A city of the first class may collect a license tax from the owners and harborers of dogs and other animals in an amount which shall be determined by the city council and enforce the license tax by appropriate penalties. Any licensing provision shall comply with subsection (2) of section 54-603 for service animals. The city may cause the destruction of any dog or other animal for which the owner or harborer shall refuse or neglect to pay such license tax. The city may regulate, license, or prohibit the running at large of dogs and other animals and guard against injuries or annoyances therefrom and authorize the destruction of such dogs and other animals when running at large contrary to any ordinance.