The phrase red in tooth and claw means characterised by savage violence or merciless competition.

It refers to In Memoriam (1850), by the English poet Alfred Tennyson (1809-92); a personified, feminine Nature asserts that the notion of spirit refers to the mere act of breathing; Man prays, and has faith in God’s love, in spite of the evidence of Nature’s brutality (i.e. “Nature, red in tooth and claw”):

(from the third edition – London, 1850)

She [= Nature] cries ‘a thousand types are gone:
I care for nothing, all shall go.

Thou makest thine appeal to me:
I bring to life, I bring to death:
The spirit does but mean the breath:
I know no more.’ And he, shall he,

Man, her last work, who seem’d so fair,
Such splendid purpose in his eyes,
Who roll’d the psalm to wintry skies,
Who built him fanes of fruitless prayer,

Who trusted God was love indeed
And love Creation’s final law—
Tho’ Nature, red in tooth and claw
With ravine, shriek’d against his creed.

The nouns tooth and claw had been used in collocation before Tennyson composed In Memoriam; for instance, the following is from the description of a combat between a jackdaw and a water rat, published in The Morning Post (London) of Tuesday 28^th January 1823:

Thus assailed, the rat immediately stood on the defensive, and, to say truth, made a sturdy show of resistance. The daw, however, was by far too nimble for him, and easily eluded the eager efforts of tooth and claw, by hoisting himself a little way into his native element. Thus poised or hoisted, he again pounced upon the enemy, and inflicted by means of his bill at least one lusty wound before the poor quadruped could rally his forces.

The earliest transferred use of red in tooth and claw that I have found is from The Downshire Protestant (Downpatrick, County Down, Ireland) of Friday 25^th September 1857, which published an article praising the expedition led by Henry Havelock (1795-1857), Adjutant-General of the Indian Army, in retaliation for a massacre that had taken place during the Indian Mutiny of 1857:

Amid the desolations and butcheries of India, amid the indescribable agonies and debasements of the Europeans, amid the fearful carnage and the cries for mercy wrung from suffering women and children, the prowess of Havelock towers up a pillar of strength and hope. He has swept across that blood-stained country like a destroying angel to inflict God’s own retribution, and pour out His vengeance upon His adversaries, performing with his small and glorious band of avengers such mighty marvels of heroism as fill all Europe with wonder and admiration;—marching, within eight days, over a hundred and thirty miles, through jungle and waste, past mosque and pagoda, over burning sands, and beneath the blazing sun of the tropics, meeting and routing four several times that dark demon Nena Sahib and his thousands,—tracking the monster while he was yet reeking with the carnage of Cawnpore, burning his lair, and still following him on towards Locknow, with fourteen hundred Europeans putting to flight ten thousand Sepoys, and so pursuing the fiendish Rajah and his murderous hordes until at last the brown waters of the Ganges roll over Nena Sahib and his family, and close for ever over the fellest monster, “red in tooth and claw,” of whom the world has any record.