(CNN) -

"We need long-term to take out ISIS' leadership, to degrade their operational capabilities, to cut off their financing sources, to go after them in a comprehensive way to cut off their ability to do the things we've seen them do."

Those were the words of State Department deputy spokeswoman Marie Harf on Monday -- suggesting the Obama Administration is preparing to do much more against the Islamic State in Syria and Iraq than deprive it of the Mosul Dam. They sounded much like the checklist used to degrade al Qaeda over a decade.

Until the sudden capture of Mosul in June, ISIS was of concern to Western governments but not a pressing priority. Since then, the threat to Baghdad, the plight of the Yazidi minority in northern Iraq, direct threats to U.S. interests and citizens and now the gruesome execution of American journalist James Foley have galvanized an unlikely coalition.

Iran, Saudi Arabia, the United States and Jabhat al Nusrah, the al Qaeda affiliate in Syria: all have the same adversary.

On Wednesday, President Obama said: "There has to be a common effort to extract this cancer so it does not spread." French President Francois Hollande concurs. In an interview with Le Monde Wednesday he called for a "comprehensive strategy against this structured group, which has access to substantial funding and to very sophisticated weapons, and which threatens countries such as Iraq, Syria or Lebanon."

The first step in taking down al Qaeda central was the invasion of Afghanistan to deprive it of living space. This time, the United States hopes others -- specifically the Kurdish Peshmerga and Iraqi armed forces -- will do that part of the job against ISIS, with a little help from U.S. drones and F-16s.

Even so, killing off an organization that is now much more potent than al Qaeda or its affiliates will depend on a lot of things going right in a region where much has gone wrong.

Here are just a few of the challenges.

1. ISIS has considerable territory

In eight months, ISIS has taken control of swathes of western and northern Iraq, and expanded its presence in northern Syria. For hundreds of miles along the Euphrates and Tigris rivers, ISIS is the power in the land; it now holds an area larger than the neighboring state of Jordan. While al Qaeda never really held territory beyond training camps and caves in remote parts of Afghanistan, ISIS controls cities (Mosul, Tikrit and Tal Afar in Iraq; Raqqa in Syria) and oil fields, main roads and border crossings. And it possesses more military hardware than some national armies after seizing both Iraqi and Syrian military bases and armories.

Critically, ISIS is able to use both Syrian and Iraqi soil in a much more muscular way than al Qaeda and the Taliban used the mountain tracks between Pakistan and Afghanistan. This gives it tactical flexibility and safe havens. Although its Syrian strongholds have come under aerial attack recently by the Syrian air force, the group retains control of Raqqa and Deir Ezzour provinces in the north and east of the country, and has in recent days seized villages close to Aleppo, some 250 miles from the border with Iraq. It also holds villages and towns along the Syrian border with Turkey.

As ISIS threatens to overwhelm other rebel groups (see below), especially the remnants of the Free Syrian Army, one critical factor will be the Syrian regime's tactics. Until recently it has focused its fire on other groups in securing Damascus and retaking Homs. There are signs it now sees ISIS as a clear and present danger; ISIS has seized several military bases in Raqqa province, and threatens to take the important Tabqa air base.

In the last week, the Assad regime has stepped up its use of air-strikes against ISIS, no doubt aware of the coincidental benefit of showing the West that Syrian help is required to tackle ISIS.

ISIS could be squeezed from several directions, but it would require co-ordinated commitment from Syria -- which has other battles to fight and may still see ISIS as a useful counterbalance against other rebel groups -- as well as the Iraqi army and the Kurds. Desperation has led Baghdad to co-operate with the Kurds. Whether that is sustainable is open to question.

2. ISIS has men, money, munitions

Unlike most jihadist groups, ISIS has some serious weaponry and plenty of seasoned fighters. In an assault on a major Syrian army base earlier this month, ISIS deployed three suicide bombers and dozens of well-armed fighters. A long battle ended with the fall of the base (one of the last held by the regime in Raqqa) and -- according to Syrian activists -- the summary execution of dozens of soldiers.

It was symbolic of ISIS' ability to conduct complex operations simultaneously in theaters hundreds of miles apart. The Syrian Observatory for Human Rights claims ISIS gained 6,300 new recruits -- 80 percent of them Syrian and the rest foreign -- in July alone. While U.S. officials say the number of active fighters probably numbers some 15,000, Iraqi analysts believe ISIS may be able to field three times that number.

A significant number are from Europe, Australia and the former Soviet Union. On Wednesday, Austrian prosecutors said nine people had been arrested on suspicion of intending to join Islamic militants in Syria, the latest indication of the stream of radicalized young Muslims lured to the promised land.

ISIS paints a picture of this land through a sophisticated outreach program on social media and through its English-language online publication, Dabiq, which is full of accounts of the coming showdown with "crusader armies," appeals to Muslims to come to the Islamic State and promises that "it is only a matter of time and patience before it reaches Palestine to fight the barbaric jews."

The aim of creating a Caliphate gives the group a mission that appeals to many young jihadists in Syria, Iraq and beyond. It's a goal that gives ISIS' campaign religious underpinning, and is constantly referred to in the group's literature.

ISIS has shown a ruthless discipline in its military tactics, forcing the Iraqi military to fight on several fronts at once and using mobile groups of a few dozen fighters as a first wave in attacking targets. It has a well-deserved reputation for accepting casualties in the pursuit of an objective and uses probing operations to test defenses (as in Mosul) and to keep opponents off-balance. In July, ISIS fighters attacked gas installations in Homs province, which diverted Syrian forces, only to then launch more concerted assaults on targets further east.

According to the Institute for the Study of War (ISW), which follows ISIS' campaign closely, "the breadth of these linked offensives across Iraq and Syria illustrate the ISIS priority objective of establishing territorial integrity for the Caliphate, and are evidence of the large military capacity ISIS still possesses nearly two months after the fall of Mosul.

"As continued military successes from increasingly unified theatres of operation fuel the ISIS war machine, a hardened ISIS exterior line is likely to allow ISIS forces to pursue further expansion," ISW says.

ISIS control of border crossings is a source of revenue, as are bank raids in the towns and cities they have seized. The group has seized oil refineries, and may make as much as $2 million a day from its control of fuel supplies in northern Iraq. They also hold the al-Omar oilfield in Raqqa.

3. ISIS is strangling the Syrian rebels