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For concreteness, let’s say it’s a limit cycle in the Rosenzweig-MacArthur model. Why do the predators and prey cycle? 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. 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. 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. 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.

The two corner balls must be a stripe and a solid. If you want to get better at the pool, then learning each of the 15 balls is very important. You should click through and read it (no training in economics required; stop when you get to the bit at the end about "concrete steppes", which is where the post segues into technical economics issues). 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. Ecology is about dynamical systems. ’t just make it hard to teach ecology. I think it also makes it hard for professionals to do ecology. 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.

Which I think makes them positively misleading in many circumstances (as I say, much more on SEMs in a future post). 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. The way your lighting resonates with your players can affect whether people use your pool tables or leave them to gather dust. Since most of us are not using a light meter for exact lighting measurements, let’s just take this to mean a comfortably bright lighting level for the human eye in indoor environments, and the brightness should be consistent across the entire table. For instance, let’s say the system is at equilibrium, meaning that predator and prey abundances aren’t changing over time. Or, let’s say the predators and prey exhibit cyclic dynamics. "If the boy does what he should, I will be able to say ‘I’ve shared a bath with a Masters winner’ - brilliant.

Danny Lee, of New Zealand, hits a drive on the second hole during the second round of the 2016 Masters Tournament at Augusta National Golf Club. The UK Championship, held annually since 1977, is considered to be the second most important ranking tournament, after the World Championship. Some players (most often amateurs) place the balls in numeric order but for the 9 ball; from the top of the triangle down and from left to right, i.e., the 1 on the foot spot, followed by the 2 then 3 in the second row, and so on. 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? 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. 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.

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