How is the IS doing in its effort to install a Caliphate?

The Economist has a new article on the Islamic State, which includes this map:

The article states:

It looks as if it has set expectations too high. Since last August its expansion has stalled, and it has been beaten back across much of Iraq (see map). That is why, in Najaf and to an extent Baghdad, the fight against IS no longer feels like a struggle for survival, more like yet another war. The revenues IS depends on have been reduced. And there is some evidence of increased unhappiness within the territory it holds, and among its own members.

The coalition against IS that was put together by America after Iraqi prime minister Nuri-al Maliki left office in August 2014 now numbers some 60 countries; it typically carries out a dozen air strikes a day. America has given weapons to the Iraqi army and the Kurdish Peshmerga; it is training Iraqi soldiers and says it is gearing up to do the same for a small force of anti-IS rebel fighters in Syria.

You might compare the areas affected by the Islamic State with the areas of low population density in these maps:

Population Density Map of Syria

Source

 Population Density Map of Iraq

Source

The point is that there is an big desert with little population that covers much of eastern Syria and western Iraq, and that influence in these areas affects relatively few people and relatively little economic activity.

The population density is greater along the Euphrates and the Tigris Rivers, and indeed the IS control is most notable along the Euphrates.

I suspect that a part of what is happening is a battle of Sunni forces against the Shiite controlled government in Iraq and the Alawite (school of the Shiite Islam) controlled Syrian government. Thus areas of these countries with relatively large Sunni populations tend to be shown as supportive of the IS.

In most of Iraq, though, the bulk of the fighting is being carried out by Iranian-backed Shia militias. When the IS onslaught was at its height Grand Ayatollah Ali al-Sistani, one of Shia Islam’s leading clerics, issued a fatwa calling on Shia men to join the Hashid al-Shabi, an umbrella organisation of mainly Shia volunteer militias. At least 100,000 have signed up. 

Iran has given cash and weapons to the Hashid al-Shabi; though the militia group nominally answers to the Iraqi government it is to a great extent an Iranian concern. Iran also has a lot of influence on the American-trained army, grossly run down and rendered sectarian under Mr Maliki, and has sent it advisers including Qassem Suleimani, the head of its elite Revolutionary Guard. Thanks to General Suleimani’s handiwork, Baghdad is now well fortified, and aircraft can fly in and out in safety. A 12-year curfew has been lifted, allowing Iraqis to linger on the banks of the Tigris smoking shisha pipes into the early hours. Malls and cafés are buzzing. The atmosphere is more relaxed than at any time since the American-led invasion of 2003........ 

The Kurds have taken back everything they consider Kurdistan. Their front line is supported by air power and well fortified: “It’s like world war one along that 1,000km-long border,” says one diplomat. Sorties by IS sometimes penetrate the line—there was a ferocious attack on Kirkuk in January—but they rarely get more than five kilometres into Kurdish territory. 

All told, IS has been stripped of some 13,000 square kilometres of land, reducing by a quarter what it held at its peak. American officials reckon some 1,000 fighters were killed in just the battle for Kobane, a Kurdish town on the Syrian-Turkish border that IS tried to take for months without quite managing it. Seventeen of its top 43 commanders have been felled, according to Hisham al-Hashimi, an Iraqi analyst of IS in Baghdad. But for all the losses fighters on the front line say there is no sign that IS is running short of men. Recruitment seems to be keeping up.