It is not true that a continuous-time model can not be chaotic.
This is very true. But it is true in the one-dimensional case, which I claim is the case for the dynamics of the population of humans.
All that being said, why does the model for the population have to be one-dimensional? This is a reasonable objection. An answer to that is, no matter how many dimensions the system has, there should be a way to coarse-grain it and get an essentially 1-D system.
For example, let the population of humans be P(t). If we make the assumption that the growth rate of humanity depends only upon the number of people alive, then it can be written
dP/dt = f(P) P,
where f(P) is some unknown function which absorbs all of the possible variables. For example, we could say that there are thousands of factors which impinge on the growth rate of people, like availability of food, prevalence of pollution, etc. Now, our only assumption is that all of these factors depend on the size of the population. Then we can say that the growth rate is really a function only of P.
Now, to be fair, it is possible that there are factors which don't depend on the size of the population. But I think it is reasonable, in that all of the standard constraints to growth, like prevalence of pollution, limits of food, etc., can depend only on population.
Now, if this model is correct, then it is true that this system is not chaotic at all, but that there is complete regularity. For example, as long as it is true that for sufficiently large P, this function f(P) is negative, then there is a number, called the carrying capacity for the system, and any initial population will tend towards it.
I do agree that it is possible that this function f can depend on some other things, for example imagine that for some strange reason, tomorrow the availability of the food supply takes a big hit, and that f becomes much smaller without P changing. Then we could have a crash of some sort, but my feeling is that although this is possible, it's not reasonable. I think that on the other hand it is reasonable to expect that this function f will depend only on P.
I agree with your first comment completely, and I wouldn't even think it was tongue-in-cheek... If we see a phenomenon in nature, and the model doesn't predict that, then the model is not good. Certainly.
What makes life complicated is there are tons of models out there, and who knows which one goes with which real system? That's why scientists get paid the "big bucks". Now that is tongue-in-cheek.
We can look forward to wars over resources in the relatively near future
I think they haven't "already started", they've been going on for at least 30 years since the first Petroleum crisis of the seventies.
I'd go even further. I'd say almost all wars in human history were, at least to some degree, a fight over limited resources. At the very least, once colonization became a major factor in, say, 1500 or so, that's all it's been about. The big boys fighting over the resources...
Actually, it depends pretty radically on which type of model you're considering.
For example, if you assume that the population is governed by a continuous-time model, i.e. by a differential equation, then it is not really possible for a population to exceed a carrying capacity, and then crash. What happens is that the population asymptotically approaches the carrying capacity, but can never go above it.
I think it is reasonable to put humans in this case, as our growth rate is a smooth frunction of time (no breeding season, for example).
Aside note: for those who may not know, the term "carrying capacity" is a term used in population dynamics which sort of represents the available resources. In most models, what happens is that there is some amount of population which can be supported by the existing resources, and if the population is below that, it should grow, and above that, it should shrink. Most "reasonable" models of population dynamics have such a carrying capacity, and I can even state a theorem: if you have any model where the growth rate of a species depends on its size, AND it is true that this growth rate becomes negative for some sufficiently large value of the population, then you will have a carrying capacity. Furthermore, if nothing in the system changes, the population will approach this value and stay there forever.
Now, I'm not saying a crash is impossible, but you need a more complicated system. There are several ways to add complexity to the system. One way is to consider a predator-prey type of system, but of course humans have nothing which can really be called a predator. The only thing I can think of is some sort of disease, but this leads to a different model altogether (some sort of "epidemological model"), and these models rarely predict population crashes, as they have a different character, which is disease needs to be carried by disease-carrying individuals (ok, duh) but then these tend to die out. So the predator carries its own destruction around with it, in some sense, and it corrects itself.
Another postulate one can make, and I think this is somewhat reasonable, is that the carrying capacity of the earth might change radically in the future (and of course, radically downward would be the interesting case in this discussion). This could happen any number of ways. And if it turns out that the carrying capacity moves on some very quick timescale (much more rapid than the change in growth of the population), then we could see a "crash". For example, if it turned out that our ability to grow food took a big hit for some reason or another, then this could happen.
One last way to get population crashes is to consider the case of the discrete system. For example, this does apply to species which have a discrete (say, yearly) breeding system. The population does not change smoothly over time, but is simply a function of one year to the next. It is somewhat surprising, but true, that the dynamics of a population with a discrete model can be much more complicated than those with a continuous model. In fact, a discrete model can actually have what satisfies the mathematical definition of "chaos". Thus you can see any type of behaviour you might imagine, including crashes, but also including periodicity (say, a 17-year cycle for population values). I do not think it is reasonable to assume that humanity can be modeled by this sort of model, even in a coarse-grained sense, because we breed day in and day out all the time. This (and this is somewhat surprising) makes our population a much more stable quantity.
Ok, fine, I guess most people can't get to the original Science article. I had to do some funky proxy shit with my university's library server. So it should be somewhat forgiven.
But I just wanted to point out that the ABC article is somewhat misleading. The original research article at no point addresses or attempts to refute the mass-suicide myth. Because, honestly, no scientist believed that was possible. The question they considered was much more reasonable: do the large deviations come from predators eating lemmings, or from a lack of vegatation for the lemmings to eat? It seems as though they have resolved that the crashes in population come from predator over-population, not from food scarcity.
This article will probably not shake the foundations of population dynamics. As some other posters have pointed out, it is not so surprising that one sees immense highs and massive crashes in a predator-prey system, because these phenomena exist even in simple mathematical models of pred-prey systems. So for a mathematician this should fly right under the radar.
On the other hand, to a population dynamics guy, this is somewhat interesting, as in that field it is typically considered hard to model these dynamics accurately. It seems as though these guys have determined some parameters in the population dynamics model experimentally, and this is what it is interesting.
You take a job at a company, knowing you're going to get paid by the hour, knowing the guy on top of you will be an MBA, and you're surprised that you have to deal with political bullshit?
Please.
Re:A related and interesting article
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· Score: 1
Yeah, the fact that the rest of America isn't interested in the same topics as you is a clear sign that the country is going to hell.
I started to calcute how fast glaciers will increase 1m the sea level etc. but then I noticed that the same trend of 0.1mm yearly contributions will not stay if the temperature stays the same. The area of melting will slowly become smaller leading to smaller contributions.
But there is another, increasing, effect. As the amount of surface area of the earth which is covered by ice decreases, the amount of sunlight absorbed at sea level will increase. This is because ice reflects almost all of the sunlight which hits it (it has a very low albedo) but most other surfaces don't.
So, in fact, you might end up with a phenomenon which gets worse as the ice melts.
Slashdot Mirror
We would always do better to at least pay attention to what they're doing over there, the benefits would easily pay back careful study.
What evidence is this you speak of? Sci-fi novels don't count.
What? You don't know any scientists, do you? Trust me on this, fame and fortune is not the reason a person goes into that business.
This is very true. But it is true in the one-dimensional case, which I claim is the case for the dynamics of the population of humans.
All that being said, why does the model for the population have to be one-dimensional? This is a reasonable objection. An answer to that is, no matter how many dimensions the system has, there should be a way to coarse-grain it and get an essentially 1-D system.
For example, let the population of humans be P(t). If we make the assumption that the growth rate of humanity depends only upon the number of people alive, then it can be written
dP/dt = f(P) P,
where f(P) is some unknown function which absorbs all of the possible variables. For example, we could say that there are thousands of factors which impinge on the growth rate of people, like availability of food, prevalence of pollution, etc. Now, our only assumption is that all of these factors depend on the size of the population. Then we can say that the growth rate is really a function only of P.
Now, to be fair, it is possible that there are factors which don't depend on the size of the population. But I think it is reasonable, in that all of the standard constraints to growth, like prevalence of pollution, limits of food, etc., can depend only on population.
Now, if this model is correct, then it is true that this system is not chaotic at all, but that there is complete regularity. For example, as long as it is true that for sufficiently large P, this function f(P) is negative, then there is a number, called the carrying capacity for the system, and any initial population will tend towards it.
I do agree that it is possible that this function f can depend on some other things, for example imagine that for some strange reason, tomorrow the availability of the food supply takes a big hit, and that f becomes much smaller without P changing. Then we could have a crash of some sort, but my feeling is that although this is possible, it's not reasonable. I think that on the other hand it is reasonable to expect that this function f will depend only on P.
What makes life complicated is there are tons of models out there, and who knows which one goes with which real system? That's why scientists get paid the "big bucks". Now that is tongue-in-cheek.
I think they haven't "already started", they've been going on for at least 30 years since the first Petroleum crisis of the seventies.
I'd go even further. I'd say almost all wars in human history were, at least to some degree, a fight over limited resources. At the very least, once colonization became a major factor in, say, 1500 or so, that's all it's been about. The big boys fighting over the resources...
For example, if you assume that the population is governed by a continuous-time model, i.e. by a differential equation, then it is not really possible for a population to exceed a carrying capacity, and then crash. What happens is that the population asymptotically approaches the carrying capacity, but can never go above it. I think it is reasonable to put humans in this case, as our growth rate is a smooth frunction of time (no breeding season, for example).
Aside note: for those who may not know, the term "carrying capacity" is a term used in population dynamics which sort of represents the available resources. In most models, what happens is that there is some amount of population which can be supported by the existing resources, and if the population is below that, it should grow, and above that, it should shrink. Most "reasonable" models of population dynamics have such a carrying capacity, and I can even state a theorem: if you have any model where the growth rate of a species depends on its size, AND it is true that this growth rate becomes negative for some sufficiently large value of the population, then you will have a carrying capacity. Furthermore, if nothing in the system changes, the population will approach this value and stay there forever.
Now, I'm not saying a crash is impossible, but you need a more complicated system. There are several ways to add complexity to the system. One way is to consider a predator-prey type of system, but of course humans have nothing which can really be called a predator. The only thing I can think of is some sort of disease, but this leads to a different model altogether (some sort of "epidemological model"), and these models rarely predict population crashes, as they have a different character, which is disease needs to be carried by disease-carrying individuals (ok, duh) but then these tend to die out. So the predator carries its own destruction around with it, in some sense, and it corrects itself.
Another postulate one can make, and I think this is somewhat reasonable, is that the carrying capacity of the earth might change radically in the future (and of course, radically downward would be the interesting case in this discussion). This could happen any number of ways. And if it turns out that the carrying capacity moves on some very quick timescale (much more rapid than the change in growth of the population), then we could see a "crash". For example, if it turned out that our ability to grow food took a big hit for some reason or another, then this could happen.
One last way to get population crashes is to consider the case of the discrete system. For example, this does apply to species which have a discrete (say, yearly) breeding system. The population does not change smoothly over time, but is simply a function of one year to the next. It is somewhat surprising, but true, that the dynamics of a population with a discrete model can be much more complicated than those with a continuous model. In fact, a discrete model can actually have what satisfies the mathematical definition of "chaos". Thus you can see any type of behaviour you might imagine, including crashes, but also including periodicity (say, a 17-year cycle for population values). I do not think it is reasonable to assume that humanity can be modeled by this sort of model, even in a coarse-grained sense, because we breed day in and day out all the time. This (and this is somewhat surprising) makes our population a much more stable quantity.
But I just wanted to point out that the ABC article is somewhat misleading. The original research article at no point addresses or attempts to refute the mass-suicide myth. Because, honestly, no scientist believed that was possible. The question they considered was much more reasonable: do the large deviations come from predators eating lemmings, or from a lack of vegatation for the lemmings to eat? It seems as though they have resolved that the crashes in population come from predator over-population, not from food scarcity.
This article will probably not shake the foundations of population dynamics. As some other posters have pointed out, it is not so surprising that one sees immense highs and massive crashes in a predator-prey system, because these phenomena exist even in simple mathematical models of pred-prey systems. So for a mathematician this should fly right under the radar.
On the other hand, to a population dynamics guy, this is somewhat interesting, as in that field it is typically considered hard to model these dynamics accurately. It seems as though these guys have determined some parameters in the population dynamics model experimentally, and this is what it is interesting.
Please.
Yeah, the fact that the rest of America isn't interested in the same topics as you is a clear sign that the country is going to hell.
But there is another, increasing, effect. As the amount of surface area of the earth which is covered by ice decreases, the amount of sunlight absorbed at sea level will increase. This is because ice reflects almost all of the sunlight which hits it (it has a very low albedo) but most other surfaces don't.
So, in fact, you might end up with a phenomenon which gets worse as the ice melts.