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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. 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. In fact, there’s a lot happening. Births and deaths are happening instantly and continuously. That does not mean nothing’s happening! "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.

If you’re setting up a pool table in your bar, pub or even your rec room, then don’t underestimate the power of good pool table lighting. The game continues until every red ball has been potted and only the six colours and the cue ball are left on the table. A player wins the game if that player legally pockets the 8 ball into a designated pocket after all of their object balls have been pocketed. Carom games such as straight-rail, three-cushion, balkline, and five pins make use of only three balls: a red object ball, one solid white cue ball for player one, and another cue ball that is white with a dot on it, or yellow, for player two. The make up was a good representation of the mix of players. The world's top 10 association croquet players as of October 2023 were Robert Fletcher (Australia), Robert Fulford (England), Paddy Chapman (New Zealand), Jamie Burch (England), Reg Bamford (South Africa), Matthew Essick (USA), Mark Avery (England), Simon Hockey (Australia), Harry Fisher (England), and Jose Riva (Spain). Challenge yourself and others and win against everyone to reach the top. For concreteness, let’s say it’s a limit cycle in the Rosenzweig-MacArthur model.

Why do the predators and prey cycle? 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. No. A predator birth or death? 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. 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.

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. A prey individual was born, which caused prey abundance to increase by one, which caused… Predators convert consumed prey into new predators, and they die. You’ve got some prey that reproduce and die, and some of those deaths are due to predators. They are generally very old, always dirty, while the butler is as ancient as the bungalow. There are no sequences of events here. You cannot think about this dynamical system in terms of sequences of causal events. 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. Choose from 7 variations in this one game if you love real life looking billiards. 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.

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