There are many possible solutions to this problem; here is one example.

If the duck uses polar coordinates $\left(r\,\mathrm{theta;}\right)$, she knows that to be as far away from the fox as possible, she will ideally have the opposite phase angle that the fox has:

${\mathrm{\θ}}_{\mathrm{duck}}$${}$$\={\mathrm{\θ}}_{\mathrm{fox}}plus;\mathrm{pi;}$.

She also knows that the fox would prefer that their phase angles are the same, so to achieve her ideal situation, she needs to be able to adjust her phase angle faster than the fox can adjust his. In other words, she has to be able to swim with angular speed faster than the fox can run. Either animal's angular speed is the ratio of its linear speed to the radius of the circle that it's navigating:

$\mathrm{\ω}equals;\frac{v}{r}$.

So, as long as the duck is swimming within $1m$ of the center of the pond, she will be able to position herself directly opposite the center from the fox.

At that point, and knowing that swimming in a wider circle won't help, the obvious thing to try is to swim as fast as possible directly towards the bank that is $3m$ away. It will take her $3s$ to get there, and the fox will take $\mathrm{\π}s$ ($3.14s$)$\,$just slightly longer, so the duck will be able to escape!

Can you discover a better solution?