Von Neumann was in many ways a traditional mathematician, who (like Turing) believed he needed to turn to partial differential equations in describing natural systems. I’ve been told that at Los Alamos von Neumann was very taken with electrically stimulated jellyfish, which he appears to have viewed as doing some kind of continuous analog of the information processing of an electronic circuit. In any case, by about 1947, he had conceived the idea of using partial differential equations to model a kind of factory that could reproduce itself, like a living organism.
