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Now I can hear some of you saying, ok, that’s true of the math we use to describe the world, but it’s not literally true of the real world. For the rest of this post I'll assume that's true. This post was inspired by a post on the same topic by Nick Rowe. Getting everybody in the world to jump at the same time. World Professional Billiards and Snooker Association. If you watch ESPN then you have seen at some point in time a billiards tournament. While these have one long shade that hangs above the table, they still contain multiple bulbs and come in several different lengths. Hall's theorem needs. For multiple collisions interacting there are some issues and I didn't do the calculations. And at every instant in time, predators are being born and predators are dying, and those two rates are precisely equal in magnitude but opposite in sign. At every instant in time, prey are being born, and prey are dying, and those two rates are precisely equal in magnitude but opposite in sign. And again for the sake of simplicity, let’s say it’s a constant environment and there’s no particular time at which organisms reproduce or die (e.g., there’s no "mating season"), so reproduction and mortality are always happening, albeit at per-capita and total rates that may vary over time as prey and predator abundances vary.

Now, you could try to drill down even further, down to the underlying physiological (or whatever) causes of individual births and deaths, and the underlying mechanisms linking per-capita birth and death probabilities to species’ abundances. No. What that increase in prey abundance did was slightly change the expected time until the next birth or death event, by increasing prey abundance and (in any reasonable model) feeding back to slightly change the per-capita probabilities per unit time of giving birth and dying. In the real world one could in principle write down, in temporal order of occurrence, all the individual birth and death events in both species. Individual World Championships usually take place every two or three years. At the Golf Croquet World Team Championships, eight nations contest the Openshaw Shield. In 1868 a song titled Croquet (essentially anonymous: by M.B.C.S and W.O.F.) was included in a popular song book by W. O. Perkins, what is billiards The Golden Robin (Pub. However, there is no evidence that pall-mall involved the croquet stroke, which is the distinguishing characteristic of the modern game. The pool is a fun game you can play with friends or even alone! "The prey go up, which causes the predators to go up, which causes the prey to crash, which causes the predators to crash." In lecture, even I’ve been known to slip and fall back on talking this way, and when I do the students’ eyes light up because it "clicks" with them, they feel like they "get" it, they find it natural to think that way.

It gets better. Even assuming the Earth did move by some significant distance when everybody jumped, just think about it: it'd move right back again! You cannot think about equilibria in terms of sequences of causal events, it’s like trying to think about smells in terms of their colors, or bricks in terms of their love of Mozart. It’s just a bit of ecology I know well. This is a case where it’s sooo tempting to think in terms of sequences of events; I know because my undergrad students do it every year. So now you know that white balls are heavier in the pool than snooker.What difference would you notice in playing with a larger cue ball? Beyond pockets, the playing surface is the next biggest difference. The example is predator-prey dynamics. But you’re never going to find something that lets you redescribe predator-prey dynamics in terms of sequences of events, each causing the next. 3: And to clarify further, no, I’m not trying to argue against the notion that population dynamics are ultimately a matter of individual organisms giving birth, dying, and moving around. Here’s an example. It’s a population ecology example, but not because population ecology is the only bit of ecology that’s about dynamical systems.

Purely for the sake of simplicity (because it doesn’t affect my argument at all), let’s say it’s a closed, deterministic, well-mixed system with no population structure or evolution or anything like that, so we can describe the dynamics with just two coupled equations, one for prey dynamics and one for predator dynamics. For instance, to preview a future post, much of the appeal and popularity of structural equation models (SEMs) that they let researchers take causal diagrams (variables connected by arrows indicating which ones causally affect which others) and turn them directly into fitted statistical models. For example, placing a wood table in a room with high humidity could lead to warping over time, and extreme temperature changes can affect the table’s playability and longevity. Placing a pool table in a garage is indeed an appealing and practical idea for many billiard enthusiasts. The best thing you can do to help maintain the pool table in your garage is to place a cover on it. You can get down on your hands and knees to look at the underside of the table. This system is reversible, and you can approximate it efficiently (by fixed or floating point arithmetic). But my point would still hold.