I was mowing in Cairns until late 2013.
It's mostly buffalo and the odd Queensland blue couch, not much else.
Don't worry about the competition and bother undercutting their prices, just do a good job.
The turnover for mowing contractors is large, many find the work too hard and give in or fall in love with cute backpackers and lose interest.
Palm fronds will be a huge part of your business. Get a long handled 2 or 3 tined fork for stock piled wet and rotting fronds.
Cairns has several types of suburb/localities or 'block types' you can usually tell what it will be like just from the address.
Each of these has different set of service requirements.
• Sand areas with low growth rate - you can afford to go cheaper on these.
• Low flat areas are usually wet and fast growing, a lot of times emptying the catcher you will need to charge more for these.
• Slopes/steep areas - Name your price, these will wear you out quickly, are usually very irregular in shape and covered in fronds and palm nuts.
Price for small to average blocks back then (2013) were $40-60, corner blocks $50-80, but the locality will affect the price due to the level of work involved.
Good luck with your new endeavour.
PM me if you would like any hints or further discussion, I mowed from Gordonvale to the Northern Beaches and up to Kuranda for more than 10 years.
EDIT: Never complain about the prices at the Portsmith dump, it is $35 per cubic meter in Melbourne!