Population dynamics 
I don't understand why we make the three assumptions underlined above.
 A: A differential equation involves real numbers, that might take any value between 0 and 1.  Populations, on the other hand, are whole numbers.   They increase by 1 for each birth, and decrease by 1 for each death.
Difference equations describe the situation better, but differential equations are easier mathematics.
To make the differential equation a good match for population, the +1 and -1 must be a small fraction of the population, so the relative difference between +100 and +99.64 is small.  It's like treating water as continuous flow, although it is made of water molecules.
The other assumptions are the same: A pride of lions might eat zebras, so the feed comes in large units; but we suppose the supply of zebra meat has a continuous rate so our Differential Equation theory applies.
Lastly, if predator and prey both have five-year life cycles (synchronized generations) then large variations, of bust and boom, can happen within a generation, that affects long-term stability, but might not be predicted by the differential equation.
