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"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, what is billiards they find it natural to think that way. What’s the next domino to fall in the sequence? 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. 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. 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. A prey individual was born, which caused prey abundance to increase by one, which caused… You’ve got some prey that reproduce and die, and some of those deaths are due to predators.

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. For instance, let’s say the system is at equilibrium, meaning that predator and prey abundances aren’t changing over time. 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.

Another prey birth? No. A prey death? No. A predator birth or death? Death Parade (Japanese: デス・パレード, Hepburn: Desu Parēdo) is a Japanese anime series created, written, and directed by Yuzuru Tachikawa and produced by Madhouse. Finally, the temperature in the garage may be uncomfortable (or unbearable) during much of the year, depending on where you live. Considering that the minimum size of a 2-car garage is typically 22’ X 20’, it’s crucial to take a hard look at the available space. 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. What "sequence of events" keeps the system in equilibrium? 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. Billiards is all about sequences of causal events. What happens if you play billiards on an elliptical table, with no friction? Billiards and snooker share similarities but also have distinct differences in terms of table size, rules, and gameplay strategies. 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. 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. 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. But this is impossible: as we noted above, the square with side length is the smallest square that can be tiled in this way. These rubber shims can be cut to fit the legs of the table, helping to hide them from sight. These colored balls can vary, but they are typically yellow, blue, pink, green, brown, and black.