Video: Georgia: Baby Shot and Killed Mom Wounded by Two Boys

The case of a 13-month-old boy shot dead in his stroller in this tiny waterfront town in southeast Georgia left authorities groping for more leads Friday as they searched for the killer.

"Right now, we are not ruling anyone or anything out," said Todd Rhodes, a spokesman for the Brunswick Police Department.

Police have received more than 30 leads and have offered a $10,000 reward for information leading to the killer, he told reporters.

Sherry West told CNN affiliate WAWS that she was pushing her son in the stroller at 9 a.m. Thursday when two boys approached and demanded money, then wounded her and fatally shot her boy when she failed to give them any.

"He said, 'I'm gonna kill you if you don't give me your money,' and I said, 'I swear, I don't have any,' " West said. "I put my arms over my baby and he shoves me and he shot my baby right in the head."

Both boys then ran into a residential neighborhood of this seaside Georgia town about 30 miles north of Florida, she told police.

The woman told reporters that one of the attackers appeared to be 13 to 15 years of age, and the other was as young as 10.

Rhodes said police were checking attendance at nearby schools for students who were absent Thursday and fit the descriptions provided by West.

"We are aware there is some speculation being circulated throughout the rumor mill," Rhodes said, without specifying what he was referring to. "However, let me assure you -- as I previously indicated -- that we are thoroughly investigating this case and we will not, I repeat, will not leave any stone unturned."

The baby was shot with a handgun, which has not been recovered, he said.

Though no one has reported seeing the shooting, several people called 911 after hearing it, he said.

"What we heard was that there were shots that went out in the area," Rhodes said.

After stating that "there is no clear motive right now," the public information officer was asked about West's account of having been asked for money. "That is what the mother said," he said. "We as law enforcement, we can't go off of what anyone says. We have to thoroughly investigate this case."

Rhodes said crime is not common in the residential neighborhood where West said her child was shot. "It's a nice area, it's a clean area, it's an area where law-abiding citizens not only live, they also work and play.

"Much like every other city in these United States, from time to time we will have a glitch. Unfortunately, this happened here in Brunswick, Georgia."

Meanwhile, investigators returned Friday to the crime scene. "We want to make sure we haven't overlooked anything," Rhodes said.

He said West has been cooperating with authorities.

Asked whether people in the neighborhood should fear for their safety, Rhodes said they should be alert and report any suspicious activity. "But no, they have no reason to live in fear," he added.

"But why my little one?" asked a distraught Louis Santiago, the father, in an interview Thursday with WAWS. "You know, you should have just taken the pocketbook and go."

"This is obviously just a terrible day in the city of Brunswick," Mayor Bryan Thompson told reporters Thursday, "and made even more terrible by the senseless killing of a young child. And it appears perpetrated by children themselves."

The city's website describes Brunswick, with a population of about 15,000, as "one of the most unique, historic and visually stunning places in the world."

It boasts 19th-century Victorian-style homes, century-old oaks and magnolia tree-lined streets.

CNN's Nick Valencia, Chandler Friedman and Dave Alsup contributed to this report.