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## University of Groningen

Skip to 0 minutes and 10 seconds In this lecture, I will try to convince you that not only humans, but all other biological organisms, from bacteria to elephants, are continually facing a complex and uncertain environment. This complexity is, to a large extent, caused by the fact that even the most basic and seemingly simple interactions among organisms can lead to quite complicated dynamics. In this lecture, I will illustrate this by showing that even the interactions of the tiniest of organisms– algae in a droplet of water – are highly chaotic and unpredictable. From this, I will conclude that in the living world, including human societies, such complicated dynamics– so-called non-equilibrium dynamics– is probably the rule, rather than the exception.

Skip to 1 minute and 3 seconds Let me first introduce the fundamental difference between equilibrium and non-equilibrium dynamics. Here, you see two graphs showing the change of two components of a system– for example, the abundance of two species in the course of time. The system in the top graph settles at an equilibrium after some time. This means that after some time, it reaches a state that does not change anymore. In contrast, the system in the bottom graph is a non-equilibrium system. The system changes indefinitely and never approaches a stationary state.

Skip to 1 minute and 41 seconds In all sciences, be it in the natural sciences, the life sciences, the social sciences, or the economical sciences, equilibrium arguments are very common. By this, I mean arguments that are based on the assumption that the system under scrutiny is in equilibrium. The reason for the popularity of equilibrium arguments is related to the fact that equilibria can be characterised in a relatively easy and straightforward manner. By definition, a system in equilibrium does not change anymore, implying that all tendencies leading to an increase of a system component must be exactly balanced by the tendencies leading to a decrease.

Skip to 2 minutes and 28 seconds In other words, equilibria can be characterised by a balance equation. The influx of a substance is equal to its outflux. Immigration is balance but emigration. The birth rate is equal to the death rate. The costs are balanced by the benefits.

Skip to 2 minutes and 47 seconds By means of a prominent example from biology, I will now demonstrate that such equilibrium arguments can be very misleading.

Skip to 2 minutes and 57 seconds Competition among individuals, companies, or nations are among the most important processes in economics. Likewise, competition among organisms is perhaps the most important type of interaction in biology. Biological competition theory makes one clear-cut prediction called the principle of competitive exclusion. This principle states that at equilibrium, no more competitors can survive the competition and stably coexist with each other than there are so-called limiting factors. A limiting factor is anything of importance for the growth and reproduction of organisms that is in short supply. As an example, take algae, minuscule, unicellular organisms that inhabit all natural waters, such as oceans or lakes.

Skip to 3 minutes and 55 seconds Each individual algae is very small, but since the algae occur in enormous quantities, they are of crucial importance for life on Earth– for example, for the CO2 balance of all planets in times of global warming.

Skip to 4 minutes and 12 seconds These algae need only a few resources to survive and reproduce– sufficient light and a handful of nutrients, like nitrogen and phosphate. These are the potential limiting factors. And the principle of competitive exclusion, therefore, predicts that no more algae species can stably coexist with each other than the number of resources in short supply. Since these algae need only a handful of resources, only a handful of algae species should be able to coexist next to each other.

Skip to 4 minutes and 47 seconds This prediction is not at all in line with empirical evidence. If you take a droplet of water from any pool or lake and put it under the microscope, you will most probably see something like this. Even the non-biologist will clearly see that a droplet of water contains hundreds of different algae species, many, many more than the principle of competitive exclusion seems to predict.

Skip to 5 minutes and 14 seconds This discrepancy between theoretical prediction and empirical findings is known as the paradox of the algae.

Skip to 5 minutes and 23 seconds To understand what is going on, we need to realise that the principle of competitive exclusion is an equilibrium prediction. At equilibrium, no more species can coexist than there are limiting factors.

Skip to 5 minutes and 38 seconds Is it perhaps possible that other communities are not in equilibrium?

Skip to 5 minutes and 44 seconds A long-term experiment conducted by the German biologist [INAUDIBLE] and his group indicates that the equilibrium assumption is indeed problematic.

Skip to 5 minutes and 55 seconds The graph [INAUDIBLE] shows the change in the densities of various species of algae, indicated by different colours, over a period of 10 years. All this time, the algae were kept under constant conditions in the lab. It is amazing that in this very simple environment and under constant conditions, the system does not convert to equilibrium. Is this something very special going on?

Skip to 6 minutes and 24 seconds Originally, most biologists thought so, since competition for resources was considered one of the most boring processes on this planet. The reason is that one assumed that competitive ability is transitive. If A is a better competitor than B, and if B is a better competitive than C, then A is a better competitor than C as well. As a consequence, one assumed that there is a hierarchy of competitors and that the dynamics of competition will lead to the extinction of most inferior competitors and survival of a few top competitors.

Skip to 7 minutes and 4 seconds However, as explained in more detail in the reading material, it has recently been shown that competitive interactions are often not transitive, and hence, not boring at all.

Skip to 7 minutes and 19 seconds The children’s game, rock, scissors, paper, is a prototype example for a non-transitive interaction. In this hand game, which is played all around the world, there are three strategies. Each is superior to one other strategy but inferior to the other one. Rock defeats scissors, since scissors are blunted by a rock. Scissors defeat paper, since paper is cut by scissors. And paper defeats rock, since rock is wrapped by paper. Hence, the fact that A beats B and B beats C does not imply that A beats C. Together with my colleague, Jeff [INAUDIBLE], I studied the dynamics of competition in detail, both theoretically and experimentally. We could show that competition is indeed boring, if there are only one or two limiting factors.

Skip to 8 minutes and 20 seconds In such a case, a competitive system will converge to equilibrium, and nothing really interesting will happen. However, if there are three or more limiting factors, a competitive system will often have ingredients resembling the rock, paper, scissors game. In such cases, the system will typically not settle at equilibrium.

Skip to 8 minutes and 46 seconds And if a competitive system does not converge to equilibrium, the basic premise of the principle of competitive exclusion does not hold anymore. In fact, under non-equilibrium conditions, many more than three species can stably coexist when competing for three limiting resources. This is illustrated in this graph, where nine competing species are able to exist because they go on oscillating indefinitely.

Skip to 9 minutes and 18 seconds I could show examples with many more coexisting species, yet I find the following example even more thought provoking. Here, the principle of competitive exclusion is violated, since six species coexist on only three limiting resources. The interesting thing is that the system very much looks like a system in equilibrium. If these were empirical data, the small fluctuations, which are crucial for the system behaviour, would most probably be considered noise. Hence, the smallest fluctuations around equilibrium can be very important.

Skip to 9 minutes and 59 seconds As shown in the following example, it often happens that the dynamics of a competitive system is highly regular for some time, followed by a rather chaotic period, followed by regularity, and so on. This happens despite of the fact that, in these simulations, the environment is kept constant, and chance does not play a role at all.

Skip to 10 minutes and 23 seconds In another part of the course– your third– the chaotic systems have the property that the outcome depends on the initial conditions in a very sensitive manner. The ecologist, Daniel [INAUDIBLE], used this general property to test whether the dynamics of algae competition is indeed chaotic. He took a bucket of water from a lake, mixed the water, and then took 10 samples to initiate 10 different algae cultures in the lab under conditions that are kept as constant as possible.

Skip to 11 minutes and 2 seconds In the competition– if the competition dynamics is chaotic, the outcome of the 10 experiments should be very different, in spite of the almost identical starting conditions.

Skip to 11 minutes and 15 seconds Here are the results for 3 of his 10 replicate cultures. Even to the non-biologist, it is obvious that the dynamics and outcome of competition are very different, despite of the fact that the starting conditions were virtually identical and the conditions in the lab were kept as standardised and constant as possible. In conclusion, even the most basal and simple biological interaction, competition of algae in a droplet of water, does often not lead to equilibrium, and instead exhibits complicated dynamics. This would most probably also hold for many other interactions, such as competition among companies in human societies. Non-equilibrium systems have very different properties than equilibrium systems.

Skip to 12 minutes and 5 seconds For example, hundreds of species can coexist when competing for three resources, while in an equilibrium system, not more than three species would be able to persist.

Skip to 12 minutes and 18 seconds The general insight of this lecture is never trust equilibrium arguments, unless you have very good reasons to assume that the corresponding system does indeed converge to equilibrium. Even systems looking very much like equilibrium systems can have very different properties.

# Non-equilibrium dynamics

This video introduces the concept of non-equilibrium dynamics in evolutionary systems as the rule rather than the exception. Even the smallest of organisms, algae, live in a highly chaotic and unpredictable world. What are non-equilibrium and equilibrium systems? How does this relate to the rock-paper-scissor game?