Hi Irene- It is difficult to know what to recommend, as this is an area that receives a good deal of interest from many disciplines (economics, anthropology, sociology, geography, comminication studies....), and you do not make the focus of your interest clear. Your example of how does the rural economy affect the urban includes directionality- the most apparent way I can think of in this sense is that the rural is a site of raw materials (i.e. farm and forest products) that may be further developed within the urban economy. But the literature is vast...
Here is a link to a comprehensive study that looks at land use changes in consideration of worldwide increasing urbanization. I hope this is useful as a starting point.
From my experience, this field is quite uncharted, but apparently of great importance for many scientific disciplines.
Generally, you can consider the relation of rural-urban areas as a relationship between centre-periphery (the wider area that surrounds a centre). Rural areas host mostly agricultural and industrial activities and therefore are essential for the survival of the urban population, as they produce many commodities that are consumed in cities (food, industrial goods etc). That is a recaltionship. Another one, is that rural areas provide many recreational opportunities of the cities' population. Of course, examples mentioned above, differ from country to country, as in each case rural areas play a quite different role.
Finally, I must stress the lack of extensive quantitive and qualitive research for these links. At least this is the rule in Greece.
Your question is a very good and profound one! It is indeed two-fold: the first, relates to the dynamics of rural-urban linkages; and the second deals with how "does the rural economy affect urban economies". Available literature on the former is ubiquitous, which among others, include:
(i)UN Habitat. (1999). Strengthening rural-urban linkages 80-89. Nairobi: UN Habitat.
(ii)UN Habitat. (2005). Urban-rural linkages approach to sustainable development. Nairobi, Kenya: UN Habitat.
Onyebueke U. V. , and Ezeadichie, N H.. (2011) "Rural–Urban ‘Symbiosis’, community self-help, and the new planning mandate: Evidence from Southeast Nigeria." Habitat International 35.2 (2011): 350-360.[If you like this and are unable to obtain it, it could be arranged!]
The second aspect, which is quite critical, remain largely under-researched. Nigerian urbanists like Akin Mabogunje and Andrew Onokerhoraye have commented on the colonial-induced structural segregation between foreign-owned industries in the formal sector (UAC, John Holt, Patterson Zochanis, etc.) and traditional craft industries in the informal sector contributed to the neglect and consequent restricted growth of the latter. Related to this, low rates of backward integration in the urban-based ‘import-substitution’ industries equally created a structural gap between the productive sectors of the urban and rural economies in Nigeria.
Perhaps, the difference between the economies of countries in the Global North and South could be differentiated on the basis of how connected their rural economies are to the urban equivalent.
Besides, we have a centre working on such issues at our institute. I currently coordinate it. Pls. look at the weblink
https://www.irma.ac.in/article.php?menuid=407
It outlines our concerns with respect to developing countries in mind, more specifically, India.
There is a huge debate on issues related to rural transformation. It is however more segmented and one needs to put it together as it is spread over issues linked with the status, potential and constraints of both rural and urban.
Dear Irene, this is very important issue, especially when we speak about migration process from the villages and how it affects on agriculture development. I think that developing rural economy is highly related to develop urban economy. when the rural sector is developed productivity of country is high.
http://www.un.org/ga/Istanbul+5/86.pdf
Also, it is very important to discuss this issue in the context of sustainable development.