# Differential vs difference equations in mathematical modeling

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.

• One of the main differences, at the risk of repeating myself, that firstly comes up to mind is that of considering either continous dynamical systems, which are often modeled by differential equations, or discrete dynamical systems, which are mainly described by difference equations. Of course, this does not intend to be a rigorous answer but it may give you some hints. Cheers! – Dmoreno May 2 '14 at 21:15
• The answer to the first question always depends on the problem? No, it also crucially depends on the toolbox of the modeler. – Artem May 2 '14 at 21:22
• Differential equations are often easier to solve exactly – vonbrand May 3 '14 at 22:30
• Is this such a meaningful divide? Differential equations that can be solved explicitely using usual functions are few and, to solve a differential equation numerically, one resorts to difference equations. – Did May 4 '14 at 10:44
• Well, I don´t know if this is a meaningful divide. But my question is more about advantages and disadvantages in modeling, not only in the methods of resolution. Thank you all. – Hiperion May 4 '14 at 19:19

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.
Yes, difference equations are harder. But there is a way of solving the logistic difference equation $$x_{n+1}=r'x_n(1-x_n)$$ that gets a solution similar to solving $$x'=rx(1-x)$$. One way is given in http://www.dankalman.net/AUhome/atlanta17JMM/kalman_logisitc_paper.pdf