Analysis of long transients and detection of early warning signals of extinction in a class of predator–prey models exhibiting bistable behavior

Abstract

In this paper, we develop a method of analyzing long transient dynamics in a class of predator–prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator–prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251–5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.

Appendix

Expressions for the coefficients in system (6): The expressions for $\omega$, $F_{13}$, $F_{111}$, $H_3$, $H_{11}$, and $\alpha (h)$ in Theorem 3.1 are given below:

$$\begin{aligned} \left\{ \begin{array}{lllll} \omega &{}=\sqrt{-\bar{x} (\bar{y}\bar{\phi _y}\bar{\chi _x}+\bar{z} \bar{\phi _z}\bar{\psi _x})},\\ F_{13}&{}= \bar{x} \bar{\phi _z}(\bar{z}\bar{\psi _z}-\bar{y}\bar{\chi _y})+\frac{\bar{x}}{\bar{\phi _y}}(\bar{y}\bar{\phi ^2_y}\bar{\chi _z} -\bar{z}\bar{\phi ^2_z}\bar{\psi _y}) +\frac{\omega ^2}{\bar{\phi _{xx}}\bar{\phi _y}}(\bar{\phi _{xz}}\bar{\phi _y} - \bar{\phi _{xy}}\bar{\phi _z}),\\ F_{111} &{}= \frac{\omega ^2}{\bar{x}^2\bar{\phi ^2_{xx}}} (3\bar{\phi }_{xx}+\bar{x}\bar{\phi }_{xxx}) +\frac{\bar{x}}{\omega ^2}[\bar{y}\bar{\chi _x}(\bar{y} \bar{\phi _y}\bar{\chi _y} + \bar{z}\bar{\phi _z}\bar{\psi _y})+ \bar{z}\bar{\psi _x} (\bar{y}\bar{\phi _y}\bar{\chi _z}+\bar{z} \bar{\phi _z}\bar{\psi _z} )] \\ &{}+\frac{1}{\bar{x}\bar{\phi }_{xx}}[\bar{y}\bar{\chi _x}(\bar{\phi _y}+\bar{x}\bar{\phi }_{xy}) +\bar{z}\bar{\psi _x}(\bar{\phi _z} +\bar{x}\bar{\phi }_{xz})+ \bar{x}(\bar{y}\bar{\phi _y}\bar{\chi }_{xx}+ \bar{z}\bar{\phi _z}\bar{\psi }_{xx})], \\ H_3 &{}= \frac{\bar{x}\bar{y} \bar{z}}{\omega ^2}[\bar{\psi _x}(\bar{\phi _y} \bar{\chi _z} -\bar{\phi _z}\bar{\chi _y} ) - \bar{\chi _x}(\bar{\phi _y} \bar{\psi _z} -\bar{\phi _z}\bar{\psi _y} ) ],\\ H_{11} &{}= \frac{\bar{x}\bar{y}\bar{z}}{\omega ^4}\bar{\phi _y}(\bar{\psi _x}-\bar{\chi _x}) \left( \bar{y} \bar{\chi _x}\bar{\chi _y}+\bar{z}\bar{\psi _x}\bar{\psi _z}\right) + \frac{\bar{y}\bar{z}}{\omega ^2 \bar{x}\bar{\phi }_{xx}}\bar{\phi _y} \left( \bar{z} \bar{\psi _x}\bar{\chi }_{xx}-\bar{x}\bar{\chi _x}\bar{\psi }_{xx}\right) , \\ \alpha (h)&{}= \left\{ -\begin{pmatrix} (\bar{f}_1)_{xx} &{} (\bar{f}_1)_{xy} &{} (\bar{f}_1)_{xz} \end{pmatrix} J^{-1} \begin{pmatrix} (\bar{f}_1)_h \\ (\bar{f}_2)_h\\ (\bar{f}_3)_h \end{pmatrix} +(\bar{f}_1)_{xh}\right\} \Big (\frac{h-\bar{h}}{\zeta }\Big ) \\ {} &{} -\frac{1}{\omega ^2}\{(\bar{f}_1)_y [(\bar{f}_2)_y (\bar{f}_2)_x + (\bar{f}_2)_z (\bar{f}_3)_x] + (\bar{f}_1)_z [(\bar{f}_3)_y (\bar{f}_2)_x + (\bar{f}_3)_z (\bar{f}_3)_x) ] \}. \end{array} \right. \end{aligned}$$

Hopf bifurcation of system (3): System (3) undergoes a Hopf bifurcation at $h=\bar{h}+\zeta A+O(\zeta ^{3/2})$ for sufficiently small $\zeta >0$, where A is the solution of the equation

$$\begin{aligned} \left\{ -\begin{pmatrix} (\bar{f}_1)_{xx}&(\bar{f}_1)_{xy}&(\bar{f}_1)_{xz} \end{pmatrix} J^{-1} \begin{pmatrix} (\bar{f}_1)_h \\ (\bar{f}_2)_h\\ (\bar{f}_3)_h \end{pmatrix} +(\bar{f}_1)_{xh}\right\} A= & {} \frac{1}{\omega ^2}\{(\bar{f}_1)_y [(\bar{f}_2)_y (\bar{f}_2)_x + (\bar{f}_2)_z (\bar{f}_3)_x] \nonumber \\+ & {} (\bar{f}_1)_z [(\bar{f}_3)_y (\bar{f}_2)_x + (\bar{f}_3)_z (\bar{f}_3)_x) ] \}.\nonumber \\ \end{aligned}$$

(27)

Transformation to normal coordinates: The transformations from the original coordinates (x, y, z) to the normal form coordinates (u, v, w) are given below:

$$\begin{aligned} \left\{ \begin{array}{ll} u &{}= \frac{\bar{f_{1xx}}}{\omega } X -\delta (\bar{f_{1y}} B_1(A_1, A_2, X, Y, Z)+\bar{f_{1z}}B_2(A_1, A_2, X, Y, Z) \\ &{}\quad + \frac{\delta }{3}C_{uv} \Big [ \Big ( \frac{\bar{f_{1xx}}}{\omega } X - \delta (\bar{f_{1y}} B_1(A_1, A_2, X,Y,Z) \\ &{} \quad +\bar{f_{1z}}B_2((A_1, A_2, X, Y, Z))\Big )^2 \Big (-1+\frac{\bar{f_{1xx}}}{2\omega ^2} (\bar{f_{1y}}Y+\bar{f_1}_zZ )\Big ) \\ &{}\quad +\frac{\bar{f^2_{1xx}}}{\omega ^4} (\bar{f_{1y}}Y+\bar{f_{1z}}Z)^2 \Big ],\\ v &{}= \frac{\bar{f_{1xx}}}{\omega ^2} (\bar{f_{1y}}Y+\bar{f_{1z}}Z),\\ w &{}=- \frac{\bar{f_{1xx}} \bar{f_{1y}}}{\omega ^2 \bar{f_{1z}}} \Big [ \Big (1+ \frac{\bar{f_{2x}}\bar{f_1}_{y}}{\omega ^2}\Big ) Y + {\bar{f_{2x}}\bar{f_{1z}}}{\omega ^2} Z + \frac{\delta }{\omega }\bar{f_1}_{xx} A_1 X\Big ], \end{array} \right. \end{aligned}$$

(28)

where

$$\begin{aligned} X = \frac{x-x_0}{\sqrt{\zeta }}, \ Y = \frac{y-y_0}{\zeta }, \ Z = \frac{z-z_0}{\zeta }, \end{aligned}$$

and

$$\begin{aligned} A_1= & {} \left( 1+ \frac{\bar{f_{2x}}\bar{f_{1y}}}{\omega ^2}\right) (\bar{f_{2y}}\bar{f_{2x}} + \bar{f_{2z}}\bar{f_{3x}}) + \frac{\bar{f_{2x}}\bar{f_{1z}}}{\omega ^2} (\bar{f_{3y}}\bar{f_{2x}} + \bar{f_{3z}}\bar{f_{2x}}), \\ A_2= & {} \frac{\bar{f_{3x}}\bar{f_{1y}}}{\omega ^2} (\bar{f_{2y}}\bar{f_{2x}} + \bar{f_{2z}}\bar{f_{3x}}) + \left( 1+\frac{\bar{f_{3x}}\bar{f_{1z}}}{\omega ^2}\right) (\bar{f_{3y}}\bar{f_{2x}} + \bar{f_{3z}}\bar{f_{2x}}),\\ C_{uv}= & {} -\frac{1}{\omega ^2} \Big [ \bar{f_{1y}}(\bar{f_{2y}}\bar{f_{2x}} + \bar{f_{2z}}\bar{f_{3x}}) + \bar{f_{1z}} (\bar{f_{3y}}\bar{f_{2x}} + \bar{f_{3z}}\bar{f_{2x}})\Big ] \\ {}- & {} \frac{1}{\bar{f_{1xx}}} \Big [\bar{f_{1xy}}\bar{f_{2x}} + \bar{f_{1xz}}\bar{f_{3x}} + \bar{f_{1y}}\bar{f_{2xx}} + \bar{f_{1z}}\bar{f_{3xz}}\Big ], \\ B_1= & {} -\frac{\bar{f_{1xx}}}{\omega ^4} (\bar{f_{1y}}Y+ \bar{f_{1z}}Z) (\bar{f_{2y}}\bar{f_{2x}} + \bar{f_{2z}}\bar{f_{3x}}) +\frac{\bar{f_{1xx}}\bar{f_{2xx}}}{2\omega ^2} + \frac{\bar{f_{2y}}\bar{f_{1xx}} }{\omega ^2}\Big [\left( 1+ \frac{\bar{f_{2x}}\bar{f_{1y}}}{\omega ^2}\right) Y \\ {}+ & {} \frac{\bar{f_{2x}}\bar{f_{1z}}}{\omega ^2} Z + \frac{\delta A_1 X}{\omega } \Big ] + \frac{\bar{f_{2z}}\bar{f_{1xx}} }{\omega ^2}\Big [ \frac{\bar{f_{3x}}\bar{f_{1y}}}{\omega ^2} Y + \Big (1+\frac{\bar{f_{3x}}\bar{f_{1z}}}{\omega ^2}\Big ) Z + \frac{\delta A_2 X}{\omega } \Big ],\\ B_2= & {} -\frac{\bar{f_{1xx}}}{\omega ^4} (\bar{f_{1y}}Y+ \bar{f_{1z}}Z) (\bar{f_{3y}}\bar{f_{2x}} + \bar{f_{3z}}\bar{f_{3x}}) + \frac{\bar{f_{1xx}}\bar{f_{3xx}}}{2\omega ^2} + \frac{\bar{f_{3y}}\bar{f_{1xx}} }{\omega ^2}\Big [\left( 1+ \frac{\bar{f_{2x}}\bar{f_{1y}}}{\omega ^2}\right) Y \\+ & {} \frac{\bar{f_{2x}}\bar{f_{1z}}}{\omega ^2} Z + \frac{\delta A_1 X}{\omega } \Big ] + \frac{\bar{f_{3z}}\bar{f_{1xx}} }{\omega ^2}\Big [ \frac{\bar{f_{3x}}\bar{f_{1y}}}{\omega ^2} Y + \Big (1+\frac{\bar{f_{3x}}\bar{f_{1z}}}{\omega ^2}\Big ) Z + \frac{\delta A_2 X}{\omega } \Big ]. \end{aligned}$$

Proof of Lemma 3.1:

Let $M\ge N$ be the largest integer such that $\{u(\tau _i)\}_{i=1}^M$ decreases and $w(\tau )>0$ on $[0, \tau _M]$. By Remark 3.1, the expression for $\bar{u^2}_{\text {est}}(\tau )$, defined by (15)–(16), holds up to $O(\delta ^2)$ for all $\tau \in [\tau _1, \tau _M]$. It is clear from (P2) that $\bar{w}(\tau )$ also has a decreasing envelope for all $\tau \in [\tau _1, \tau _N]$. Estimating $\bar{w}(\tau )$ by $\bar{w}_{\text {base}}(\tau )$, where $\bar{w}_{\text {base}}(\tau )$ is defined by (17), it can be easily verified that $\bar{w}_{\text {base}}(\tau )$ has a unique critical point at $\tau =\bar{\tau }_m>0$, where

$$\begin{aligned} \bar{\tau }_m = \frac{1}{\delta H_3-b_2}\ln \Big (\frac{H_{11}b_1b_2e^{(\delta H_3-b_2) \tau _1}}{H_3(2\bar{w}(\tau _1)(\delta H_3-b_2)+\delta H_{11}b_1)} \Big ) \end{aligned}$$

if (18) holds. The critical point corresponds to a minimum of $\bar{w}_{\text {base}}(\tau )$ with

$$\begin{aligned}\bar{w}_{\text {base}}(\bar{\tau }_m)= \frac{-H_{11}b_1}{2 H_3}e^{b_2(\bar{\tau }_m-\tau _1)}>0. \end{aligned}$$

Since $\bar{w}(\tau )-\bar{w}_{\text {base}}(\tau )=O(\delta ^2)$ as long as $w(\tau ) =O(\delta )$, it follows that $\bar{w}(\tau )$ and the lower envelope of $w(\tau )$ also attain their minima. Consequently, $w(\tau )$ attains its global minimum at $\tau = {\tau }_{\textrm{min}}$, where ${\tau }_{\textrm{min}}= {\bar{\tau }}_m +O(\delta ^2)$. Finally, by (P2), since $w(\tau )$ has a decreasing envelope on $[\tau _1, \tau _N]$, it is clear that ${\tau }_{\textrm{min}}\ge \tau _N$. $\square$

Proof of Lemma 3.2:

We start by considering the relative position of $w(\tau )$ with respect to $-\frac{H_{11}}{2H_3} (u^2+v^2)(\tau )$ and show that there exists some $\tau _c> {\tau }_{\textrm{min}}$ such that $w(\tau )$ intersects with the upper envelope of $-\frac{H_{11}}{2H_3} (u^2+v^2)(\tau )$ at $\tau =\tau _c$. By (P2), $w(\tau )<-\frac{H_{11}}{2H_3} (u^2+v^2)(\tau )$ for all $\tau \in [0, \tau _{N}]$. Let $\{t_k\}_{k=1}^{M}$ and $\{s_k\}_{k=1}^{M-1}$ be an increasing sequence of locations of relative maxima and minima of $v(\tau )$ in I respectively with $t_k<\tau _k<s_k$. Since the trajectory is spiraling inwards, we have that $\{v(t_k)\}_{k=1}^M$ and $\{v(s_k)\}_{k=1}^{M-1}$ are decreasing and increasing respectively. Without loss of generality, we assume that $|v(t_k)|\le |v(s_k)|$ for all $1\le k \le M-1$. With the aid of (6), we note that the zeros of $u(\tau )$ correspond to the critical points of $(u^2 +v^2)(\tau )$. Hence the relative maxima and minima of $(u^2 +v^2)(\tau )$ occur at $s_k$ and $t_k$ respectively with values $v^2(s_k)$ and $v^2(t_k)$ for all k. Denoting the local maxima and minima of $-\frac{H_{11}}{2H_3} (u^2+v^2)(\tau )$ on the interval $[t_k, s_k]$ by $M_k$ and $m_k$ respectively, we then have that $\{M_k\}_{k=1}^{M-1}$ and $\{m_k\}_{k=1}^{M-1}$ are both decreasing. If $u_{\text {env}}(\tau )$ and $v_{\text {env}}(\tau )$ denote the upper envelopes of $u(\tau )$ and $v(\tau )$ respectively, we then have from (9) that up to $O(\delta ^2)$, $u_{\text {env}}(\tau )= Ae^{c_2(\tau -\tau _1)}$ and $v_{\text {env}}(\tau )= Be^{c_2(\tau -\tau _1)}$ for $\tau \in [\tau _1, \tau _M]$, where A is defined by (20) and

$$\begin{aligned} B= \frac{A}{c_2^2+\vartheta ^2},\ c_2=\frac{b_2}{2}=\frac{\alpha \delta }{2} +O(\delta ^k). \end{aligned}$$

(29)

Since $(u^2+v^2)(\tau )\le (u_{\text {env}}^2(\tau ) +v_{\text {env}}^2(\tau ))$ and $|w(\tau )- \bar{w}_{\text {base}}(\tau )|=O(\delta ^2)$, it suffices to show that there exists some $\tau _c \in ({\tau }_{\textrm{min}}, \tau _M)$ such that $\bar{w}_{\text {base}}(\tau )$ intersects with $\frac{-H_{11}}{2H_3} (u_{\text {env}}^2(\tau ) +v_{\text {env}}^2(\tau ))$ at $\tau = \tau _c$. To this end, we first note from (17), (19) and (29) that $\bar{w}_{\text {base}}(\tau )$ intersects with $\frac{-H_{11}}{2H_3} (u_{\text {env}}^2(\tau ) +v_{\text {env}}^2(\tau ))$ if and only if $\frac{-H_{11}(A^2+B^2)}{2H_3} > \frac{\delta H_{11}b_1}{2(b_2-\delta H_3)}$, i.e.

$$\begin{aligned} 1+\frac{1}{c_2^2+\vartheta ^2} > - \frac{1-e^{\frac{-4 \pi c_2}{\vartheta }}}{\frac{8\pi c_2}{\vartheta }\Big (\frac{2c_2}{\delta H_3}-1\Big )}. \end{aligned}$$

(30)

Since $c_2 =O(\alpha \delta /2)$ and $\vartheta \approx 1$, it then follows that the left hand side of (30) is approximately equal to 2, whereas the right hand side of (30) is less than 1/2 for $\delta$ sufficiently small, and thus (30) holds. Next, we observe from (13) and (16) that as long as $\bar{w}_{\text {base}}(\tau )$ is decreasing, $\bar{w}_{\text {base}}(\tau )\le \frac{-H_{11}}{2H_3} b_1$. Since $b_1 \approx A^2/2$ and $c_2<0$, we then have from the definition of $u_{\text {env}}$ and $v_{\text {env}}$ that

$$\begin{aligned} \bar{w}_{\text {base}}(\tau ) \le \frac{-H_{11}}{2H_3} b_1< -\frac{H_{11}}{2H_3} (u_{\text {env}}^2(\tau ) +v_{\text {env}}^2(\tau )) \ \text {on} \ [\tau _1, \bar{\tau }_m]. \end{aligned}$$

Hence, for sufficiently small $\delta >0$,

we must have $\tau _c > \bar{\tau }_m = \tau _{\text {min}} +O(\delta ^2)$.

Finally, we note that the existence of $\tau _c$ implies that $w(\tau )\ge M_i$ on $[s_i, t_{i+1}]$ for some i between 1 and N, and hence the trajectory is in ${\mathbb {R}}^3 {\setminus } \Omega _{\alpha }$ over the interval $[s_i, t_{i+1}]$. The monotonic properties of $\{M_k\}_{k=1}^{M-1}$ and $\{m_k\}_{k=1}^{M-1}$, and the fact that $\bar{w}'(\tau )>0$ for $\tau >{\tau }_{\textrm{min}}$ will then imply that the trajectory is in ${\mathbb {R}}^3 {\setminus } \Omega _{\alpha }$ over the interval $[s_i, \tau _M]$. Choosing $\tau _a=s_j$, where $1<j<M$ is the smallest integer such that $s_j\le \tau _c$ with $w(s_j)>M_j$, and $\tau _b=\tau _M$ then yields the result.

Proof of Lemma 3.3:

Let $\{\tau _i\}_{i=1}^M$ be an increasing sequence of locations of relative maxima of $u(\tau )$ such that $\{u(\tau _i)\}_{i=1}^M$ is decreasing, where M is some integer greater than N to be chosen later. By Remark 3.1, we note that the expressions for $\bar{u^2}_{\text {est}}(\tau )$ and $\bar{w}_{\text {base}}(\tau )$ defined by (15) and (17) respectively, hold up to $O(\delta ^2)$ for all $\tau \in [\tau _1, \tau _M]$. If $\bar{w}(\tau _1)< \frac{\delta H_{11}b_1}{2(b_2-\delta H_3)}$, then it follows from (17) that $\bar{w}_{\text {base}}(\tau )$ decreases and eventually becomes negative at $\tau =\tau _e$, where

$$\begin{aligned} {\tau _e} = \frac{1}{\delta H_3-b_2}\ln \Big (\frac{\delta H_{11}b_1e^{(\delta H_3 -b_2)\tau _1}}{\delta H_{11}b_2 e^{b_2 \tau _1}+2\bar{w}(\tau _1)(\delta H_3-b_2)} \Big ). \end{aligned}$$

Since $|w(\tau ) - \bar{w}_{\text {base}}(\tau )|=O(\delta ^2)$ and $\bar{w}_{\text {base}}'(\tau )<0$ for all $\tau \in [0, \tau _M]$, we must have that $w(\tau )$ also changes its sign near $\tau _e$. Let $\tilde{\tau }=\tau _e+o(1)$ be such that $w(\tilde{\tau })=0$. Choose M large enough such that $\tilde{\tau } <\tau _{M}$. Since $q_e$ is unstable along the w-direction and $w'(\tilde{\tau }) = \frac{\delta }{2} H_{11}u^2 (\tilde{\tau })\le 0$ (follows from (6)), we cannot have $w(\tau )=0$ for all $\tau \ge \tilde{\tau }$, thereby leading to $w(\tau )<0\le -\frac{H_{11}}{2H_3} (u^2+v^2)(\tau )$ for all $\tau >\tilde{\tau }$. Thus $(u(\tau ), v(\tau ), w(\tau ))\in \Omega _{\alpha }$ for all $\tau >\tilde{\tau }$. It is also evident from (6) that $w'(\tau ) < \delta H_3 w$ and therefore if $w(\bar{\tau })=-\tilde{k}$ for some $\tilde{k}>0$ and $\bar{\tau } > \tilde{\tau }$, then by Gronwall’s inequality, $w(\tau ) < -\tilde{k} e^{\delta H_3 (\tau -\bar{\tau })} \rightarrow -\infty$ as $\tau \rightarrow \infty$. Consequently, the fast variables $(u(\tau ), v(\tau ))$, governed by system (10), approach (0, 0) as $\tau \rightarrow \infty$.

Next, we will prove that $\Omega _{\alpha }$ is positively invariant with respect to $(u(\tau ),v(\tau ),w(\tau ))$ for all $\tau \ge 0$. Denoting the sequences of local minima and maxima of $-\frac{H_{11}}{2 H_3}(u^2+v^2)$ by $\{m_k\}_{k=1}^{M-1}$ and $\{M_k\}_{k=1}^{M-1}$ and their locations by $\{t_k\}_{k=1}^{M-1}$ and $\{s_k\}_{k=1}^{M-1}$ respectively with $t_k<\tau _k<s_k$, and recalling that $(u(\tau ), v(\tau ), w(\tau )) \in \Omega _{\alpha }$ on $[0, \tau _N]$, we then have that $w(\tau ) < m_k$ on the interval $[t_k, t_{k+1})$ for all $1\le k\le N-1$. As in the proof of Lemma 3.2, we have that $m_k =-\frac{H_{11}}{2H_3}v^2(t_k) =-\frac{H_{11}B^2}{2H_3} e^{b_2 (t_k-\tau _1)}$ up to $O(\delta ^2)$, where B is defined by (29), and thus we have $\bar{w}_{\text {base}}(\tau )\approx w(\tau ) <-\frac{H_{11}B^2}{2H_3} e^{b_2 (\tau -\tau _1)}$ on $[0, \tau _{N}]$, i.e.

$$\begin{aligned} \frac{\delta H_{11} b_1}{2(b_2-\delta H_3)} < -\frac{H_{11} B^2}{2H_3}. \end{aligned}$$

(31)

Since by our assumption $\bar{w}(\tau _1)< \frac{\delta H_{11}b_1}{2(b_2-\delta H_3)}$, we have from (17) in combination with (31) that $\bar{w}_{\text {base}}(\tau ) < m_k$ on $[t_k, t_{k+1})$ for all $1\le k\le M-1$, i.e. $\bar{w}_{\text {base}}(\tau )< -\frac{H_{11}}{2 H_3}(u^2+v^2)(\tau )$ for all $\tau \in [0,\tau _M]$. Finally, since $w(\tau )=\bar{w}_{\text {base}}(\tau )+O(\delta ^2)$, we can conclude that the solution is in $\Omega _{\alpha }$ for all $\tau \in [0,\tau _M]$. Combining this with the fact that $(u(\tau ), v(\tau ), w(\tau ))\in \Omega _{\alpha }$ for all $\tau >\tilde{\tau }$ proves the lemma. $\square$