The use of quinine compounds provoked large amount of agricultural products by the community, particularly the farmers.
Is this sentence okay if the is omitted?
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The adoption of quinine compounds into widespread use stumulated a big growth of agricultural output by the community, particularly the farmers.
In most cases, we cannot omit an article if the noun is singular and countable. The noun community is singular and countable (you can have "10 communities"). If you omit the, it would be ungrammatical.
If we omit the article before "farmers", it would refer to farmers more generally, more generically. If we retain the article, then in this particular context it would be understood as "the farmers that are part of this community".
I would retain the article the. Why? Because we've mentioned that the community as a whole has increased its agricultural output. The farmers of this community have made a particularly notable contribution to this increase. We have in mind a specific group of farmers, not just farmers as a category of people.
The article the stresses this specificity: the farmers of this community, as a whole, have made this contribution.
A lot of wheat was produced by the community in 2014, particularly by farmers. (sounds strange)
Why this sounds a bit strange? Because without the article, we speak of farmers generically, as in
Violinists are musicians.
We simply named a category: "violinists".
This is a generic use of farmers from a book:
Most draft-horse users in this country, particularly farmers, dislike a horse with a white face and legs. (Farmers' Bulletin, 1954)
This is a more specific use:
These results then indicate that the respondents, especially the farmers, are not aware of the concept of pest resurgence and natural enemies. (Impact of Pesticides on Farmer Health and the Rice Environment, 1995)
The authors used the to indicate that this is a particular subgroup of the responders.
Quoting from A Comprehensive Grammar of the English Language (Quirk et al. 1985) (Unit 5.56),
Generic the occurs with plural noun phrases in two special cases:
(a) Nationality nouns, ie noun phrases referring to the people of a nationality, an ethnic group, etc., eg: The Chinese, the English.
(b) Phrases with an adjective head referring to a group of people, eg: the unemployed, the blind, the rich, etc.
In other cases, the tends to make a plural count noun phrase more "specific". It's a tricky issue; the authors of the book are themselves not very sure how to treat such constructions.
(a caveat: I'm not a native English speaker, and I got tangled in this specific\generic thing a bit)