To win the Battle of the Atlantic, the Kriegsmarine had to, essentially, sink merchant ships considerably faster than they could be replaced. While there were a few months where they were able to do so, they were not able to keep this up for the sustained periods needed to win the Battle. Donitz calculated that the Kriegsmarine needed to sink 800,000 tons of shipping per month, a rate achieved only once, in November 1942.

There were two main periods of crisis in the Battle of the Atlantic. The first came following the Fall of France in 1940, while the second followed the American entry to the war. Part of the reason for these crises were that merchants were not effectively escorted. In 1940, because there were insufficient mid-ocean escorts. In 1942-3, the USN failed to effectively protect American coastal convoys. However, other factors abounded.

The Fall of France in 1940 allowed the Germans to use bases like Brest and Lorient, on the Atlantic coast of France. This let German submarines range further west, into areas which they had been unable to reach before. With few long-range escorts or aircraft available, many convoys were unprotected in these areas. In October-November 1940, the U-boats sank 43 ships from just three convoys, HX 72, SC 7 and HX 79. British losses of merchant shipping in 1940-41 were some five million tons, greatly outweighing British production, which was roughly two million tons. The Fall of France also forced the shipping lanes further north, into the teeth of the worst Atlantic weather. This meant that many ships that weren't sunk were instead temporarily forced out of the battle by weather damage. The crisis was further exacerbated by Luftwaffe attacks on British ports along the east coast, and by air and surface attacks on British coastal convoys. These forced the abandonment of the major ports of Hull and London, with all shipping going into the western ports of Liverpool, Glasgow and, to a lesser extent, Bristol. This caused congestion in the ports, lowering effective shipping capacity.

This crisis was solved through several means. Firstly, increased production of food and raw materials in the UK reduced import requirements, making it easier to absorb losses. Secondly, a lot of the administrative issues with the ports were solved as operators became more experienced, and as new infrastructure developed. British losses were compensated for by the induction of foreign ships into British service. The German invasions of Holland, Norway, Denmark and, later, Greece meant that Britain inherited these nation's large merchant fleets. This added some three million tons of shipping to the British merchant fleet. Purchases from America added a further six hundred thousand tons. New tactics, more escorts and more aircraft helped to reduce losses from the winter of 1940. The 'Destroyers for Bases' agreement gave the RN 50 obsolete American destroyers, though the need to modernise and renovate these reduced their effect on the 1940 crisis. More effective were the introduction of more corvettes of the Flower class, especially in the Royal Canadian Navy. Destroyers were freed up for escort duties as it became clear that no invasion was coming. Permanent escort groups were formed, giving convoy escorts proper leadership, and allowing for effective tactics to be developed as experience was gained. RAF's Coastal Command introduced new aircraft, and had greater freedom to act following the ebbing of the German invasion threat. This helped to prevent convoys close to the UK, allowing the escorts to concentrate in the 'air gap' further from the coasts.

The next crisis followed the American entry to the war. Over the course of 1942, the Allies would lose 7.2 million tons of shipping, with 6.1 million of these coming in the Atlantic. The Germans had greatly increased the number of available U-boats, up to a peak of ~300. Many of these were longer-ranged than their previous subs. They began to operate off the American coast, and in the Caribbean. The Allies, in particular, the USN, did not operating convoys in these areas until May 1942. The USN believed that the role of the convoy escort was to sink any attacker, and so there was no point in establishing convoys until strong enough escorts could be provided. The British, meanwhile, were aware that the role of the escort was to get the convoy through at any cost, but did not have sufficient escorts to establish convoys in the Americas. In February, 72 ships were sunk; by June, this had risen to 122. The complete establishment of convoys in the American theatre helped to reduce losses from July. However, a new threat arose, as the RCN, primarily escorting the slow convoys, began to become over-extended. They needed to withdraw ships for refit, training and modernisation, but the time to do so was not forthcoming. The Germans also introduced new tactics, forming large wolf-packs to attack convoys. This meant heavy losses were experienced by the slow convoys, with SC 107 losing 15 ships. There were other problems; with the beginning of the Pacific theatre, and the intensification of Lend-Lease to the Soviet Union, demands on shipping increased.

The simplest solution to this crisis was the vast expansion of shipbuilding in American yards. This graph shows how, from the beginning of 1943, merchant shipping began to be turned out at rates exceeding losses. Over 1943, American yards turned out 14 million tons of shipping, outweighing all previous losses. The requirements for shipping were also reduced. By switching to less bulky products, such as dried eggs or boneless beef, import requirements were further reduced. Military requirements on shipping were reduced by increased packing - shipping an unpacked 3-ton truck took 1,000 cubic feet, while when packed, it took just 503 cubic feet. More escorts and longer ranged aircraft (as well as the establishment of airbases on the Azores), helped to reduce losses. Effective aircraft patrols over the Bay of Biscay helped to reduce the number of German subs surviving the transit into the Atlantic. The introduction of new weapons, like the acoustically guided Mark 24 torpedo, and sensors, like more effective radar or the Leigh Light, helped make the escorts and aircraft more effective. The introduction of the escort carrier meant that convoys were ensured an air umbrella, even where land-based aircraft could not go. New tactics were formed and promulgated. On the intelligence front, the cracking of the Enigma code, and the prevalence of HF/DF (high-frequency direction finding) equipment allowed convoys to be routed away from U-boat concentrations. The move away from the compromised Naval Cipher No. 3, broken by the German Navy's B-Dienst, helped to prevent the Germans finding convoys.

Ultimately, the Battle of the Atlantic was won by the Allied invasion of Europe, which denied the Germans the bases they had previously been using. Before then, though, the U-boat menace had mainly been tamed, through the effective use of airpower and convoy escorts. The Germans were never able to do sufficient damage to win the Battle, especially given the sheer number of ships American yards could produce.

Sources:

History Of The Second World War (United Kingdom Civil Series): British War Economy, Keith Hancock and Margaret Gowing, HMSO, 1949

The Defeat of the German U-Boats: The Battle of the Atlantic, David Syrett, University of South Carolina Press, 1994

The Battle of the Atlantic: How the Allies Won the War, Jonathan Dimbleby, Oxford University Press, 2016

Battle of the Atlantic, Marc Milner, The History Press, 2011