How is this an autocatalytic reaction?
In your first reaction, $\ce{HIO2}$ and $\ce{HIO3}$ react to form $\ce{2IO2.}$ and $\ce{H2O}$. 

In your second reaction, $\ce{IO2\cdot}$ is being reduced back to $\ce{HIO2}$ by $\ce{Mn^{2+}}$. Since the first reaction produced $\ce{2IO2.}$, this second reaction needs to happen twice for each time the first reaction occurs. The pair of reactions is really:

$$\ce{ IO3- + HIO2 + H+ -&gt; 2IO2. +H2O }$$
$$2\left(\ce{IO2. +Mn^{2+} +H2O-&gt; HIO2 + Mn(OH)^{2+}} \right)$$

When you combine these reactions into an overall reaction, the instances of $\ce{IO2.}$ cancel and you are left with a net gain of $\ce{HIO2}$. 

$$\ce{IO3- + H+ +2Mn^{2+} + H2O-&gt; HIO2 + 2Mn(OH)^{2+}}$$

Of course, the net reaction does not hint at the autocatalytic behavior. To emphasize that the reaction is autocatalytic, you could write the overall reaction in a not quite simplest form:

$$\ce{IO3- + HIO2 +  H+ +2Mn^{2+} + H2O-&gt; 2HIO2 + 2Mn(OH)^{2+}}$$