To be fair, PEGIDA is intentionally offensive.
In Germany "patriotism" is still a touchy subject although we seem to be becoming less uncomfortable with the concept. This is a process that takes time, but there has been a clear progression over the recent decades, especially thanks to our economical endurance and, of course, football.
Likewise, "islamisation" is an overloaded term. Taken literally it just means an increasing cultural dominance of Islam. In practice, for many it's primarily a racial issue dating back to the "Gastarbeiter" fallacy several decades ago -- in fact, quite a few of the (mostly elderly) "concerned citizens" willing to explain their motivation name the uncontrolled influx of Turkish people.
There are at least three factors at play here: asylum seekers, actual first generation immigrants and "unassimilated" n-th generation immigrants.
The asylum seekers are mostly legitimate. Some of them would love to return to their country (but can't due to war / ethnic tensions), some of them would love to stay and become German citizens. Neither of them is generally allowed to seek employment and both of them continually face the risk of getting deported at a moment's notice. Both of them are portrayed as "parasites" by right-wing groups as well as some vocal members and sub-groups of PEGIDA. They also aren't exclusively or even predominantly Arabic muslisms (most are neither), contrary to what anti-asylum groups would have you believe.
The first-generation immigrants generally come to Europe because they want to live and work here. They are frequently well-educated and interested in adopting the local language and culture. Unsurprisingly that doesn't mean they want to abandon their heritage completely. Contrary to public opinion, they generally don't "bring their unfit family over who claim social benefits without ever having paid into the system". There are immigrants who abuse social benefits, but there are far more non-immigrants who do the same. Additionally, immigrants face discrimination when seeking employment or trying to establish a business; frequently any credentials are dismissed entirely, too, so they are nearly always classified as "uneducated" regardless of their actual background.
The n-th generation immigrants sometimes cause problems, frequently they do not. There is arguably a large number of adolescent or young adult criminals who are at odds with social norms (e.g. being misogynists and prone to violence). But the underlying issues are actually more complex than that.
When Germany first tried to raise immigration we did it because we needed a large number of mostly unskilled workers and Germans weren't willing to take the jobs for various reasons (my opinion on this matter is irrelevant). For economical reasons, many of the immigrants came from Turkey and the Middle East. They largely saw this as a chance to improve their situation and that of their family.
However we considered them not immigrants but "Gastarbeiter" (guest workers). The implication was that they were invited temporarily and expected to leave eventually. Because we only considered them "guests" (who eventually outstayed their welcome), we didn't come up with any plans to make sure they would integrate into society, learn the language and take our values to heart. Somehow this still worked out, generally speaking.
For some, however, it didn't work out. The first-generation immigrants would often be working low-income jobs, barely enough to sustain their family here while sending money to their relatives back "home". Like most people who have undergone hardships, they would often spoil their children (especially boys, if their culture was already strictly patriarchal). Their children, however, would face discrimination outside their "kind": to the other kids they'd just be immigrants, not "real" Germans. At the same time, when visiting their relatives, they would find that their "home country" would be entirely alien to them, they'd be foreigners in both places, nowhere at home. Add in the common other factors for youth criminality and you end up where we are today.
Sure, that's a gross oversimplification and the criminal "immigrant" youth of today is often not just in their second generation but in their third or forth, but the basic sequence of events is frequently no different.
But even disregarding the "why", what are we supposed to do with n-th generation "immigrants" we don't like? It's naive to believe that deportation is even remotely possible or sensible. They're Germans, not just "guests". They've often lived in Germany all their live and at best spent a few vacations in their "home" country. They're our problem and it is cowardly to pretend they're not.
But rehabilitation is of course a far more boring and unappealing answer than deportation.
Yes, Islamism is a problem. We need to watch certain Muslim sects very closely. Yes, youth crime is a problem. We need to take better care of our youth and protect them from becoming criminals.
But PEGIDA is no more of an answer than "Hooligans against Salafists". It's strengthening the true parasites: nationalist right-wing groups. We don't need fearmongering and chest-thumping, we need intelligent solutions. And as long as we don't have those solutions, we need to find them.
/rant