After the last coin is mined...

[Bobsurplus]

Will asic's (miners) still need to be running to secure the network and verify transactions or will just the wallet on my computer be enough?

Thanks guys.

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[OnkelPaul]

Miners will still need to run to ensure integrity of the blockchain. Their revenue will come from fees.
How many miners will run is mostly a matter of profitability - in the steady state, only miners whose resource consumption can be paid for with their mining revenue should be active. Less efficient miners will either drop out or will be replaced with more efficient miners. Difficulty will automatically adjust so that mining power will always approximately suffice to create 6 blocks per hour.

Onkel Paul

[Bobsurplus]

Miners will still need to run to ensure integrity of the blockchain. Their revenue will come from fees.
How many miners will run is mostly a matter of profitability - in the steady state, only miners whose resource consumption can be paid for with their mining revenue should be active. Less efficient miners will either drop out or will be replaced with more efficient miners. Difficulty will automatically adjust so that mining power will always approximately suffice to create 6 blocks per hour.

Onkel Paul

Thanks Onkel. You helped prove my point.

[sergio]

well after the last coin, there will be satoshis and by that time they could be very valuable, but most of it will come from transactions fees, as time goes by block rewards halves every 4 years, but transactions fees go up as more people use Bitcoin.