Yes, Mass exctinctions are in a very short span of GEOLOGICAL TIME. And they happen for a couple of reasons. The Cretaceous-Terciary exctinction was most probably one of the fastest ones. Right now there's a great debate between scientist that believe that for this exctinction to happen, it most have been very fast (most probably hundreds or thousands of years). On this extinction aprox. 75% of species become exctinct. Lots of scientists say that most species got extinct in a matter of hundreds or thousands of years. And most of species that survived had a very short population number. Plus a negative rate of population growth. But they say that for the 75% of species to dissapear it most have taken maybe millions of years. In massive exctinctions, the animals that live on the sea, are the ones that take the longer to dissapear. There's a significant amount of variability in the rate of extinction between and among different. Not only the dinosaurs got exctint, you know? Finally the invertebrates species that dissapeared most probably took a lot longer to got exctinct, but we don't know because of the incomplete fossile evidence is incomplete of this clade.

The Ordovician Silurian, The late davonian, the permian-triassic aND THE TRIASSIC-JURASSIC took longer than the K-T extinction.

Millions of people are starving in third world countries because of land mismanagement caused by corrupt governments who gained power through the populist policies of wealth redistribution. Zimbabwe is the classic example of this, but the theft of productive land and/or punitive actions by the government against "wealthy" farmers are often seen in the rapid decline in food production in countries; and often predate famine. Beyond that, many western developed nations often have very negative policies in place that create artificial limits on food production, and these tend to include subsidies to not produce food; either to artificially limit supply and increase prices, or to (foolishly) subsidize the production of other crops (often for the production of ethanol).

Often the poverty and famine make matters worse because they encourage increased birth rates because children are (effectively) an "investment" in your retirement; and if you're likely going to lose multiple children to famine you need to have more children to ensure that some survive to adulthood.

Nothing, people will just die. Look at the 3rd world countries, limited resources are the cause of many deaths.

the logistical population model relies on having a relatively small system where you can ignore many factors. just like the exponential model, over time it will fail becasue it ignores variables. keeping in mind that the logistical model was derived from the exponential model because it didnt provide adequate predictions over time. we have to do the same thing for the logistical model if we want to mode the entire planet over a length of time thats more than a few years.

we know that environmental degradation is happening and when we factor it into the equation carrying capacity is no longer a constant through time, its also a rate which will probably look somewhat like -e^x, making carrying capacity look something like K-e^t on your graph

im not an ecologist but it seems pretty evident that as environmental degredation occurs, the environment can sustain less people, not the assumed constant amount, unless we can stop environmental degredation our population will continually decrease along with the carrying capacity.

and after a little bit of research i found that, yes carrying capacity must be augmented for many systems

The population will balance out at the capacity the Earth will allow and how effectively we utilise our resources. It's no coincidence that the population grows almost symbiotically with advances in agricultural science. You can't really have significantly more people than you can sustain.

For example, right now rearing livestock consumes 30% of our land. But meat production is about to be revolutionised with things like lab grown meat. That will allow the amount of meat produced per land unit to be rapidly increased, boosting our overall production greatly. We could probably expect a similar boom to that found after the Haber-Bosch process was first used.

For over a century people have thought the world will be overpopulated in what they called the near future, but it's never happened. They often consider a future population using current technology, which I believe is the pitfall.

It will be by region. Sub saharan Africa is expected to triple in population by 2050, but anyone can tell that won't actually happen because it can't sustain itself with its current population.

You are right. Carrying capacity is not a constant value. It's always a dependant value. It depends on various factors. But it mainly depends on resources (food, water, space, etc.). In certain populations K can change from one year to another. Actually that model (the carrying capacity one) is only valid in a lab with controlled conditions.

Nevertheless it normally adjusts quite well to real and complex populations. But we have to take into accounts that humans are really a metapopulation. So the model should be applied to continents or countries. I accept that this model is quite limited to the complexity of the human population. But we can expect the human population to stop growing and get stable depending on the availability of resources (which is the main pillar of the K theory).

The key is that as countries grow more developed, population growth goes way down. The growth of these countries is not endless

The unfortunate problem of reducing birth rates is that you'll inevitably have smaller and smaller working generations supporting retiring, elderly generations. While we should be working to lower the number of children being born, we don't want to create an unsustainable, elderly welfare state. The only solution I could see is to continue increasing the retirement age and encourage lower birth rates (and, of course, this will be a natural process as more and more people move into cities).

Thankfully, most of the growth will be in places where people consume very few resources per person (primarily in Africa and Asia), but the strain on already starving and struggling populations will be terrible.