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. 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. 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. 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. Or, let’s say the predators and prey exhibit cyclic dynamics. Another prey birth? No. A prey death? 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.

No. A predator birth or death? For instance, let’s say the system is at equilibrium, what is billiards 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. The game arose from two changes made, namely that the 8 ball must be pocketed last to win, and that each player may pocket only half of the other object balls. Also, polyester film is so strong, it will not break under tension and may cause serious damage to expensive camera or projector mechanisms in the event of a film jam, whereas triacetate film breaks easily, reducing the risk of damage. In terms of lacquers and coatings, nitrocellulose dissolves readily in organic solvents, which upon evaporation leave a colorless, transparent, flexible film. 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.

And it’s wrong. Not "wrong in the details, but basically right". Not "slightly wrong, but close enough." Wrong. You cannot think about this dynamical system in terms of sequences of causal events. Which I think makes them positively misleading in many circumstances (as I say, much more on SEMs in a future post). That is, SEMs mesh with and reinforce our natural tendency to think about causality in terms of colliding billiard balls. 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 instance, you can have a sequence of causal events in which the magnitude of the effect is nonlinearly related to the magnitude of the cause. There are no sequences of events here. If you’ve stepped into a billiards showroom or even done some online browsing for pool tables, you’ve probably noticed that there are an overwhelming number of felt colors to choose from. There are 7 red balls, 7 yellow balls, and a black 8-ball. No other ball is numbered besides the 8-ball. British style billiard balls are also noticeably smaller than standard American pool balls.

You’ve got some prey that reproduce and die, and some of those deaths are due to predators. 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 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. 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. It’s just a bit of ecology I know well. But ecology is mostly not like billiards, or falling dominoes, or Rube Goldberg machines. Rube Goldberg machines are sequences of causal events. 2: Nor am I saying that ecological systems are "nonlinear" or "nonadditive". Births and deaths are happening instantly and continuously. In fact, there’s a lot happening.