No, no luck involved at all.
Vague hints:
Spoiler
Destroy filled sprayers over the void.
The holes in the islands aren't decorative.
You shouldn't have to touch any digitalis until late-game.
The creeper will only get 200 strong. Each shell from the bertha takes out 100 creeper. You have 2 berthas.
In-depth tips
Spoiler
Build a sprayer. Disarm it while it's building, let it build to about 15 ammo or more, then destroy it. Now the flip emitters on your island will give a ton of free AC. If you don't disarm it, it's just going to waste ammo on the digitalis.
Build more sprayers. Disarm + dispatch excess ac as ore. (You'll be doing this a lot - reason #1 why this map is annoying.) Destroy them over any unclaimed void on the map, i.e. the rings around the north and south islands and the holes in them.
Begin clearing the islands. Start with the north island. (It's better due to 2 potential PZs.) Land both filled sprayers on the island, fire both berthas on them. When the shells land, immediately destroy the sprayers to convert the flip emitters on the island. You may have to do this twice or so to be able to get your first foothold. Consider using one cannon to defend it.
If you took the northern island, you'll be able to have a PZ beam and PZ bertha before the 2-spore towers begin firing. Take the north-east and south-east islands (the ones with the beam techs) before the 10-spore towers fire. Some spores might make it through early on, especially from that northern 1-spore tower, but it shouldn't do too much damage with the AC on there. You might have to rebuild a collector or reactor though - that's why reactors are cheaper (reason #2 this map is annoying).
You'll need to use PZ relays to reach the north-east and south-east islands. Use the south island PZ for your bertha when you're using the north island PZ for a relay and vice versa, so that you always have 1 PZ bertha. After claiming the north-east and south-east islands, build PZ beams on both to handle the 10-spore towers.
Finally, claim the east island with the reactor tech. You can now link the east island to your base with a PZ relay, and the NE/SE islands to the E island with regular relays, freeing up both N and S island PZs for your berthas.