I'm reading a little about mathematical modeling and I've seen some population models based on differential equations. I've also seen some (not many) that can support both difference and differential equations.
My question is: in the case where a problem (not necessarily about population growth) can be modeled both ways, what advantages / disadvantages of using each method? The answer to the first question always depends on the problem?
I've read books like "Mathematical biology" by JD Murray and "Mathematical Models" by Haberman, but these authors do not mention advantages / disadvantages of using one method or another.
If you can recommend me literature on the subject would be great.
Thank you very much.
In application, differential equations are far easier to study than difference equations. I think this is because differential systems basically average everything together, hence simplifying the dynamics significantly. On the other hand, discrete systems are more realistic.
A great example of this is the logistic equation. The differential version $x' = rx(1-x)$ is simple to study and solved, even for students in an introductory course. On the other hand, $x_{n+1} = r'x_n(1-x_n)$ (1) not only generates interesting dynamics and chaotic behavior but also ties with open theoretical problems.
As a trade-off, the numerical aspects of difference equation is far easier.