Since static characters remain the same throughout a narrative, they do not change beyond the way in which they are first presented. In order for a character to be dynamic, there must be situations in the narrative in which this personage meets with some conflict and reacts in an unpredictable manner. Therefore, if Santiago, the protagonist, is a dynamic character, the reader must find in him a change in personality or attitude; that is, his behavior will not always be predictable.
In examining Santiago as potentially a dynamic character, the reader seeks scenes in which this protagonist engages in internal or external conflict since there is usually some character change after conflict. Of course, the main conflict in the novella is Man vs. Nature. And, since Santiago perceives the sea as "La Mer," his conflict transcends one merely with a fish, but, rather, becomes one of man versus woman in a sort of marriage of lives. For, Santiago is not merely trying to catch a fish after having gone without doing so for eighty-four days; he is fighting for his very way of life, for what gives it meaning: his relationship with "La Mer."
So, the character change in Santiago--who is, indeed, a dynamic character--centers around his interactions with the marlin and the sea while he struggles to bring in this great fish and fight off the sharks. Thus, his interior dialogue is paramount to his developing attitudes. And, these attitudes develop from venturing out to make a catch to a conflict with "La Mer," and then to an existential contemplation of life. For instance, after successfully having outdone the fish by living through their combat, Santiago heads back to shore, but the sharks come, and he begins to despair,
"It was too good to last,....I wish it had been a dream now and that I had never hooked the fish and was alone in bed on the newspapers."
Nevertheless, Santiago rebounds spiritually,
"But man is not made for defeat....A man can be destroyed but not defeated....But I must think....Because it is all I have left....I am still an old man. But I am not unarmed."
Santiago realizes that he is higher than the fish since he can always think and have resolve. With the theme of the importance of dreams which give man a spiritual dimension, Santiago dynamically rises to the challenge of the sea and, although he loses the marlin to sharks, he is unconquered spiritually as he can still dream of the lions. He has changed from his earlier despair from not catching fish to a state of hope.
[Please see the link to another question about Santiago's character that is cited below]
The problem lies with the question. While the idea of round and flat and dynamic characters works well with short stories and some other literary forms with some stereoptypes that we .all can identify, in "The Old Man and the Sea" the characters besides being individual figures are expressing some archetypal or basic images from the collective or universal mind. For example, one can think of Odysseus or any wise old man figure, and then the child who in this case is a caring figure. The sea adventure reaches back to Melville. The characters are best understood - or at least one way of understanding the story - as having archetypal dimensions, thus providing more complexity and psychological interests than other terms or ways of identifying character in the story.