In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Answering your question, the General Iftaa’ Department in Jordan, states the following:
Islam is the religion of mercy, and Almighty Allah has sent Prophet Muhammad (peace be upon him) as a mercy for all creatures where He said: “We sent you not, but as a mercy for all creatures.” (Al-Anbia` 22:107)
Therefore, this mercy is not only for man and animals; rather, it is for all creatures.
The Shari`ah scripts teach that kindness and forbearance should be observed in everything, even with animals, since the Prophet was reported to have said, “A woman got into Hell-Fire because of a cat whom she had tied, and thus it could not eat, and she did not let it free so that it could devour the vermin of the earth, until it died.” (Al-Bukhari)
He (peace be upon him) also said, “While a man was walking he felt thirsty and went down a well and drank water from it. On coming out of it, he saw a dog panting and eating mud because of excessive thirst. So he (went down the well), filled his shoe with water, caught hold of it with his teeth and climbed up and watered the dog. Allah thanked him for his (good) deed and admitted him to heaven.” (Ibn Hibban)
Among the key objectives of the Shari`ah and on top of its priorities is preserving man`s life, wealth and honor, as a means of honoring him/her.
On this basis, the Shari`ah has allowed killing some predatory animals that pose a threat to man`s life. The Prophet (peace be upon him) said, “Five are the beasts in killing which or their being killed, there is no sin: crow, kite, scorpion, rat and voracious dog.” (Al-Bukhari)
Al-Imam An-Nawawi said, “According to the majority of the Muslim scholars, predatory dog refers to every ferocious animal, such as the jaguar, the tiger, the wolf and the like.” (Sharh An-Nawawi Ala Muslim 8/114-115)
In conclusion, it is permissible to ward off the harm of predatory dogs and similar animals as a means of honoring man whose life and wealth are preserved by the Shari`ah. But, if their harm couldn’t be prevented save by putting them down, then doing so is permissible.
However, man should observe kindness when doing that because the Prophet (peace be upon him) said, “Verily Allah has prescribed perfection in all things. So if you kill, then kill well. If you slaughter, then slaughter well. Let each one of you sharpen his blade and let him spare suffering to the animal he slaughters.” (Muslim)
Almighty Allah knows best.