By Oliver Barth

DRESDEN, Germany (Reuters) - Protesters marched in several German cities on Monday against higher levels of immigration and what they see as the growing influence of Islam, in defiance of an appeal from Chancellor Angela Merkel to spurn rallies she views as racist.

The rallies, organized by a new grassroots movement known as PEGIDA, or Patriotic Europeans Against the Islamization of the West, have become an almost weekly event in the east German city of Dresden in recent months.

Some 18,000 people, the biggest number so far, turned out in Dresden on Monday but similar rallies in Berlin and the western city of Cologne were heavily outnumbered by counter-protesters who accuse PEGIDA of fanning racism and intolerance.

The PEGIDA protesters waved Germany's black, red and gold flag and brandished posters bearing slogans such as "Against religious fanaticism and every kind of radicalism".

One poster in Cologne called for "potatoes rather than doner kebabs", a swipe at ethnic Turks who at around three million represent Germany's largest immigrant community.

Germany has some of the world's most liberal asylum rules, partly due to its Nazi past. The number of asylum seekers arriving in Germany, many from the Middle East, jumped to around 200,000 last year -- four times as many as in 2012.

In her New Year address last week, Merkel urged Germans to shun the anti-Muslim protesters, saying their hearts were full of hatred.

"We need to ... say that right-wing extremism, hostility towards foreigners and anti-Semitism should not be allowed any place in our society," Merkel said on Monday in the eastern town of Neustrelitz.

In Cologne, home to a large Muslim population, there were 10 times as many counter-demonstrators as PEGIDA protesters. In similarly multi-ethnic Berlin, some 5,000 counter-demonstrators swamped around 400 anti-Muslim protesters, local police said.

"Germany is a country where refugees are welcome and the silent majority must not remain silent but rather go out onto the streets and show itself," Justice Minister Heiko Maas said at the Berlin counter-demonstration.

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Cologne Cathedral, one of Germany's most famous landmarks, switched its lights off to protest against the anti-Muslim rallies. Berlin's Brandenburg Gate turned off its floodlights in a similar gesture of solidarity.

PEGIDA has unsettled Germany's political establishment and at first looked likely to help the Euroskeptic party Alternative for Germany (AfD). But the party, already suffering an internal power struggle, is split over how to deal with the movement.

(Reporting by Oliver Barth in Dresden; additional reporting by Boris Berner and Erik Kirschbaum in Berlin; Writing by Michelle Martin in Berlin; Editing by Gareth Jones)