The number of deaths does not follow a linear line.
So? Are people in the future suddenly going to be dying faster to make sure the young can continue paying for the old? Or what do you mean?
I think that we are using a different meanings of stability.
Explain what yours is then.
If a nation used a strict protocol in preventing unneeded immigrants from coming in, much false hope would be prevented.
You'll first have to define then what immigrants are needed. As I already explained, if we want to keep certain economic systems exactly the same as they are now, we'll be needing a lot more immigration than ever will be available. If we only allow immigrants in who can fill specific shortages in the labour market, we'll still be allowing in a lot more people than today.
And there is also the issue of people fleeing from war, persecution and hunger. I think it is hard to justify letting our needs prevail above theirs.
So then what? continue to shoe-horn in squillions of immigrants (who will also get old)?
I already said that immigration cannot be a preventive measure against the collapse of state pension systems. There is no way you can keep up the exponential growth needed for them. Such systems will have to be reformed.
In case you havent noticed, there is a finite amount of space- in the case of the uk, its very finite indeed.
If population growth of the native population is slowing, or the native population is even shrinking, if the immigrant population is not significantly large, or if it is the only reason the total population isn't shrinking, then space is really not relevant. If there are space problems at all, you would have had the exact same space problems if the population growth was caused by childbirths in the native population.
There is of course a finite space in every country, but there is also a finite number of immigrants. If the UK would open its borders completely, there wouldn't suddenly be 6000 million people wanting to get in. It will be hundreds of thousands, a few million at most.
People want to move to the UK because they think it is an attractive place and they think they'll have a better future there. If accessive immigration would make the UK a less attractive place, less people will go there. Let's also not forget that these people move not only because the UK is an attractive place, but also because the country they come from is an unattractive one. Immigration will slow down without strict immigration laws if these countries can become more prosperous themselves. If globalisation and the opening of markets to world trade is done right, that's going to happen too.
are they going to stand on each other's heads?
Not literally, but you'll probably need a bit more highrise than you currently have. One or two of those kilometers tall mega-skyscrapers that the Japanese have on their drawing boards will probably solve any supposed 'space problem' you think you have.