The doctrine of the “communication of idioms” states that what can be predicated of Jesus’ respective natures can also be predicated of his Person, though not necessarily predicated to the other nature. What can be said of the human nature can be said of the Person, and what can be said of the divine nature can be said of the Person, though what can be said of the divine nature may or may not be said of the human nature (and vice versa).
In other words, the human Jesus experienced hunger with respect to his human nature, but not with respect to his divine nature. But the experience of hunger is not something that belongs to natures but to persons. And so it was the Person of the Son who hungered, because he had a human nature.
This is the Christology of Cyril and Chalcedon and the Scriptures.
But this creates some wonderfully paradoxical scenarios. Here is one example:
Did Jesus die? Yes.
Did Jesus die? No.
The proposition “Jesus died” is undisputed Christian orthodoxy and the testimony of Scripture (1 Cor. 15:3). But it was the human nature that Jesus assumed precisely so that he could die (Heb. 2:14). Death, Biblical speaking, is the severing of the soul from the body (death is not nonexistence), and by definition is something the nature of the immortal and invisible God cannot experience. So Jesus experienced death clearly with respect to his human nature and not his divine, and yet it was the Person who experienced death (lest we end up with Nestorianism).
At the same time, the proposition “the Son of God did not die” is true with respect to Christ’s divine nature, which, according to the communication of idioms, can be predicated of his Person. So the statement “Jesus did not die” is also true.
And so we have one of hundreds of illustrations of why the Incarnation is more mysterious than anything a human mind can comprehend, including (to compound the mystery) the human mind of Christ.