Systems Paleoecology – r, R, and Bifurcations

Tags

attractor, bifurcation, ecology, mathematical model, paleoecology, population growth, resilience, theoretical ecology

WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series:

1. Welcome Back Video
2. Introduction
3. Malthusian Populations

4. Logistic Populations
5. Logistic Populations II
6. Deviations from Equilibrium

In chaos, there is fertility. Anais Nin

The importance of r (and R)

The previous post outlined the circumstances in which an intrinsically stable logistic population can deviate from equilibrium, or its attractor, when perturbed by the external environment. Those deviations are brought about by either direct perturbation of the population, or an alteration of the environment’s carrying capacity (for that species). There is a third parameter, however, that determines dynamics in our models, and that is the rate of increase (r or R). It is a life-history trait determined by the evolutionary history of the species (and population), and interaction of that trait with the environment. Its influence on X(t) is generally to accelerate (or decelerate) the overall rate of population growth, with higher values causing higher overall rates. We can see this by repeating the earlier perturbation example, but with a smaller value of r (Fig. 1). The rate at which a population recovers from a disturbance is therefore determined by its intrinsic rate of increase. This is one measure of resilience. Resilience has (too many) varied meanings in ecology and other sciences, and is typically applied to communities or ecosystems, i.e. multi-population systems. In this instance, however, resilience means specifically the time taken for the system to return to equilibrium, and can therefore be applied to our population. Holling (Holling, 1973) has termed this type of resilience engineering resilience, as the concept has broad application in physics and engineering. Under this definition, populations that recover more quickly are considered to be more resilient. Thus, the rate at which a population recovers from a negative perturbation is directly proportional to its intrinsic rate of increase.

Two populations with different intrinsic rates (blue, R=0.25; orange, R=0.5; K=100) recovering from simultaneous and numerically equal direct perturbations. The population with the higher r recovers faster to equilibrium, and thus has greater engineering resilience.

Importantly, however, a population isolated from conspecific populations can never grow faster than its intrinsic rate of increase.

Bifurcations

The intrinsic rate of increase can also be a source of dynamics more complex than those presented so far. This is particularly acute in the discrete time, or difference, models because of the recursive feedback loop present in those models (i.e. X(t + 1) is a direct function of X(t)). May (1976) highlighted this using a discrete logistic model.

EQ. 1: (future population size) = [(intrinsic growth rate) x (current population size)] x (growth limited by carrying capacity)

where x is population size standardized to a carrying capacity of 1 and is restricted to the interval 0 < x < 1, and r is the intrinsic growth rate.

May showed that very complex dynamics, such as chaos, can emerge from this very simple model of population growth with non-overlapping generations, as r is increased. The same holds true for the discrete Ricker logistic model presented earlier (Eq. 1). In that model, values of R < 2.0 yield the expected equilibrium logistic growth, but even at values as low as 1.8 < R ≤ 1.9, interesting behaviours begin to emerge — approaching the carrying capacity, population size will overshoot K very slightly before converging to it (Fig. 2A). This is a transient, pre-equilibrium excursion. At R = 2.0 the system undergoes a dramatic shift from the single-valued equilibrium point to an oscillation between two values around the carrying capacity (Fig. 2B). You will notice that the transient overshoot is preserved, and in fact the amplitude of the oscillation is initially large, but the system eventually converges to two fixed values. Those values represent a new attractor, because the system will always converge to an oscillation between them. The value R = 2.0 is a critical point at which the system is said to undergo a bifurcation, with the equilibrium now consisting of two population sizes.

Transitions of a discrete logistic function with increasing R. Values of R, from upper plot to lower: 1.9, 2.0. K=100, and X(0)=1.0. The upper plot illustrates a quasiperiodic series, while the lower plot is chaotic. Each series was iterated for 30 generations. Plots on the left show population size, while on the right they plot the attractor for the entire series.

The amplitude of the oscillations grows as R increases, and the system eventually undergoes further bifurcations, e.g. where the population oscillates between four fixed points. Is the population still stable? The determination of stability now depends on two factors, the first of which is the timescale at which the population is observed. Population sizes and the attractor are repeating cycles, with X(t) cycling (or “orbiting”) between an ordered set of points. Therefore, if the length of time over which X is observed exceeds the period of the attractor, one will observe the system repeating itself, but if it is shorter, the question of stability remains open unless the underlying dynamical law is known. Second, the observation of multiple cycles allows a complete description of the system’s dynamics, and one could then conclude that the system is confined to a compact subset of the phase space. Most importantly, one would conclude that the system is deterministic and predictable. Recall that deterministic means that the entire future trajectory of the system is knowable, given the law by which the system evolves or unfolds over time, i.e. the dynamic equation and the initial condition of the system (X(0)). This is a very Newtonian system which will continue in this manner unless or until acted upon by an external force. The system is as stable as it was when it possessed a simple equilibrium, the only difference being that the attractor now traces a fixed trajectory in phase space comprising multiple values rather than occupying a single point. We can therefore refine our definition of stability.

Intrinsic stability: An intrinsically stable population expresses a finite set of infinitely repeating values.

This definition encompasses both our earlier simple equilibrium, and our newer oscillatory equilibria. It also encompasses further bifurcations that the system undergoes as R is increased, e.g. to a four point attractor.

Vocabulary
Bifurcation — The point at which a (nonlinear) dynamic system develops twice the number of solutions that it had prior to that point.
Engineering resilience — The time taken for a system displaced from equilibrium to return to equilibrium.

References
Holling, C. S. (1973). Resilience and stability of ecological systems. Annual review of Ecology and Systematics, 4(1):1–23.
May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560):459–467.

Community Stability

Tags

dynamics, extinction, food webs, Permian-Triassic extinction, stability, theoretical ecology

A community stability “landscape”. Green depressions represent regions of stability (the basins). There are two stable communities (balls) in the basin. The one on the left is disturbed, and returns smoothly to its original position. The one on the right amplifies the displacement, either returning eventually to its original position, or possibly transitioning to another basin, or alternate state.

One of the central questions of our paper was, “How stable are ecological communities during a mass extinction?” This might seem a bit of a silly question at first glance, with the obvious answer “Not stable at all!” But that is not necessarily the case. Consider yourself standing on the deck of a leaky shop which is filling gradually with water. You know that the ship is going down, but  your situation is stable as long as the deck remains level, or at least until the water begins to lap around your knees. We often tend to think of mass extinctions as chaotic dramas, perhaps being influenced by the end Cretaceous event, 66 million years ago (mya), when a 10 kilometer asteroid collided with the Earth and much hell really did break loose. There is also a lot of talk these days about collapsing ecosystems, because we continue to warm up the planet, eat all the fish we can eat, and so on. But what would a Sixth Mass Extinction really look like? Would ecosystems collapse, or wind down slowly to shadows of their former selves? Did the citizens of a Roman city in Gaul turn out the lights one night in the 5th century CE, bid the ancient world farewell and lay out their clothes for the next morning’s Middle Ages? Or did they rather one day, in corner market conversation, question how the heck all those Germans wound up in government anyway? A little bit of both I suspect.

So getting back to our question of mass extinctions at the end of the Permian, some 252 mya, were ecosystems stable before the extinction, collapsing as species extinctions spiralled out of control, or were they whittled down to a hardy core? Did they become more sensitive to smaller insults, such as storms or droughts, or were they hardy cores? Answering these questions depends surprisingly on what you mean by “stability”. The term is used in various ways in ecology, and I’ve even been accused of using it in a rather narrow sense, in contrast to others who believe that there are many kinds of stability. I am not convinced that the latter is really the case, and even if it is, I would argue that there is only one important type of stability, and that is the likelihood that the community will persist, that is, continue to exist in pretty much the same form, under non-extreme environmental conditions. The conditions that have prevailed during the history of a stable community, including seven year droughts, megastorms, the occasional disease epidemic, etc., did not cause the community to collapse or its species to become extinct. This definition encompasses many aspects of stability. Consider again our boat, this time with no leaks. Whether it is at anchor in a calm bay, sailing steadily on smooth seas, heaving rhythmically on rolling waves, or pitching about chaotically in a storm, the most important question is, are you and the boat still afloat the next day? I therefore do not believe that there are many different kinds of community stability, but instead different aspects to the likelihood of persistence, and different ways to measure it.

In our paper we looked at one particular aspect of stability, commonly termed “local”. Let me explain why. Imagine our community is represented by a small ball, and its state is represented by its position on a landscape (Fig. 1; scientists love to imagine states as positions on an imaginary landscape). The landscape is rugged and hilly, and is shaped by the environment. If our ball is on a slope, it won’t stay there for very long, and its state will change. It is unstable. If it is located at the bottom of a basin though, then it will remain there, as long as nothing disturbs it. It is stable. If it is displaced by a small amount, remaining in the basin’s depression, then it will roll downhill and return to the bottom of the basin as soon as the displacing force is removed. Interestingly, with a little care one could also balance the ball on one of the peaks, and it will remain there, but that position is precarious and fragile. Any relatively minor force would serve to start a downhill roll. The basin is an “attractor“.

Now, there are a number of limitations to using local stability to describe the behaviours (dynamics) of which your community is capable. A perhaps obvious one is what happens as you increase the distance by which the ball is displaced. One possibility is that the community does not return to the basin of origin, but specifically what does happen to it depends on the topography of the landscape. A slightly more subtle set of questions, and the ones which we pursued, is what happens to the community between the time at which it is displaced (a little), and its return to the bottom of the basin? Is it a simple, Sisyphusean roll back down to the bottom of the basin? Does it happen quickly? What if the ball is kicked again before it’s finished rolling? These are important questions to ask when the planet is undergoing a slow, persistent environmental meltdown as it did 252 mya.

There are probably many interesting and important transient dynamics between departure and return. These can be very difficult to predict. To appreciate this, let us agree that our community really isn’t a ball at all, but is better described as a large collection of balls (species populations), many of which are connected to each other with ropes, pulleys and springs. The contraption now could even amplify a displacement, weaving about the slope, perhaps shifting to a new basin, or losing species along the way. These transient dynamics might be fairly common in real communities, and communities might in fact never really spend any time at the bottoms of basins, instead rolling about, tracing out complicated pathways in response to displacing forces, according to their system of species, ropes and springs.

So, what did our South African ecosystem do 252 mya as the planet became less and less hospitable?