Differential equations come naturally in natural sciences when the time period can be done infinitely small. This is because any quantum of time is much smaller than the scale of macro phenomena in physics that we model.

Contrary to that, in social sciences an elementary period which cannot be decomposed into smaller without changing the structure is usually rather large. Also, we typically have empirical observations to test models on daily, monthly or yearly basis. That is why many dynamic models in social  sciences are derived like difference equations (Cobweb model for price reaction, overlapping generation model, etc). Clearly, one can go further to differential equation as mathematical limit if \delta t goes to zero. In dynamic optimization models indeed differential equations are used but only to study the behaviour of Hamiltonian.

Maybe I do not get what you mean by dynamic system. System of differential equations? Indeed, time scales are very important but as far as I know only people studying self-organization of complex systems are doing that.

Many branches of mathematics (algebra, analysis[complex, real, functional) geometry  and others serve for investigations of differential and difference equations.

Most models of physical phenomena are given in terms of differential equations (ordinary or partial), hene their importance in many branches of science and engineering. Difference equations are much more rare in natural phenomena, but of course appear every time an approximate numerical approximation of differential equations is needed.  Mixed differential and difference equations are quite unusual to my best knowledge, except in control theory. But even there using dynamic equations on time scales doesn't seem to be worth the added technicalities. Maybe, dynamic equations on time scales are also too recent, and still have to prove their usefulness.

Dear Prof Yuri Yegorov

Dear Prof Philippe Martin

Dear Prof Gevorg Grigorian

 A lot of thanks for your answers .

 Best Regards ,

Dr/Tarek Ibrahim ( T. F. Ibrahim )

Associate professor

¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia

²Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

E-mails: [email protected]

[email protected]

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Personal Websites :

http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=A/

http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=E

https://www.researchgate.net/profile/Tarek_Ibrahim2/?ev=hdr_xprf

http://scholar.google.com/citations?hl=en&user=K0uEihsAAAAJ&sortby=pubdate&view_op=list_works&pagesize=100

Ordinary differential equations are widely used for quantitative characteristic of the biological processes. It is still one of the best modelling approach. It provides a very realistic environment.

All the equations can not be unified. So Even if time scale calculus is ready,there is a sigificance of differential equations and difference equations separately.

There are alternative approaches appart from time scales calculus. There are mathematical techniques allowing to combine (relatively) slow and fast processes in a single sistem, that means a time scales model. Please, find below references related to difference and differential equations (ODE - autonomous/nonautonomous and PDE).

Reduction of discrete dynamical systems with applications to dynamics population models. Mathematical modelling of natural phenomena. Vol 8(6). 2013. 107–129

Aggregation of variables and applications to population dynamics. In: P. Magal, S. Ruan (Eds.), Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936, Mathematical Biosciences Subseries. Springer, Berlin, 2008, p. 209-263.

Reduction of nonautonomous population dynamics models with two time scales. Acta biotheoretica. DOI 10.1007/s10441-014-9221-0