The brief temptation narrative in Mark 1:12-13 recalls the Garden of Eden.
At once the Spirit sent him out into the wilderness, and he was in the wilderness forty days, being tempted by Satan. He was with the wild animals, and angels attended him.
Instead of a garden, there is a wilderness, a desert, the land of dust and thorns into which Adam and his descendants were driven (ἐκβάλλω in the LXX). The beasts (θηρίον) are there, as they were in the garden. In the present wilderness, however, the beasts are more of a threat than the garden companions envisioned in Genesis 2:18-19 (θηρίον in the LXX).
Temptation occurs in both the garden and the wilderness. The tempter, now, is no mere serpent but Satan himself. If Adam could not stand against the serpent in the lush and peaceful garden, how can Jesus possibly hope to stand in the anti-garden, where everything in the environment weighs against human flourishing.
Despite the desert, the beasts and the temptations of Satan himself, the new Adam is driven (ἐκβάλλω) by the Spirit and defended by angels. He carries the battle with Satan out of the desert wilderness into towns and villages of Galilee.
Mitchell Lewis 
Is the serpent not satan?
Great question. I’m going to limit my response to what I understand Mark to have been thinking.
I think the linguistic parallels between Mark 1 and Genesis 2-3 indicate that Mark saw Jesus’ temptation in the wilderness as a kind of reversal or undoing of the Genesis story. That Mark would have seen the serpent’s temptation as satanic is one of those parallels. Mark’s gospel, however, is filled with numerous satanic voices that don’t directly belong to Satan himself.
Jesus is an exorcist. Casting out demons or unclean spirits is one of the core activities of Jesus’ kingdom ministry in Mark’s gospel. Jesus says that in casting out of demons he is overpowering Satan and plundering Satan’s dominion (Mark 3:22-27). The demons whom Jesus casts out, then, indirectly represent and speak for Satan.
Other creatures can also give voice to satanic temptation. When Simon Peter tried to dissuade Jesus from taking the way of the cross, Jesus replied, “Get behind me, Satan.”
The question arises, then, how Mark would have read Genesis 3. In the serpent, did Mark see Satan himself (a direct presence, i.e. the serpent IS Satan), or did he see the serpent more like one of the unclean spirits who act as Satan’s intermediaries, or as a mere creature whose temptation was ultimately Satanic in origin (an indirect presence, i.e., the serpent speaks for Satan).
In any case, where the presence of Satan in the serpent is implied but the nature of that presence is ambiguous, Satan’s presence in the wilderness with Jesus is explicit. Jesus was tested by Satan.
My point in drawing that contrast is this: since Mark can speak so freely about the presence of evil without invoking the presence of Satan himself, the explicit mention of Satan’s presence here carries remarkable weight. Satan’s work can be seen indirectly in many places, but in the wilderness its force is directly and exceptionally present.