{002}$ is well defined in this region.

Second, by using the same techniques, we analyse the value range of $\eta _{00}(u_*,\,\alpha )$ in $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$. Through a series of calculations, we have

$$\begin{aligned} \begin{array}{ll} \frac{\partial \eta _{00}(u_*,\,\alpha )}{\partial \alpha }=\frac{4u_*^{3}\eta _{003}}{\eta _{002}^{2}},\quad \frac{\partial \eta _{00}(u_*,\,\alpha )}{\partial u_*}=\frac{-2u_*(\alpha +u_*)\eta _{004}}{\eta _{002}^{2}}, \end{array} \end{aligned}$$

where $\eta _{003}$ and $\eta _{004}$ are polynomials of $(u_*,\,\alpha )$, and $\textrm{lcoeff}(\eta _{003},\,\alpha )= 729{u}^{6}-9207{u}^{5}+45790{u}^{4}-109538{u}^{3}+131768{u}^{2}-74136u+13408\ne 0$ in $[u_{00}, u_{01}]$. By using Sturm’s theorem again, we can get that $\textrm{res}(\eta _{003},\,\eta _{004},\,\alpha )$ has no real roots in $[u_{00}, u_{01}]$.

Combining the above analysis, we have

$$\begin{aligned} \begin{array}{ll} \textrm{V}(\eta _{003},\,\eta _{004})=\textrm{V}(\eta _{003},\,\eta _{004},\,\textrm{res}(\eta _{003},\,\eta _{004},\,\alpha ))=\emptyset \end{array} \end{aligned}$$

in $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$, which shows that $\eta _{00}(u_*,\,\alpha )$ has no critical point in the interior of $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$. Therefore, the maximum and minimum of $\eta _{00}(u_*,\,\alpha )$ can only be achieved on the boundary of this region.

By using Sturm’s theorem, we can get that

$$\begin{aligned} \begin{aligned} \eta _{00}(u_{00},\,\alpha ),\ \ \eta _{00}(u_{01},\,\alpha ),\ \ \frac{d(\eta _{00}(u_{00},\,\alpha ))}{d\alpha },\ \ \frac{d(\eta _{00}(u_{01},\,\alpha ))}{d\alpha } \end{aligned} \end{aligned}$$

have no roots in $[\alpha _{00}, \alpha _{01}]$, moreover, $\frac{d(\eta _{00}(u_{00},\,\alpha ))}{d\alpha }<0\ \textrm{and}\ \frac{d(\eta _{00}(u_{01},\,\alpha ))}{d\alpha }<0$ in $[\alpha _{00}, \alpha _{01}]$. Similarly,

$$\begin{aligned} \begin{aligned} \eta _{00}(u_*,\,\alpha _{00}),\ \ \eta _{00}(u_*,\,\alpha _{01}),\ \ \frac{d(\eta _{00}(u_*,\,\alpha _{00}))}{du_*},\ \ \frac{d(\eta _{00}(u_*,\,\alpha _{01}))}{du_*} \end{aligned} \end{aligned}$$

have no roots in $[u_{00}, u_{01}]$, moreover, $\frac{d(\eta _{00}(u_*,\,\alpha _{00}))}{du_*}>0\ \textrm{and}\ \frac{d(\eta _{00}(u_*,\,\alpha _{01}))}{du_*}>0$ in $[u_{00}, u_{01}]$.

Hence, the minimum and maximum of $\eta _{00}(u_*,\,\alpha )$ in $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$ are $\eta _{00}(u_{00},\,\alpha _{01})\doteq 0.000920$ and $\eta _{00}(u_{01},\,\alpha _{00})\doteq 0.000943$, respectively. Therefore, we have $\eta _{00}(u_{*1},\,\alpha _0(u_{*1}))\in [0.000920, 0.000943]$. Similarly, we have $\eta _0\doteq 0.000285$ in $[u_{00}, u_{01}]$. Therefore, we can get that $\eta _{00}(u_{*1},\,\alpha _0(u_{*1}))>\eta _0$. That is, $\big (u_{*1},\,\alpha _0(u_{*1}),\,\eta _{00}\big (u_{*1},\,\alpha _0(u_{*1})\big )\big )\notin \Omega _{24}$.

Summarizing the above analysis, we can obtain that

$$\begin{aligned} \begin{array}{ll} \mathrm{V_1}&{}= \textrm{V}(M,\,M_1,\,M_2)\cap \big \{(u_*, \alpha ):\ (u_*,\,\alpha )=(u_{*1},\,\alpha _0(u_{*1}))\big \}\cap \Omega _{24}\\ &{}\subseteq \textrm{V}(w_{10},\,M_2)\cap \big \{(u_*, \alpha ):\ (u_*,\,\alpha )=(u_{*1},\,\alpha _0(u_{*1}))\big \}\cap \Omega _{24}\\ &{}= \textrm{V}(M_2)\cap \big \{(u_*, \alpha , \eta ):\ (u_*,\, \alpha ,\, \eta )=\big (u_{*1},\,\alpha _0(u_{*1}),\,\eta _{00}(u_{*1},\,\alpha _0(u_{*1}))\big )\big \}\\ {} &{}\quad \, \cap \Omega _{24}\\ &{}= \emptyset . \end{array} \end{aligned}$$

Step 4. Prove that $\textrm{V}_3=\emptyset$ in (A11).

In the following, we use the same method as Step 3 to analyse $\textrm{V}_3$. We first simplify the algebraic variety $\textrm{V}(M,\,M_1)$ in (A11), from (A13) and (A14) we know that

$$\begin{aligned}{} & {} \textrm{V}(M,\,M_1)\cap \big \{(u_*, \alpha ):\ (u_*,\,\alpha )=(u_{*3},\,\alpha _0(u_{*3}))\big \}\cap \Omega _{24} \nonumber \\{} & {} \quad \subseteq \textrm{V}(w_{10})\cap \big \{(u_*, \alpha ):\ (u_*,\,\alpha )=(u_{*3},\,\alpha _0(u_{*3}))\big \}\cap \Omega _{24} \nonumber \\{} & {} = \big \{(u_*, \alpha , \eta ):\ (u_*,\, \alpha ,\, \eta )=\big (u_{*3},\,\alpha _0(u_{*3}),\,\eta _{00}(u_{*3},\,\alpha _0(u_{*3}))\big )\big \}\cap \Omega _{24}.\qquad \end{aligned}$$

(A16)

We next prove that $\big (u_{*3},\,\alpha _0(u_{*3}),\,\eta _{00}\big (u_{*3},\,\alpha _0(u_{*3})\big )\big )\in \Omega _{24}$. In fact, we have $(u_{*3},\,\alpha _0(u_{*3}))\in I_3\times [\frac{48549816647411}{10^{15}}, \frac{48549816647417}{10^{15}}]\triangleq [u_{10}, u_{11}]\times [\alpha _{10}, \alpha _{11}]$, where $I_3$ is shown in (A5).

Similarly, we can also show that $\eta _{00}(u_*,\,\alpha )$ is well defined and

$$\begin{aligned} \begin{array}{ll} \eta _{00}(u_{*3},\,\alpha _0(u_{*3}))\in [0.00026494773853172, 0.00026494773853175] \end{array} \end{aligned}$$

(A17)

in $[u_{10}, u_{11}]\times [\alpha _{10}, \alpha _{11}]$, and $\eta _0\doteq 0.004283$ in $[u_{10}, u_{11}]$. Hence, we can get that $\eta _{00}(u_{*3},\,\alpha _0(u_{*3}))<\eta _0$. That is, $\big (u_{*3},\,\alpha _0(u_{*3}),\,\eta _{00}\big (u_{*3},\,\alpha _0(u_{*3})\big )\big )\in \Omega _{24}$.

Therefore, we can obtain that

$$\begin{aligned} \begin{array}{ll} &{}\textrm{V}(M,\,M_1)\cap \Omega _{24}\supseteq \big \{(u_*, \alpha , \eta ):\ (u_*,\, \alpha ,\, \eta )\\ &{}\quad =\big (u_{*3},\,\alpha _0(u_{*3}),\,\eta _{00}(u_{*3},\,\alpha _0(u_{*3}))\big )\big \}\cap \Omega _{24}\ne \emptyset , \end{array} \end{aligned}$$

(A18)

then we can get that $E_*$ is a cusp of codimension at least 4.

Second, we continue to analyze the algebraic variety $\textrm{V}(M,\,M_2)$ in (A11), by using pseudo-division once again, we can get one pseudo-remainder,

$$\begin{aligned} \begin{array}{ll} w_{11}=\textrm{prem}(M_2,\,M,\,\eta ), \end{array} \end{aligned}$$

(A19)

where $w_{11}$ is 1-order function of $\eta$, and the coefficients are functions of $(u_*,\,\alpha )$. Then we have

$$\begin{aligned} \begin{array}{ll} \textrm{V}(M_2,\,M)=\textrm{V}(M,\,w_{11})\subseteq \textrm{V}(w_{11}), \end{array} \end{aligned}$$

(A20)

and from $w_{11}=0$ we can get a unique root $\eta =\eta _{01}(u_*,\,\alpha )$.

Similarly, we can also show that $\eta _{01}(u_*,\,\alpha )$ is well defined and $\eta _{01}(u_{*3},\,\alpha _0(u_{*3}))\in [0.014132345, 0.014132346]$ in $[u_{10}, u_{11}]\times [\alpha _{10}, \alpha _{11}]$. Hence, we can get that $\eta _{01}(u_{*3},\,\alpha _0(u_{*3}))>\eta _0$, that is, $\big (u_{*3},\,\alpha _0(u_{*3}),\,\eta _{01}\big (u_{*3},\,\alpha _0(u_{*3})\big )\big )\notin \Omega _{24}$.

Therefore, we can obtain that

$$\begin{aligned} \begin{array}{ll} \mathrm{V_3}&{}= \textrm{V}(M,\,M_1,\,M_2)\cap \big \{(u_*, \alpha ):\ (u_*,\,\alpha )=(u_{*3},\,\alpha _0(u_{*3}))\big \}\cap \Omega _{24}\\ &{}\subseteq \textrm{V}(w_{11},\,M_1)\cap \big \{(u_*, \alpha ):\ (u_*,\,\alpha )=(u_{*3},\,\alpha _0(u_{*3}))\big \}\cap \Omega _{24}\\ &{}= \textrm{V}(M_1)\cap \big \{(u_*, \alpha , \eta ):\ (u_*,\, \alpha ,\, \eta )=\big (u_{*3},\,\alpha _0(u_{*3}),\,\eta _{01}(u_{*3},\, \alpha _0(u_{*3}))\big )\big \}\\ {} &{}\quad \, \cap \Omega _{24}\\ &{}= \emptyset . \end{array} \end{aligned}$$

Combining the above analysis and from (A3) and (A11), we can get that

$$\begin{aligned} \begin{array}{ll} \textrm{V}(M,\,M_1,\,M_2)\cap \Omega _{24}=\emptyset , \ \ \textrm{V}(M,\,M_1)\cap \Omega _{24}\ne \emptyset , \end{array} \end{aligned}$$

which shows that $E_*$ is a cusp of codimension exactly 4.

It is easy to know that $\Omega _{25}\subseteq \textrm{V}(M,\,M_1)\cap \Omega _{24}$, where $\Omega _{25}$ is shown in (1.5), and when $(u_*,\,\alpha ,\,\eta )\in \Omega _{25}$ we have $M=M_1=0$ and $M_2\ne 0$, that is $E_*$ is a cusp of codimension exactly 4.

Appendix B: The proof of $\overline{d}_{20}<0$ and $\overline{d}_{41}<0$ in (3.26) of Theorem 2

From (3.26), when $\mu =0$ we have

$$\begin{aligned} \begin{array}{ll} \overline{d}_{20}=\frac{(\eta +2 u_*^3-u_*^2)\overline{d_{200}}}{u_*^2 \left( \eta +u_*^2\right) \left( -\eta +2 \alpha u_*+u_*^2\right) {}^2},\\ \overline{d}_{41}=\frac{\overline{d_{410}}}{144 u_*^5 \left( \alpha +u_*\right) {}^4 \left( \eta +u_*^2\right) {}^4 \left( \eta +2 u_*^3-u_*^2\right) \left( \eta -2 \alpha u_*-u_*^2\right) \overline{d_{200}}^3}, \end{array} \end{aligned}$$

(B1)

where

$$\begin{aligned} \begin{array}{ll} \overline{d_{200}}=\alpha \eta ^2-3 \alpha ^2 \eta u_*+\alpha ^2 u_*^3-\alpha \eta u_*^3-3 \alpha \eta u_*^2+3 \alpha u_*^5-3 \eta u_*^4+u_*^6, \end{array} \end{aligned}$$

and we omit the expression of $\overline{d_{410}}$. Notice that, $(u_*,\, \alpha ,\, \eta )\in \Omega _{25}$, $\Omega _{25}$ is shown in (1.5), that is $(u_*, \alpha )=(u_{*3},\,\alpha _0(u_{*3}))\in I_3\times [\frac{48549816647411}{10^{15}}, \frac{48549816647417}{10^{15}}]\triangleq [u_{10}, u_{11}]\times [\alpha _{10}, \alpha _{11}]$, and $\eta =\eta _{00}(u_*,\, \alpha )$, where $\eta _{00}(u_*,\, \alpha )$ is shown in (A15).

We first calculate $\textrm{sign}\overline{d_{200}}$. Substitute the parameter $\eta =\eta _{00}(u_*,\, \alpha )$ into $\overline{d_{200}}$, we can get that $\overline{d_{200}}\mid _{\eta =\eta _{00}(u_*,\, \alpha )}=\frac{4u_*^3\alpha (u_*+\alpha )^2\overline{d_{201}}}{\eta _{002}^2}$, where $\overline{d_{201}}$ is a polynomial of $(u_*,\, \alpha )$, we omit the detailed expression. Therefore, we have

$$\begin{aligned} \begin{array}{ll} \textrm{sign}\Big (\overline{d_{200}}\mid _{\eta =\eta _{00}(u_*,\, \alpha )}\Big )=\textrm{sign}\overline{d_{201}}. \end{array} \end{aligned}$$

By using the same method in Steps 3.1–3.2 of Appendix 1, we can get that $\overline{d_{201}}\in [4.7852072357$ $861\times 10^{-16},\,4.7852072357899\times 10^{-16}]$ in $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$. Moreover, combining condition (3.5), we know that $\overline{d_{20}}<0$ for small $\mu$.

Similarly, we can get that $\overline{d_{410}}\mid _{\eta =\eta _{00}(u_*,\, \alpha )}=\frac{-16384{u_*}^{21}\alpha \left( \alpha +u_* \right) ^{11}\overline{d_{411}}}{\eta _{002}^{12}}$, where $\overline{d_{411}}$ is a polynomial of $(u_*,\, \alpha )$, and we omit the detailed expression. Therefore, we have

$$\begin{aligned} \begin{array}{ll} \textrm{sign}\Big (\overline{d_{410}}\mid _{\eta =\eta _{00}(u_*,\, \alpha )}\Big )=-\textrm{sign}\overline{d_{411}}, \end{array} \end{aligned}$$

moreover, we can get that $\overline{d_{411}}\in [2.113486566436\times 10^{-96},\,2.113486566447\times 10^{-96}]$ in $[u_{00}, u_{01}]\times [\alpha _{00}, \alpha _{01}]$, which shows that $\overline{d_{41}}<0$ for small $\mu$.

Appendix C: The proof of nondegeneracy condition (3.29) of Theorem 2

In order to show that the nondegeneracy condition (3.29) holds when $(u_*,\, \alpha ,\, \eta )\in \Omega _{25}$, where $\Omega _{25}$ is shown in (1.5), we just need to show that $\overline{f}_{1}\overline{f}_{21}\overline{f}_{22}\ne 0$. Notice that, $\Omega _{25}\subseteq \textrm{V}(M,\,M_1,\,r_{11},\,r_{12},$ $r_{22})\cap \Omega _{24}$, so that we first prove $\textrm{V}(M,\,M_1,\,r_{11},\,r_{12},\,r_{22},\,\overline{f}_{1})\cap \Omega _{24}=\emptyset$.

By eliminating variables in the order $\eta \prec \alpha \prec u_*$, we have

$$\begin{aligned} \begin{aligned} \textrm{res}(M,\,\overline{f}_1,\,\eta )=&\ 18239578490850508800 \alpha ^{10} u_*^{96} (3 u_*-1){}^6 (\alpha +u_*){}^{40} \\ {}&\ (\alpha +u_*^2){}^{21}r_{31},\\ \textrm{res}(r_{11},\,r_{31},\,\alpha )=&\ C_0(u_*-3){}^{18} (u_*-1){}^{17} u_*^{407} (u_*+5) (3 u_*-1){}^{18}r_{32},\\ \textrm{res}(r_{22},\,r_{32},\,u_*)\ne&\ 0, \end{aligned} \end{aligned}$$

where $C_0$ is a positive constant, $r_{31}$ and $r_{32}$ are polynomials of $(u_*, \alpha )$ and $u_*$, respectively, and we omit their complicated expressions.

From Appendix 1Step 1, we know that $\textrm{lcoeff}(M, \eta )> 0$ and $\textrm{lcoeff}(r_{11}, \alpha )< 0$ in $\Omega _{25}$. Similarly, from Theorem 1 in Chen and Zhang (2009) we can get that

$$\begin{aligned} \begin{array}{ll} &{}\textrm{V}(M,\,M_1,\,r_{11},\,r_{12},\,r_{22},\,\overline{f}_{1})\cap \Omega _{24}\\ &{}\quad =\textrm{V}(M,\,M_1,\,r_{11},\,r_{12},\,r_{22},\,\overline{f}_{1},\,r_{31},\,r_{32})\cap \Omega _{24}=\emptyset , \end{array} \end{aligned}$$

(C1)

which shows that $\overline{f}_{1}\ne 0$ in $\Omega _{25}$.

Second, we prove $\textrm{V}(M,\,M_1,\,r_{11},\,r_{12},\,r_{22},\,\overline{f}_{21} \overline{f}_{22})\cap \Omega _{24}=\emptyset$. Similarly, by eliminating variables in the order $\eta \prec \alpha \prec u_*$, we have

$$\begin{aligned} \begin{aligned} \textrm{res}(M,\,\overline{f}_{21}\overline{f}_{22},\,\eta )=&\ 21743271936 \alpha ^7 u_*^{48} (3 u_*-1){}^3 (\alpha +u_*){}^{21} (\alpha +u_*^2){}^{11}r_{41},\\ \textrm{res}(r_{11},\,r_{41},\,\alpha )=&\ 3676258543978604182634496000 (u_*-3)^8 (u_*-1)^7 u_*^{175} \\&\ (3 u_*-1)^9r_{42}r_{22}=0,\\ \textrm{res}(r_{22},\,r_{42},\,u_*)\ne&\ 0, \end{aligned} \end{aligned}$$

where $r_{41}$ and $r_{42}$ are polynomials of $(u_*, \alpha )$ and $u_*$, respectively, and we omit their complicated expressions. From $\textrm{V}(r_{11},\,r_{41})\ne \emptyset$, we have

$$\begin{aligned} \begin{array}{ll} \textrm{V}(M,\,\overline{f}_{21}\overline{f}_{22})\cap \Omega _{24}=\big (\textrm{V} (M,\,\overline{f}_{21})\cup \textrm{V}(M,\,\overline{f}_{22})\big )\cap \Omega _{24}\ne \emptyset . \end{array} \end{aligned}$$

(C2)

By using pseudo-division, we can get that

$$\begin{aligned} \begin{array}{ll} \textrm{prem}(M,\,\overline{f}_{21},\,\eta )=-u_* \left( \alpha +u_*\right) {}^2 \left( \alpha +u_*^2\right) \left( u_*^2-3 \eta \right) , \end{array} \end{aligned}$$

and from (3.29) we have $\textrm{lcoeff}(\overline{f}_{21},\,\eta )=\alpha \ne 0$, then we can get that

$$\begin{aligned} \begin{array}{ll} \textrm{V}(M,\,\overline{f}_{21})=\textrm{V}(\overline{f}_{21},\,(u_*^2-3 \eta ))\subseteq \{\eta :\ \eta =\frac{u_*^2}{3}\}. \end{array} \end{aligned}$$

Moreover, from (A17), when $u_*\in [u_{10}, u_{11}]$ we have $\frac{u_*^2}{3}\doteq 0.00166258>\eta _{00}(u_{*3},\,\alpha _0(u_{*3}))$, which shows that

$$\begin{aligned} \begin{array}{ll} &{}\textrm{V}(M,\,M_1,\,r_{11},\,r_{12},\,r_{22},\,\overline{f}_{21})\cap \Omega _{24}\\ &{}\quad =\textrm{V}(M,\,M_1,\,r_{11},\,r_{12},\,r_{22},\,\overline{f}_{21},(u_*^2-3 \eta ))\cap \Omega _{24}=\emptyset . \end{array} \end{aligned}$$

(C3)

By using pseudo-division again, we can get that

$$\begin{aligned} \begin{array}{ll} \textrm{prem}(\overline{f}_{22},\,M,\,\eta )=-\frac{9216 u_*^{19} \left( \alpha +u_*\right) {}^6 \left( \alpha +u_*^2\right) {}^4}{\alpha ^7 \left( \alpha +3 u_*\right) {}^{11}}w_{12}, \end{array} \end{aligned}$$

where $w_{12}$ is 1-order function of $\eta$, and the coefficients are functions of $(u_*,\,\alpha )$. Then we can get that

$$\begin{aligned} \begin{array}{ll} \textrm{V}(\overline{f}_{22},\,M)=\textrm{V}(M,\,w_{12})\subseteq \textrm{V}(w_{12}). \end{array} \end{aligned}$$

and from $w_{12}=0$ we can get a unique root $\eta =\eta _{02}(u_*,\,\alpha )$.

By using the method in Appendix 1 Steps 3.1–3.2, we can also show that $\eta _{02}(u_*,\,\alpha )$ is well defined and $\eta _{02}(u_{*3},\,\alpha _0(u_{*3}))\in [0.01413234,$ 0.01413235] in $[u_{10}, u_{11}]\times [\alpha _{10}, \alpha _{11}]$. Hence, we have $\eta _{02}(u_{*3},\,\alpha _0(u_{*3}))>\eta _0$, that is, $\big (u_{*3},\,\alpha _0(u_{*3}),\,\eta _{02}\big (u_{*3},\,\alpha _0 (u_{*3})\big )\big )\notin \Omega _{24}$.

Therefore, we can get that

$$\begin{aligned} \begin{array}{ll} &{}\textrm{V}(M,\,M_1,\,r_{11},\,r_{12},\,r_{22},\,\overline{f}_{22})\cap \Omega _{24}\\ &{}\quad \subseteq \textrm{V}(M_1,\,r_{11},\,r_{12},\,r_{22},\,w_{12})\cap \Omega _{24}\\ &{}\quad = \textrm{V}(M_1)\cap \{(u_*, \alpha , \eta ):\, (u_*,\, \alpha ,\, \eta )\\ &{}\quad =(u_{*3},\, \alpha _0(u_{*3}),\,\eta _{02}(u_{*3},\,\alpha _0(u_{*3})))\}\cap \Omega _{24}= \emptyset . \end{array} \end{aligned}$$

(C4)

From (C1), (C3) and (C4), we can obtain that $\overline{f}_{1}\overline{f}_{21}\overline{f}_{22}\ne 0$ when $(u_*,\, \alpha ,\, \eta )\in \Omega _{25}$, that is the nondegeneracy condition (3.29) holds.

Appendix D: The proof of $0<\hat{b}_{11}\mid _{\lambda =0}<2\sqrt{2}$ in (3.55)

Notice that, in (3.55) we have

$$\begin{aligned} \begin{array}{ll} \widehat{d}_1=&{} \alpha ^2 \eta ^3-4 \alpha ^3 \eta ^2 u^*-7 \alpha ^2 \eta ^2 (u^*)^2+2 (u^*)^7 (\alpha ^2+2 \eta )+4 \eta (u^*)^5 (2 \alpha ^2+\alpha \\ &{} -\eta )+2 \alpha \eta (u^*)^3 (2 \alpha ^2-\alpha \eta -2 \eta )-\alpha (u^*)^6 (\alpha -8 \eta )+\alpha \eta (u^*)^4 (7 \alpha -8 \eta ),\\ \widehat{d}_2=&{} 3 \alpha ^4 \eta ^4+2 (u^*)^{11} (39 \alpha ^3-50 \alpha \eta )+6 \alpha ^2 \eta ^3 (u^*)^2 (\eta -13 \alpha ^2)+(u^*)^{12} (88 \alpha ^2\\ &{} -2 \alpha -24 \eta )+\alpha ^3 \eta ^3 u^* (11 \eta -12 \alpha ^2)+\alpha (u^*)^9 (28 \alpha ^3+\alpha ^2 (68 \eta +1)\\ &{} -172 \alpha \eta +42 \eta ^2)+\alpha (u^*)^8 (\alpha ^3 (48 \eta +5)-254 \alpha ^2 \eta -6 \alpha \eta (8 \eta +1)\\ &{} +106 \eta ^2)+2 \alpha ^2 (u^*)^7 (3 \alpha ^3-40 \alpha ^2 \eta -\alpha \eta (29 \eta +24)+144 \eta ^2) -4 (12 \alpha ^3\\ &{} +3 \alpha ^2 \eta -50 \alpha \eta +21 \eta ^2)\alpha ^2 \eta (u^*)^5-2 \alpha ^2 \eta ^2 (u^*)^4 (12 \alpha ^3-112 \alpha ^2+17 \alpha \eta \\ &{} +21 \eta )+6 \alpha ^3 \eta ^2 (u^*)^3 (13 \alpha ^2-18 \eta )+2 (u^*)^{10} (6 \alpha ^4+11 \alpha ^3-36 \alpha ^2 \eta \\ &{} -15 \alpha \eta +18 \eta ^2)+2 \alpha \eta (u^*)^6 (12 \alpha ^4-3 \alpha ^3 (2 \eta +15)+101 \alpha ^2 \eta +29 \alpha \eta \\ &{} -21 \eta ^2)+34 \alpha (u^*)^{13}+4 (u^*)^{14}. \end{array}\nonumber \\ \end{aligned}$$

(D1)

First, we prove $\hat{b}_{11}\mid _{\lambda =0}>0$. Substituting $\eta =\overline{\eta }$ in (1.7) into $\widehat{d}_1$, we have

$$\begin{aligned} \begin{array}{ll} \widehat{d}_1=&{}\frac{ \big (\hat{d}_{11}+\hat{d}_{12} \sqrt{(u^*)^2 ((3 \alpha ^2+\alpha (u^*+3) u^*+3 (u^*)^3)^2-4 \alpha u^* (\alpha ^2+3 \alpha (u^*)^2+(u^*)^3))}\big )}{-2 \alpha ^2}\\ {} &{}\times (u^*)^2 (\alpha ^2+\alpha (u^*)^2+\alpha u^*+(u^*)^3), \end{array} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} \hat{d}_{11}=&{}9 \alpha ^5 u^*+6 \alpha ^4 (u^*)^3+34 \alpha ^4 (u^*)^2+36 \alpha ^3 (u^*)^4+45 \alpha ^3 (u^*)^3 +10 \alpha ^2 (u^*)^6\\ &{}+66 \alpha ^2 (u^*)^5+24 \alpha ^2 (u^*)^4+33 \alpha (u^*)^7+52 \alpha (u^*)^6+36 (u^*)^8+\alpha ^3 (u^*)^5,\\ \hat{d}_{12}=&{}-3 \alpha ^3-\alpha ^2 (u^*)^2-9 \alpha ^2 u^*-7 \alpha (u^*)^3-8 \alpha (u^*)^2-12 (u^*)^4. \end{array} \end{aligned}$$

Through simple calculation, we have

$$\begin{aligned} \begin{aligned}&\hat{d}_{11}^2-\hat{d}_{12}^2 (u^*)^2 ((3 \alpha ^2+\alpha (u^*+3) u^*+3 (u^*)^3)^2-4 \alpha u^* (\alpha ^2+3 \alpha (u^*)^2+(u^*)^3))\\&\quad =-32 \alpha ^2 (u^*)^4 (\alpha +u^*)^3 (\alpha ^3+6 \alpha ^2 (u^*)^2+\alpha (u^*)^4+4 \alpha (u^*)^3+4 (u^*)^5)<0, \end{aligned} \end{aligned}$$

and $\hat{d}_{12}<0$, which show that $\widehat{d}_1>0$.

Similarly, we can prove that $\widehat{d}_2<0$. Combining the condition $\eta =\overline{\eta }$ and $0<u^*<\frac{1}{3}$, we have $\hat{b}_{11}\mid _{\lambda =0}>0$.

Second, we prove that $\hat{b}_{11}\mid _{\lambda =0}<2\sqrt{2}$, from (3.55), which is equivalent to proving that

$$\begin{aligned} \begin{array}{ll} &{}72 \hat{d}_1^3 u^* (\alpha +u^*) (\eta +2 (u^*)^3-(u^*)^2)+\hat{d}_2^2 (\eta +(u^*)^2)^2\\[0.5ex] &{}\quad =-\frac{ \big (\hat{d}_{21}+\hat{d}_{22} \sqrt{(u^*)^2 ((3 \alpha ^2+\alpha (u^*+3) u^*+3 (u^*)^3)^2-4 \alpha u^* (\alpha ^2+3 \alpha (u^*)^2+(u^*)^3))}\big )}{2 \alpha ^7}\\ &{}\qquad \times 9 (u^*)^9 (\alpha +u^*)^5 (\alpha +(u^*)^2)^3 <0, \end{array} \end{aligned}$$

where

$$\begin{aligned} \hat{d}_{21}=&-6561 \alpha ^{17} u^*-729 \alpha ^{16} ((u^*)^2 (27 u^*+67))-81 \alpha ^{15} ((u^*)^3 (297 (u^*)^2\\ {}&+2121 u^*+1385))-9 \alpha ^{14} ((u^*)^4 (1755 (u^*)^3+27819 (u^*)^2+54939 u^*\\&-13213))+\alpha ^{13} (-6075 (u^*)^4-196857 (u^*)^3-902097 (u^*)^2+173511 u^*\\&+1295173) (u^*)^5-\alpha ^{12} (1377 (u^*)^5+90441 (u^*)^4+883203 (u^*)^3\\&+311171 (u^*)^2-5142519 u^*-3530471) (u^*)^6+\alpha ^{11} (-171 (u^*)^6\\&-24339 (u^*)^5-501571 (u^*)^4-934249 (u^*)^3+8332381 (u^*)^2\\&+16569133 u^*+5489045) (u^*)^7+\alpha ^{10} (-9 (u^*)^7-3561 (u^*)^6\\&-165393 (u^*)^5-913593 (u^*)^4+6980391 (u^*)^3+32872783 (u^*)^2\\&+29149703 u^*+5440095) (u^*)^8+\alpha ^9 (-219 (u^*)^7-29363 (u^*)^6\\&-441427 (u^*)^5+3095111 (u^*)^4+35618269 (u^*)^3+66318797 (u^*)^2\\&+32229157 u^*+3444736) (u^*)^9+\alpha ^8 (-2169 (u^*)^7-106569 (u^*)^6\\&+611581 (u^*)^5+22621213 (u^*)^4+83767639 (u^*)^3+82507335 (u^*)^2\\&+22611288 u^*+1290240) (u^*)^{10}+\alpha ^7 (-10251 (u^*)^7-489 (u^*)^6\\&+8326479 (u^*)^5+63373303 (u^*)^4+118457877 (u^*)^3+64564384 (u^*)^2\\&+9351168 u^*+221184) (u^*)^{11}+\alpha ^6 (-12315 (u^*)^6+1609645 (u^*)^5\\&+28651757 (u^*)^4+103141317 (u^*)^3+104191032 (u^*)^2+29649136 u^*\\&+1769472) (u^*)^{13}+\alpha ^5 (120231 (u^*)^5+7137447 (u^*)^4+54545143 (u^*)^3\\&+102913984 (u^*)^2+53464944 u^*+6225920) (u^*)^{15}+\alpha ^4 (749493 (u^*)^4\\&+16252821 (u^*)^3+62466696 (u^*)^2+59444000 u^*+12527104) (u^*)^{17}\\&+3 \alpha ^3 (705861 (u^*)^3+7236576 (u^*)^2+13658976 u^*+5208064) (u^*)^{19}\\&+216 \alpha ^2 (15687 (u^*)^2+75738 u^*+56192) (u^*)^{21}+3888 \alpha (765 u^*\\&+1408) (u^*)^{23}+1119744 (u^*)^{25}, \end{aligned}$$

and $\hat{d}_{22}$ is also a polynomial of $(\alpha ,\,u^*)$, we omit its expression, and $\hat{d}_1$ and $\hat{d}_2$ are showing in (D1).

By simple calculation, we have

where

$$\begin{aligned} \begin{array}{ll} \hat{d}_{23}&{}= \alpha ^5 (-3 u^*-5)+\alpha ^4 (-3 (u^*)^2-30 u^*+13) u^*+\alpha ^3 (-(u^*)^2+81 u^*\\ &{}\quad +10) (u^*)^2+4 \alpha ^2 (9 u^*+35) (u^*)^4+2 \alpha (37 u^*+32) (u^*)^5+64 (u^*)^7. \end{array} \end{aligned}$$

Notice that, $\hat{d}_{21}$ is a 17-order polynomial of $\alpha$ with coefficients are functions of $u^*$, where the coefficients of (0–13)-order terms are positive, (15-17)-order terms are negative and 14-order term is uncertain when $0<u^*<\frac{1}{3}$. By using the Descartes’ rule of signs, we know that $\hat{d}_{21}$ has a unique positive real root $\widehat{\alpha }_{1}$, and

$$\begin{aligned} \begin{array}{ll} \hat{d}_{21}>0\ (=0,\ \textrm{or}\,<0)\quad \textrm{if}\quad 0<\alpha <\widehat{\alpha }_{1}\ (\alpha =\widehat{\alpha }_{1},\ \textrm{or}\ \alpha >\widehat{\alpha }_{1}). \end{array} \end{aligned}$$

Similarly, $\hat{d}_{22}$ (or $\hat{d}_{23}$) also has a unique positive real root $\widehat{\alpha }_{2}$ (or $\widehat{\alpha }_{3}$), and

(D2)

Hence, by using the same method as in the analysis of $\textrm{sign}(J)$ in proof (II) of Theorem 3, we can get that $\widehat{\alpha }_2<\widehat{\alpha }_3<\widehat{\alpha }_1$. Therefore, we can get that $\hat{d}_{21}+\hat{d}_{22} \sqrt{(u^*)^2 ((3 \alpha ^2+\alpha (u^*+3) u^*+3 (u^*)^3)^2-4 \alpha u^* (\alpha ^2+3 \alpha (u^*)^2+(u^*)^3))}>0,$ then we have $72 \hat{d}_1^3 u^* (\alpha +u^*) (\eta +2 (u^*)^3-(u^*)^2)+\hat{d}_2^2 (\eta +(u^*)^2)^2<0$, that is $\hat{b}_{11}\mid _{\lambda =0}<2\sqrt{2}$. We finish the proof.