How do researchers affect the funding opportunities of their department I am mathematician, and I am interested how math departments view my activities. 

Clearly, they view me as somebody who can carry teaching load. But I am also a researcher. What and how much I publish has an impact on my own career and my opportunities in getting grants and funding.

But what about the department? How does my work contribute to the department's possibilities when it comes research grants for individual researchers of the faculty, grants by several researchers on a project, or maybe huge grant schemes? Do I enter a statistics anywhere?

To be more concrete: when a researcher gets employed at a department, they not only do it for the sake of science. For example, they have keep an eye on their departments competitiveness. When I coauthor work with people at the department, then this certainly helps the individuals involved, e.g. in terms getting cited.

However, there are grants on a larger scale. For example, a department may apply to fund a huge center of research by the NSF. How does being productive at the department help in acquiring such a grant?
A few quick points:


Many grants have evaluation criteria that incorporate the research environment of the investigators. Thus, the research ranking and status of researchers in the department and the university of the investigator can influence the chance of getting a grant.
Researchers within a department can help each other do good research in a wide range of ways. Senior researchers often mentor and teach more junior researchers. Researchers often collaborate within a department. This is both easy, because you are collocated. It also often aligns well. I.e., an academic is not only helping his own research track record, but he or she is helping the research output and capacity of the department. 
If you do good research and increase the prestige of the school, this can have a range of other beneficial effects for the department and other researchers. Academic visitors may be more keen to visit. Good PhD students may be more attracted to the department. And so on.


So in general, the contribution of a single academic to a department's research status is often only modest, but it is still meaningful. And in extreme cases, where the department recruits a research superstar, the effects can be profound.