Persistence of Pollination Mutualisms in the Presence of Ants

Abstract

This paper considers plant–pollinator–ant systems in which the plant–pollinator interaction is mutualistic but ants have both positive and negative effects on plants. The ants also interfere with pollinators by preventing them from accessing plants. While a Beddington–DeAngelis (BD) formula can describe the plant–pollinator interaction, the formula is extended in this paper to characterize the pollination mutualism under the ant interference. Then, a plant–pollinator–ant system with the extended BD functional response is discussed, and global dynamics of the model demonstrate the mechanisms by which pollination mutualism can persist in the presence of ants. When the ant interference is strong, it can result in extinction of pollinators. Moreover, if the ants depend on pollination mutualism for survival, the strong interference could drive pollinators into extinction, which consequently lead to extinction of the ants themselves. When the ant interference is weak, a cooperation between plant–ant and plant–pollinator mutualisms could occur, which promotes survival of both ants and pollinators, especially in the case that ants (respectively, pollinators) cannot survive in the absence of pollinators (respectively, ants). Even when the level of ant interference remains invariant, varying ants’ negative effect on plants can result in survival/extinction of both ants and pollinators. Therefore, our results provide an explanation for the persistence of pollination mutualism when there exist ants.

Appendices

Appendix 1: The Computation of $e_{12}^0,e_{13}^0$ and Interior Equilibria of (1) and (3)

Let $g_1(x_1) = l_1(x_1)$. Then, we have $\bar{A} x_1^2 + \bar{B} x_1 + \bar{C} =0$ with

$$\begin{aligned} \bar{A} = - d_1, \quad \bar{B} = r_1 + \frac{e_{12}r_2}{e_{21} \beta x_1^0},\quad \bar{C} = - \frac{e_{12}r_2}{e_{21} \beta }. \end{aligned}$$

Let

$$\begin{aligned} \bar{B}^2 - 4 \bar{A}\bar{C} =0. \end{aligned}$$

(15)

We can obtain two roots $e_{12}^\pm$ of (15) and have $e_{12}^0 := e_{12}^+$. When $e_{12} \ge e_{12}^0,$ we obtain

$$\begin{aligned} x_1^\pm = \frac{-\bar{B} \pm \sqrt{\bar{B}^2 -4 \bar{A}\bar{C}} }{2\bar{A}},\quad x_2^\pm = \frac{1}{\beta x_1^0} (x_1^\pm -x_1^0). \end{aligned}$$

Denote Let $g_2(x_1) = l_2(x_1)$. Then, we have $\hat{A} x_1^2 + \hat{B} x_1 + \hat{C} =0$ with

$$\begin{aligned} \hat{A} = - d_1 -\frac{\bar{\gamma }}{\bar{\beta } \bar{x}_1^0}, \quad \hat{B} = r_1 +\frac{\bar{\gamma }}{\bar{\beta } } + \frac{e_{13}r_3}{e_{31} \bar{\beta } \bar{x}_1^0},\quad \hat{C} = - \frac{e_{13}r_3}{e_{31} \bar{\beta } }. \end{aligned}$$

Let

$$\begin{aligned} \hat{B}^2 - 4 \hat{A}\hat{C} =0. \end{aligned}$$

(16)

We can obtain two roots $e_{13}^\pm$ of (16) and have $e_{13}^0 := e_{13}^+$. When $e_{13} \ge e_{13}^0,$ we obtain

$$\begin{aligned} \bar{x}_1^\pm = \frac{-\hat{B} \pm \sqrt{\hat{B}^2 -4 \hat{A}\hat{C}} }{2\hat{A}},\quad x_3^\pm = \frac{1}{\bar{\beta } \bar{x}_1^0} (\bar{x}_1^\pm -\bar{x}_1^0). \end{aligned}$$

Appendix 2: The Proof of Lemma 4.1

From the first equation of (2), we have

$$\begin{aligned} \frac{dx_1}{dt} \le x_1 (r_1+\frac{e_{12}}{\beta }+\frac{e_{13}}{\bar{\beta }} - d_1x_1) \end{aligned}$$

so that the comparison principle (Cosner 1996) implies that

$$\begin{aligned} \limsup _{t \rightarrow \infty }x_1(t) \le \frac{ r_1 \beta \bar{\beta }+ e_{12}\bar{\beta }+e_{13} \beta }{d_1 \beta \bar{\beta }}. \end{aligned}$$

Then, for $\epsilon >0$ small, we have $x_1(t) \le \epsilon + (r_1 \beta \bar{\beta }+ e_{12}\bar{\beta }+e_{13} \beta )/d_1 \beta \bar{\beta }$ when $t$ is sufficiently large. Let $r_0 = min\{r_2,r_3 \}$. From the three equations in (2), we have

$$\begin{aligned}&\frac{d}{dt} \left( x_1 + \frac{e_{12}}{e_{21}} x_2+ \frac{e_{13}}{e_{31}} x_3\right) \\&\quad \le x_1(r_1 - d_1x_1) + \frac{2e_{12}x_1x_2}{1\!+\!\alpha x_1\!+\!\beta x_2+ \gamma x_3}+ \frac{2e_{13}x_1x_3}{1+\bar{\alpha } x_1+\bar{\beta } x_3} - \frac{e_{12}}{e_{21}} r_2 x_2 - \frac{e_{13}}{e_{31}} r_3 x_3\\&\quad < x_1(r_1 + \frac{2e_{12}}{\beta }+ \frac{2e_{13}}{\bar{\beta }}) - r_0 (\frac{e_{12}}{e_{21}} x_2+ \frac{e_{13}}{e_{31}} x_3 )\\&\quad \le \left( \frac{r_1 \beta \bar{\beta }+ e_{12}\bar{\beta }+e_{13} \beta }{d_1 \beta \bar{\beta }} \!+\! \epsilon \right) \left( r_0+r_1 + \frac{2e_{12}}{\beta } +\frac{2e_{13}}{\bar{\beta }} \right) -r_0\left( x_1 \!+\! \frac{e_{12}}{e_{21}} x_2+ \frac{e_{13}}{e_{31}} x_3\right) \!. \end{aligned}$$

Using the comparison principle a second time, we have

$$\begin{aligned} \limsup _{t \rightarrow \infty }\left( x_1 \!+\! \frac{e_{12}}{e_{21}} x_2\!+\! \frac{e_{13}}{e_{31}} x_3\right) \!&\le \! \frac{1}{r_0}\left( \frac{r_1 \beta \bar{\beta }\!+\! e_{12}\bar{\beta }\!+\!e_{13} \beta }{d_1 \beta \bar{\beta }} \!+\! \epsilon \right) \\&\quad \times \,\left( r_0\!+\!r_1 \!+\! \frac{2e_{12}}{\beta } \!+\!\frac{2e_{13}}{\bar{\beta }}\right) \end{aligned}$$

which implies that system (2) is dissipative.

Appendix 3: Proof of Lemma 4.2

Let $P^*(x_1,x_2,x_3)$ be an interior equilibrium of (2). Then, $P^*$ satisfies

$$\begin{aligned} x_3(x_1)&= \frac{1}{ \bar{\beta } \bar{x}_1^0} \left( x_1 - \bar{x}_1^0\right) ,\quad x_2(x_1) = \frac{1}{\beta x_1^0} \left( x_1 -x_1^0 - \gamma x_1^0 x_3\right) = a x_1 - b\qquad \end{aligned}$$

(17)

$$\begin{aligned} a&= \frac{1}{\beta \bar{\beta } \bar{x}_1^0} \left( \gamma ^* -\gamma \right) , \quad b = \frac{1}{\beta \bar{\beta }} \left( \bar{\beta } -\gamma \right) \end{aligned}$$

(18)

and $g(x_1) = l(x_1)$ with

$$\begin{aligned} g(x_1) = x_1 \left[ r_1- d_1 x_1 - \bar{\gamma } x_3(x_1)\right] +\frac{e_{13}r_3}{e_{31}} x_3(x_1),\quad l(x_1) = -\frac{e_{12}r_2}{e_{21}} x_2(x_1)\quad \end{aligned}$$

(19)

where the parabolic curve $v=g(x_1)$ satisfies $g(\bar{x}_1^\pm ) =0$ when $\bar{x}_1^\pm$ exists. The line $v=l(x_1)$ passes through $(b/a,0)$ as shown in Fig. 3a. The equation $g(x_1) = l(x_1)$ can be rewritten as $G(x_1) = L(x_1)$ with

$$\begin{aligned} G(x_1) = x_1 [ r_1- d_1 x_1 - \bar{\gamma } x_3(x_1)] +\frac{e_{12}r_2}{e_{21}} x_2(x_1),\quad L(x_1) = -\frac{e_{13}r_3}{e_{31}} x_3(x_1)\quad \end{aligned}$$

(20)

where $v=G(x_1)$ is a parabolic curve and the line $v=L(x_1)$ passes through $(\bar{x}_1^0,0)$ as shown in Fig. 3b. The slopes of lines $v=l(x_1)$ and $v=L(x_1)$ are, respectively, denoted by

$$\begin{aligned} k_l = -\frac{a e_{12}r_2 }{e_{21}}, \quad k_L = -\frac{e_{13}r_3}{e_{31}\bar{\beta } \bar{x}_1^0}.\quad \end{aligned}$$

From $\lambda _{13}^+<0$, we have $l(\bar{x}_1^+) >0$. We also have $\gamma ^* < \bar{\beta }$. Indeed, if $\gamma ^* \ge \bar{\beta }$, then there is $\gamma$ such that $\bar{\beta } \le \gamma \le \gamma ^*$, which implies $a \ge 0$ and $b \le 0$. Since $l(\bar{x}_1^+) >0$ and $k_l \le 0$, we obtain $l(0)>0$ as shown in Fig. 3a, which implies $b > 0$. This is a contradiction. From $\gamma ^* < \bar{\beta }$ and (18), we obtain $\bar{x}_1^0 < x_1^0$.

If $\gamma < \gamma ^*$, then $a > 0,b> 0$ and $k_l <0.$ From $l(\bar{x}_1^+) >0$ and $k_l <0$, we have $b/a >\bar{x}_1^+ > \bar{x}_1^0$. Since $\lim _{e_{12} \rightarrow +\infty } k_l = -\infty$, there is $e_{12}^* >0$ such that when $e_{12} = e_{12}^*$, the curves $v=g(x_1)$ and $v=l(x_1)$ are tangent in the region $x_1 > b/a$, while the computation of $e_{12}^*$ is given in “Appendix 4.” Thus, if $e_{12} > e_{12}^*$, then $g(x_1)$ and $l(x_1)$ have two intersection points in the region $x_1 > b/a$ as shown in Fig. 3a, which correspond to two interior equilibria of (2). If $e_{12} < e_{12}^*$, then $g(x_1)$ and $l(x_1)$ have no intersection point in the region $x_1 > b/a,$ which implies that there is no interior equilibrium of (2).

If $\gamma ^* \le \gamma \le \bar{\beta }$, then $a \le 0,b \ge 0$. From (17), we have $x_2 \le 0$ and there is no interior equilibrium of (2).

If $\gamma > \bar{\beta }$, then $a < 0,b < 0, k_l >0$ and $b/a < \bar{x}_1^0$. From (17), we have $x_2 < 0$ as $x_1 > \bar{x}_1^0$. Thus, there is no interior equilibrium of (2). Therefore, Lemma 4.2 is proved.

Appendix 4: The computation of $e_{12}^*$ and $e_{13}^*$

Let $g(x_1) = l(x_1)$. Then, we have $\tilde{A} x_1^2 + \tilde{B} x_1 + \tilde{C} =0$ with

$$\begin{aligned} \tilde{A} = - d_1 -\frac{\bar{\gamma }}{\bar{\beta } \bar{x}_1^0},\quad \tilde{B} = r_1 +\frac{\bar{\gamma }}{\bar{\beta } } + \frac{e_{13}r_3}{e_{31} \bar{\beta } \bar{x}_1^0}+\frac{e_{12}r_2}{e_{21}\beta x_1^0} ,\quad \tilde{C} = -\frac{e_{13}r_3}{e_{31} \bar{\beta } } -\frac{e_{12}r_2}{e_{21} \beta }+\frac{e_{12}r_2 \gamma }{e_{21} \beta \bar{\beta } }. \end{aligned}$$

Let

$$\begin{aligned} \tilde{B}^2 - 4 \tilde{A} \tilde{C} =0. \end{aligned}$$

(21)

When $\gamma < \bar{\beta }$, we can solve two roots $e_{12}^\pm$ of (21) and obtain $e_{12}^* := e_{12}^+$.

Similarly, we can solve two roots $e_{13}^\pm$ of (21) and obtain $e_{13}^* := e_{13}^+$.

Appendix 5: Proof of Theorem 4.3

It follows from $\lambda _1^{(2)}>0$ and $\lambda _1^{(3)}>0$ that $P_{12}^+$ (respectively, $P_{13}^+$) is globally asymptotically stable in the interior of the $(x_1,x_2)$-plane (respectively, the $(x_1,x_3)$-plane). Since $\lambda _1^{(3)}>0$ and $x_1^+>r_1/d_1$, we have $\lambda _{12}^+>0$ by the monotonicity of function $x_1/(1+\bar{\alpha } x_1)$.

(i) It follows from $\lambda _{12}^+ >0$ and $\lambda _{13}^+ >0$ that $P_{12}^+$ (respectively, $P_{13}^+$) is unstable in the $x_3$-direction (respectively, the $x_2$-direction). Thus, the boundary equilibria $O, P_1,P_{12}^+$ and $P_{13}^+$ are hyperbolic and can not form a heteroclinic cycle, which means that hypotheses of (H-1) to (H-4) in the acyclicity theorem (Butler et al. 1986; Butler and Waltman 1986) are satisfied. Therefore, system (2) is uniformly persistent.

(ii) It follows from $\lambda _{13}^+ <0$ that $P_{13}^+$ is locally asymptotically stable in $R_+^3$. Since $\omega _{13}$ is the basin of attraction of $P_{13}^+$ in $R_+^3$, $\omega _{13}$ is open and forward invariant and $R_+^3 - \omega _{13}$ is closed and forward invariant in $R_+^3$. From $\gamma < \gamma ^*$ and $e_{12} \ge e_{12}^*$, Lemma 4.2 shows that (2) has interior equilibria, so that the set int$R_+^3 - \omega _{13}$ is not empty. Then, orbits of (2) in int$R_+^3 - \omega _{13}$ will not converge to $P_{13}^+$ because they are not in the basin of attraction of $P_{13}^+$. Let $(x_1(t), x_2(t), x_3(t))$ be a solution of (2) with $(x_1(0), x_2(0), x_3(0))\in$ int$R_+^3 - \omega _{13},$ then we have $\limsup _{t \rightarrow \infty } x_i(t)>0,i=1,2,3.$ Indeed, suppose $\lim _{t \rightarrow \infty } x_2(t)=0,$ then the $\omega$-limit set of the orbit lies on the $(x_1,x_3)$-plane. On the $(x_1,x_3)$-plane, $P_{13}^+$ is globally asymptotically stable while $O$ and $P_1$ are hyperbolic saddle points. From the result of Thieme (1992) and Thieme (1993), we conclude that this orbit converges to $P_{13}^+$, which forms a contradiction. Similar discussions could show that $\limsup _{t \rightarrow \infty } x_i(t)>0,i=1,3.$ Thus, system (2) is weakly persistent on $R_+^3 - \omega _{13}$. Because the boundary equilibria are hyperbolic and cannot form a heteroclinic cycle, hypotheses of (H-1) to (H-4) in the acyclicity theorem (Butler et al. 1986; Butler and Waltman 1986) are satisfied on $R_+^3 - \omega _{13}$. Thus, solutions of (2) with $x(0) \in$ int$R_+^3 -\omega _{13}$ satisfy $\lim \inf _{t \rightarrow \infty } x_i(t) \ge \delta _0$ for some $\delta _0>0, i=1,2,3.$

(iii) It follows from $\lambda _{13}^+ <0$ that $P_{13}^+$ is locally asymptotically stable in $R_+^3$ with a basin of attraction $\omega _{13}$. If int$R_+^3 - \omega _{13}$ is not empty, a discussion similar to that of (ii) could show that system (2) restricted on $R_+^3 - \omega _{13}$ is uniformly persistent. As a result of Butler et al. (1986), Butler and Waltman (1986), there exists an interior equilibrium of (2) in $R_+^3 - \omega _{13}$. Since $\lambda _{13}^+<0$, $\gamma < \gamma ^*$ and $e_{12} < e_{12}^*$ (or $\lambda _{13}^+<0$, $\gamma \ge \gamma ^*$), Lemma 4.2 shows that (2) has no interior equilibrium, which forms a contradiction. Therefore, int$R_+^3 - \omega _{13}$ is empty and $P_{13}^+$ is globally asymptotically stable in int$R_+^3$.

Appendix 6: Proof of Theorem 4.4

When $\lambda _1^{(2)}<0$ and $e_{12} \ge e_{12}^0$, $P_{12}^-(x_1^-, x_2^-,0)$ and $P_{12}^+(x_1^+, x_2^+,0)$ are boundary equilibria of (2). From $\lambda _1^{(3)}>0$ and $x_1^\pm >r_1/d_1$, we have $\lambda _{12}^{+}>\lambda _{12}^{-}>0.$ When $\lambda _1^{(2)}<0$ and $e_{12} < e_{12}^0$, system (1) has no interior equilibrium and $P_1$ is globally asymptotically stable in the interior of the $(x_1,x_2)$-plane. By a proof similar to that of Theorem 4.3, we obtain the results in Theorem 4.4.

Appendix 7: Proof of Lemma 4.5

Since $\lambda _1^{(2)}>0$, $P_{12}^+(x_1^+, x_2^+,,0)$ is globally asymptotically stable in the interior of the $(x_1,x_2)$-plane. From $\lambda _1^{(2)}>0$, we have $x_1^+ > r_1/d_1 > x_1^0$. Assume $\lambda _{12}^+ <0$. Then, we have $\bar{x}_1^0>x_1^+,$ which implies $g(\bar{x}_1^0) - l(\bar{x}_1^0) = G(\bar{x}_1^0)<0$. From $\lambda _1^{(3)}<0$, we have $\bar{x}_1^0>r_1/d_1.$ Since $\bar{x}_1^0 > r_1/d_1 > x_1^0,$ we have $G(x_1^0)>0$ by (20). From $\bar{x}_1^0 > x_1^0$, we obtain $\gamma ^* > \bar{\beta }$ by (18).

If $\gamma \le \bar{\beta },$ then $a > 0, b \ge 0$ and $b/a < \bar{x}_1^0$. Since $G(x_1^0)>0$ and $G(\bar{x}_1^0)<0$, The roots of $G(x_1)=0$ satisfy $x_1 < \bar{x}_1^0$ as shown in Fig. 3b. Since $\lim _{e_{13} \rightarrow +\infty } |k_L| = +\infty$, there is $e_{13}^* >0$ such that when $e_{13} = e_{13}^*$, the curves $G(x_1)$ and $L(x_1)$ are tangent in the region $x_1 > \bar{x}_1^0,$ while the computation of $e_{13}^*$ is given in “Appendix 4.” Thus, when $e_{13} > e_{13}^*$, $G(x_1)$ and $L(x_1)$ have two intersection points in the region $x_1 > \bar{x}_1^0$, which correspond to two interior equilibria of (2). When $e_{13} < e_{13}^*$, $G(x_1)$ and $L(x_1)$ have no intersection point in the region $x_1 > \bar{x}_1^0,$ which implies that there is no interior equilibrium of (2).

If $\bar{\beta } < \gamma \le \gamma ^*,$ then $a \ge 0, b < 0$ and $x_2>0$ by (17). Since $G(0)>0$ and $G(\bar{x}_1^0)<0$, the roots of $G(x_1)=0$ satisfy $x_1 < \bar{x}_1^0$. By a proof similar to that in (a), we conclude that system (2) has interior equilibria if and only if $e_{13} \ge e_{13}^*$.

If $\gamma > \gamma ^*,$ then $a< 0, b< 0, k_l > 0$ and $b/a > \bar{x}_1^0$ by (18). Since $G(x_1^0)>0$ and $G(\bar{x}_1^0)<0$, the roots of $G(x_1)=0$ satisfy $x_1 < \bar{x}_1^0$. By a proof similar to that in (a), $G(x_1)$ and $L(x_1)$ have two intersection points in the region $x_1 > \bar{x}_1^0$ when $e_{13} \ge e_{13}^*$. The two points are also in the region $x_1 < b/a.$ In fact, the equation $G(x_1)=L(x_1)$ can be rewritten as $g(x_1)=l(x_1)$. Thus, $g(x_1)$ and $l(x_1)$ have two intersection points in the region $x_1 > \bar{x}_1^0$ when $e_{13} \ge e_{13}^*$. Since $g(\bar{x}_1^0) < l(\bar{x}_1^0)$, $g(x_1)$ and $l(x_1)$ have intersection points in the region $x_1 > \bar{x}_1^0$ only if the maximum point $(x_1^\#, g(x_1^\#))$ of $g(x_1)$ satisfies $x_1^\# > \bar{x}_1^0$, as shown in Fig. 3c. Since $k_l >0$ and $l(\bar{x}_1^0) = - a_{12} d_2 (\gamma ^* - \bar{\beta })/a_{21} \beta \bar{\beta }<0,$ the intersection points of $g(x_1)$ and $l(x_1)$ are in the region $x_1 <b/a.$ Hence, there are two interior equilibrium of (2) when $e_{13} \ge e_{13}^*$. When $e_{13} < e_{13}^*$, $G(x_1)$ and $L(x_1)$ have no intersection point in the region $x_1 > \bar{x}_1^0$, which implies that there is no interior equilibrium of (2).

Appendix 8: Proof of Lemma 4.9

Since $\lambda _1^{(2)}<0$ and $e_{12} \ge e_{12}^0$, $P_{12}^+(x_1^+, x_2^+,0)$ and $P_{12}^-(x_1^-, x_2^-,0)$ are boundary equilibria of (2). From $\lambda _{12}^- > 0,$ we have $x_1^- > \bar{x}_1^0$ and $\lambda _{12}^+ > 0.$ Denote

$$\begin{aligned} f_1(x_1)= x_1 (r_1- d_1 x_1)+ \frac{e_{12}r_2}{e_{21}\beta x_1^0} \left( x_1-x_1^0\right) \!,\quad f_2(x_1)= \frac{\bar{\gamma }}{\bar{\beta } \bar{x}_1^0}\left( x_1 - \bar{x}_1^0\right) \end{aligned}$$

then $f_1(x_1^-)=f_1(x_1^+)=0$. Since $\bar{x}_1^0 < x_1^-,$ there is $\bar{\gamma }_0>0$ such that when $\bar{\gamma } < \bar{\gamma }_0$, the parabola $f_1(x_1)$ and line $f_2(x_1)$ have two intersection points in the region $x_1 > \bar{x}_1^0 .$ Denote

$$\begin{aligned} \tilde{g}(x_1)= f_1(x_1)-f_2(x_1)\!,~ \tilde{l}(x_1)= \frac{e_{12}r_2\gamma }{e_{21} \beta \bar{\beta } \bar{x}_1^0}\left( x_1 - \bar{x}_1^0\right) \end{aligned}$$

then $G(x_1)=\tilde{g}(x_1) - \tilde{l}(x_1)$, and equation $\tilde{g}(x_1)=0$ have two roots in the region $x_1 > \bar{x}_1^0$ when $\bar{\gamma } < \bar{\gamma }_0$.

If $\gamma ^* >\bar{\beta }$, then $\bar{x}_1^0 > x_1^0$. Let $k_{\tilde{l}}$ be the slope of $\tilde{l}$. Assume $\bar{\gamma } < \bar{\gamma }_0$. Since $k_{\tilde{l}}>0,$ there exist $\gamma _0>0$ such that when $\gamma \le \gamma _0,$ $\tilde{g}(x_1)$ and $\tilde{l}(x_1)$ have two intersection points in the region $x_1 > \bar{x}_1^0$, which correspond to two roots of $G(x_1)=0$ in the region $x_1 > \bar{x}_1^0$. Since $k_L <0$, $G(x_1)$ and $L(x_1)$ have two intersection points in the region $x_1 > \bar{x}_1^0$ as shown in Fig. 3d. Thus, there are interior equilibria of (2). When $\bar{\gamma } \ge \bar{\gamma }_0$ or $\gamma > \gamma _0$, by a proof similar to that of Lemma 4.5, we conclude that system (2) has interior equilibria if and only if $e_{13} \ge e_{13}^*$.

If $\gamma ^* \le \bar{\beta }$, then $\bar{x}_1^0 \le x_1^0$ and $g(\bar{x}_1^0)<0.$ By a proof similar to that of Lemma 4.7, we conclude that there are interior equilibria of (2) if and only if $\gamma < \gamma ^*$ and $e_{12} \ge e_{12}^*$. Therefore, Lemma 4.9 is proved.