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. 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. 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. You cannot think about this dynamical system in terms of sequences of causal events. When humans think about causality, they find it natural to think in terms of sequences of events. "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.

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. No. A predator birth or death? For instance, let’s say the system is at equilibrium, meaning that predator and prey abundances aren’t changing over time. 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. Predators convert consumed prey into new predators, and they die. Another prey birth? No. A prey death? 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.

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. The series spawned from a short film, Death Billiards, which was originally produced by Madhouse for the Young Animator Training Project's Anime Mirai 2013 and released in March 2013. The series aired between January and March 2015. It is licensed in North America by Funimation and in the United Kingdom by Anime Limited. Birmingham Post. March 17, 1964. p. World Pool-Billiards Association (March 15, 2016). "Pool Billiards - The Rules of Play" (PDF). 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. Wu, Olivia. "Now Google's cooking." San Francisco Chronicle. 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. It’s just a bit of ecology I know well. Like history, ecology is (mostly) not "just one damned thing after another." But it’s hard not to think of it that way, and to teach our students not to think of it that way.

Ecology is about dynamical systems. 2: Nor am I saying that ecological systems are "nonlinear" or "nonadditive". UPDATE: I’m not saying that ecology, or dynamical systems in general, aren’t causal systems. 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. At the start of the game, the red balls are racked into a tightly packed equilateral triangle and the six colours are positioned at designated spots on the table. Nowadays people are becoming more and more self-conscious, they are of the view that they shall start a healthy lifestyle. Births and deaths are happening instantly and continuously. That does not mean nothing’s happening! In fact, there’s a lot happening. There are no sequences of events here. They are, but that’s not my point here. Traditionally, the 1 ball is placed on the rack’s right corner, and the 5 ball on the left corner from the racker's vantage point to maximize contrast between the corner balls and the background, as the 1 and 5 are the brightest colored balls, however, the world standardized rules do not require this.

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