In Hatcher's Algebraic Topology book it is noted after 4.61 that: 


fiber preserving map + homotopy equivalence $\Rightarrow$ fiber homotopy equivalence.


Question:
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Could there be two fibrations over the **same** base space where the total spaces ***are*** homotopy equivalent, but there is **no** fiber homotopy equivalence between them? (and therefore also no fiber preserving map)


If so, I would be glad to have a simple example.



A homotopy equivalence between total spaces in a (Hurewicz) fibration which is not a fiber homotopy equivalence
You can fiber a circle over a circle in many ways.


