When discussing Castro's rise to power it occurred to me that he, Castro, was the one who initiated the dual power from January 1959 to November 1959, I have been struggling to find sources as to why he would do this? Why wouldn't he take on full power from the beginning, furthermore as he initiated it himself what was the apparent advantages? - as I only can see disadvantages.
Summary: Because you can't just march into a capital city with the support of a couple of hundred guys with machine guns and declare yourself supreme ruler for life, tell them all that they from now on will have no money or freedom, and expect everyone to just accept that.
I'm not sure what you mean with "dual power". I will assume that you mean that Cuba had both a Prime Minister and a President. This however was in no way initiated by Castro, but was a part of the Cuban constitution of 1940, a constitution Castro claimed to want to reinstate.
Castro took power as the prime minister of Cuba on February 16, 1959. Castro had already before pushed for Manuel Urrutia Lleó to be president. Perhaps because he thought Urrutia would be easy to control or because Urrutia had the trust of the non-socialist revolutionaries. You'd have to ask Castro about the exact reason. ;-)
Urrutia wanted to restore democracy and elections and restore the 1940 constituation, but Castro blocked this despite his earlier claims to want exactly this. Urrutia was then the victim of a smear campaign, leading to his resignation. Fidel then instead put in fellow communist Osvaldo Dorticós Torrado instead, and it remained like that until 1976, when the office of Prime Minister was abolished and Castro became president.
This shows that there was never any actual "dual power". Castro was in practical control, and Urrutia had no real power.
Perhaps your question then can be worded as why Castro didn't put in a communist President from the start, or why he bothered having a President at all. And the reason is that he simply did not have the complete control and power as he needed to do that. He still needed to appease both the Cuban anti-communist revolutionaries, get the popular support of the Cuban people and appease foreign governments (especially the US) by pretending to not be a communist and pretending to be a democrat.
And since he claimed that he wanted to restore the democratic 1940 constitution, if he had simply put himself into solitary dictatorial power directly after the revolution he would have been quickly ousted. Not only was not the aim of the few revolutionary forces directly to set up a dictatorship, the forces was very small, less than 500 people. They had popular support in ousting Batista and creating a democracy, they did not have popular support in replacing one murderous dictator with one that was even worse. He had to consolidate his power first, and get complete military and police control over Cuba, so that he could resist any opposition.
For a number of reasons Bolshevik inspired revolutionaries perceive that two "revolutions" were necessary in 20th century society: a "democratic" revolution to fulfill the bourgeois revolution's political aims, and then a socialist revolution to place "the working class" into power.
From the ideological perspective, Castro's actions look informed by this "two-stage"-ism still present in Bolshevism's attitude towards backwards bourgeois, colonial or semi-feudal states. Installing a "bourgeois-democratic" president prior to moving forward was seen by bolsheviks (cf: Central Europe's integration into the soviet system) as fulfilling the democratic revolution.
There was a strong desire to meet these ideological formalisms by bolshevik inspired revolutionaries, partly because of the strength of ideological binding that bolshevik type parties endured. Partly because these parties were often isolated from working class experience and fixated on ideology instead of practice.
There are a whole set of major problems with these conceptions. Chief, in my mind, is formalism in that bolshevik inspired revolutionaries read apparent signs about state formations as if they represented substantive economic relationships. Putting some chap in power and calling them Prime Minister doesn't mean that the working class has now started developing a solid critique of an ideally functioning bourgeois state. Revolutions aren't made by parties (even if parties are essential to making revolutions); and they're not over in a nice neat package after 9 months. Ask the Mountain or the Gironde if changing a formal element of government resolved a revolution.
From a raw power perspective it looks like sensible consolidation by a Prince.
Sources: The Spartacist's theoretical paper on Constituent Assemblies. Two-stage / permanent revolution debates. Popular & United front debates.