With no natural predators and expanses of green suburban neighborhoods that allow no hunting, wildlife officials estimate the New Jersey deer population to have grown to exceed 175,000.

Ask yourself this question, who or what has no natural predators
Wildlife officials? Wildlife official's estimate? or the deer population.

(C) is the correct answer, focus on the correct subject the modifier is describing

Marger , I do not understand exactly what you are asking.

You write that with expanses of green suburban neighborhoods where there is no hunting "seems" as if it is referring to the deer population.
That phrase indeed refers to the deer population.

What do you mean by "this" in "Is there a specific rule for this?" This what?
I wrote a post about introductory phrases that begin with prepositions here.
In that instance the preposition was in. In this instance the preposition is with.
That post contains guidelines. Introductory prepositional phrases differ from five other kinds of introductory phrases.
Stricter rules govern the latter.

And what part, logically, does not "make sense"?

In this statement, both prepositional phrases (with + phrase) modify the subject of the sentence.

Both of the with phrases give us information that helps to unravel the meaning of the sentence.

• Meaning: The deer population in New Jersey does not contend with forces such as natural predators or hunters in hunting areas that might curb the deer population's growth, and officials estimate that the population has grown to exceed 175,000.

• WITH + phrase is a prepositional modifier

The target noun deer population in New Jersey is preceded by its prepositional descriptors (the "with" statements).

What is true of the deer population in New Jersey?
-- The deer population in New Jersey has no natural predators that would curb deer population growth.
-- The deer population in New Jersey has expanses of suburban neighborhoods in which to roam. Hunting is not allowed
in those neighborhoods, hunting that also would curb deer population growth.
-- Without curbs, the deer population has grown. Officials estimate . . .

• WITH?

Oxford Online U.S. Dictionary, here, defines with as possessing (something) as a feature or accompaniment.

Cambridge Online Dictionary, here, defines with as having or possessing (someone or something).

In this official question, here, just as is the case in this question, an introductory phrase headed by the preposition with modifies a sensible noun.

The deer population in New Jersey has no curbs on its growth; the population has no natural predators and has roaming areas in which hunting is not allowed.

-- Awkward rewrite: Wildlife officials estimate that the deer population in New Jersey, with no natural predators [that might curb population growth] and with habitats in which to roam where no hunting is allowed [hunting that might curb population growth], has grown to exceed 175,000.
-- The with-clauses are up front to avoid splitting two different subjects: wildlife officials (estimate that . . . ) and the deer population (which has grown . . .).

Sometimes "with" modification is easier to understand if the with phrase follows the subject.

Example from Oxford dictionary cited above:
After all, people with money and status employ other people to clean up after them.
After all, people who have money and status employ other people to clean up after them.

In the official question cited above, the first part of the sentence, not underlined, is:
With its plan to develop seven and a half acres of shore land, Cleveland is but one of a large number of communities . . .
Cleveland has a plan to develop seven and a half acres of shore land . . . .

• No other subject makes sense
Whose population is growing?
What noun has no natural predators in New Jersey?

Wildlife officials, as in options A, B, and E?
No.

New Jersey, as in option D?
No.

The deer population in New Jersey has no natural predators (and has areas in which to roam where hunting is not allowed).
The deer population has grown because it has no or few curbs on its growth.

Hope that explanation helps.
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