From Math Images
- This is a section of a bifurcation diagram. It shows the relationship between a population's potential for growth and its size over time.
Basic DescriptionBifurcation diagrams are key to understanding how dynamic systems change as the parameters used to model them grow and shrink. The bifurcation diagram of logistic systems shows how changes in conditions can lead population sizes toward stability or chaos.
The Logistic Map and Logistic Systems
Consider a school of fish living in a pond. Two factors affecting the population size are fixed and easy to determine – the initial population size, which is directly measurable, and the school’s maximum rate of change. We can think of the latter as the fecundity of the species, a constant potential for growth that is based on the specific group of animals.
A third factor must be acting on the population, though, because as the fish population expands, it will begin to run out of space and food in the pond, leading to a decrease in its rate of growth. There is, essentially, a maximum population density that the pond can sustain. When the population is low, it can operate close to its full fecundity, but as it approaches that maximum density, its rate of reproduction drops.
This sort of constraint exists for any population, and it causes some distinctive behaviors that we can model using the logistic map. The logistic map is not a "map" in the way that we usually use the term – it is not an image that conveys many pieces of information at once – but rather a function that takes the population size at the current time and returns what the population size will be after one time interval (usually a year) has passed.
In this sense, the logistic map is not so different from the paper maps we use to follow paths through the woods. The main difference is that the logistic map only defines one step of a path; to see the whole path, we apply the logistic map to our starting point (the initial population size), then observe where that step has taken us, and apply the logistic map again to our new location (the current population size), and repeat the process. The path that appears as we do this is called a logistic system – a population, considered over an extended time period, whose size after every successive time interval is determined by applying the logistic map to its previous size. A logistic system is shown in Image 1. Each point in the system is the result of inputting the previous point into the logistic map.
It is important to note that the logistic map is not just any function that models population change, but is in fact a very specific function. It multiplies three factors to calculate where a population size will be after the passage of one time interval:
- The current population size.
- The fecundity of the population. Recall that this is the maximum rate of change – the speed at which the species can reproduce under ideal conditions.
- The difference between the current population size and the maximum population size.
Think about this third factor for a moment. Notice that it is smaller when the population is larger and larger when the population is smaller. In this way, it allows us to mathematically represent the manner in which a population’s rate of change is fast when it has more room to grow, but slows or reverses as it approaches its maximum sustainable value.
This model, of course, is highly simplified – in a real-world animal population, myriad other variables affect population changes. But the logistic map and the logistic systems it generates provide a general framework for considering the overall shape or movement of population growth in most animal communities. To read more about the relationship between the logistic map and real animal populations, jump to The Issue of Real-World Applications.
Bifurcation and Chaos
Bifurcation occurs when changing a parameter causes a dynamic system to "branch" into multiple values. In the case of logistic bifurcation, we are considering the limits or end behaviors of logistic systems. Recall that a logistic system is different from the logistic map – the logistic map only describes the end behavior between the current population size and the subsequent one, and therefore has no meaningful "end behavior." A logistic system, on the other hand, changes over time to approach either a steady value, a stable oscillation (such as the one we saw in Image 1), or chaos.
In order to better understand this idea, let us consider how a bifurcation diagram is plotted. To create a bifurcation diagram, we generate a logistic system for every value of a range of fecundities, let those systems run over many iterations of the logistic map to see their end behaviors, then plot their limits as a set of points over the axis of their fecundities. The method of plotting is similar to that for creating a scatter plot, but what we observe is far from scattered.
Consider Image 2, where I used the fecundity range 2.5 to 4 and displayed the limits of the logistic systems generated by those parameters. The branching visible in the result indicates that, as fecundity increases, the end behaviors of the population sizes of these systems cease to be constant and begin fluctuating between multiple values – first two values, then four, then eight, etc. As fecundity continues to grow, the diagram appears grey as the "branches" fill the range of possible values, showing that the system has become chaotic.
However, to read this diagram, do not think of the branching action as a continuous motion. Instead consider a single vertical line through the image. Such a line captures exactly one system – that is, one animal community – with one fecundity rate, where each black point that the vertical line intersects is a population size that the system yields after an infinite passage of time intervals. Thus the portion of the diagram with two "branches" appears over the range of fecundities that create systems that oscillate between two population sizes; the portion with eight "branches" shows the range of systems that oscillate among eight population sizes; and the grey portions show the ranges of systems that are chaotic and oscillate among all possible values.
This indicates an interesting property of logistic systems: While both the initial population size and the fecundity of that population are variables in the logistic map, the fecundity is more mathematically powerful when the map is iterated to generate a system; except for in very special circumstances, it is this value that determines whether the size of the population settles to a specific value, oscillates between two or more values, or becomes chaotic.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *undestanding Iterated Functions
Deriving the Logistic Map
The logistic map is a function that defines the amount of change a system goes through in exactly one time interval. When we iterate it, we generate a logistic system that models population growth as discussed above. So to find the logistic map, let us simply start with a function we can iterate to generate basic, unrestricted population growth:
Where xn+1 is the population size or density at time (n + 1). If we iterate this function to generate a system, we will see a pattern of infinite, linear growth. But, as discussed above, indefinite and steady growth is not a realistic model of population growth in the real, ecological world. To account for the changing rate of change, P, of an actual population, we construct a new P:
Where r is a parameter for the fecundity or maximum rate of change of the population. Here we have set the maximum carrying capacity for the population (discussed above in terms of maximum population density) to 1. We can think of xn as a percent population density or a population size set to a scale where the unit measure is the carrying capacity of the environment. Either way, we set 0 < x0 < 1.
In this way, the (1 - xn) factor means that the overall rate of change, P, is higher when xn is lower and lower when xn is higher. This fits our earlier discussion of population change; fluctuations in rate of change are directly and inversely related to population size, because as the population grows, it runs out of space and resources, and its growth decreases.
Re-inserting this P in our initial representation of growth, we have:
This is the logistic map. When we iterate it, we generate a logistic system that simulates changes in population size over time.
A Mathematical View of Bifurcation
What occurs, mathematically, in logistic systems to cause bifurcation? To understand, let us consider first what condition is necessary to prevent bifurcation; what occurs in the range 0 < r < 3, where systems yield a single value?
Since a logistic system is created by iterating the logistic map, we can answer that question easily. In order for any iterated function to yield only one value, it must be generating that value over and over at every step. That is, it must have the form:
Inputting the logistic map, this means that, where the bifurcation diagram has a single line showing the range of logistic systems that yield a single value, we have:
This gives us a system of equations,
This system of equations is represented graphically in Image 2 for the system at r = 2.9, where the intersection of the black line and the red curve is the solution to the system. (The intersection at (0,0) is a trivial solution.) We can use a web diagram to see that this intersection is indeed the final xn value for the system.
Web diagrams, such as the one in Image 3 (also for r = 2.9), will be integral to our discussion of bifurcation, so let us take a moment to consider how they work. Such diagrams trace the evolution of iterated functions by representing each iteration as a pair of lines on a graph. They are laid out on the framework of the functions
where f is the iterated function’s relationship between each step.
The "web" portion of the diagram, shown in blue in Image 3, begins at some point, (x0, 0), then proceeds in a series of perpendicular lines that move between the graphed curve and graphed line so that the lines have consecutive endpoints,
That is, starting at a point x0 on the horizontal xn axis, we move vertically until we meet the curve of f(xn). In this way, we apply the function f to our original x0. To set this new value as our new xn, as we do when iterating a function, we must move horizontally until we meet the line xn + 1 = xn. From that point we can repeat the process – move vertically to the curve, move horizontally to the line – to find the values of further iterations.
Thus every two consecutive vertices of the web diagram show the progression through one iteration of a function. In the case of the r = 2.9 diagram here, the lines of the web spiral in to the intersection of the line and the curve, showing us that the limit of the system is to a single point – the same intersection that we found as the solution to the system of equations above. (In this case, as both the system of equations and web diagram show, that value is ~0.6552.)
At this point, we have found, analytically, the point at which logistic systems approach a single value. Using a web diagram, we confirmed that, for the example of r = 2.9, the point we can find analytically is in fact the point the system approaches. This is called the period-one fixed point. The bifurcation diagram supports what we found here: r values less than 3 create logistic systems that converge to single values.
What about r values greater than 3? The bifurcation diagram shows us that such values generate systems that either oscillate among many values or are chaotic. However, If we examine systems with r values greater than 3, we will find that they, too, have period-one fixed points where the line xn + 1 = xn intersects the curve xn + 1 = f(xn). But as we can see in the web diagram for r = 3.4 in Image 4, these systems yield oscillations that never intersect the systems' period-one fixed points.
What happens at bifurcation points of the logistic map to stop systems from converging to their period-one fixed points? The phenomenon is clearer if we think of bifurcation not as "branching" but as period doubling – a doubling of the number of iterations necessary to return the system to any previous value. Systems that yield a single value have a period of 1, as we have seen above represented by x(n+1) = xn. And systems that yield an oscillation between two values – such as the one we observe in the web diagram for r = 3.4 – have a period of 2, where
To input the logistic map in this expression, we must find an expression for xn + 1 in the logistic map. We can do this following the rules of iteration, where
If we express the logistic map as a function l where
then we have
Returning to Eq. 6, we find that this, like a system of period 1, creates a system of equations:
This system is shown in Image 5 for r = 3.4. Here we see the system has two solutions other than the trivial point (0, 0) and the fixed point. Comparing the web diagram in Image 4 and the system in Image 5, we see that these solutions lie at points corresponding to the two values between which the system oscillates. We can call these solutions period-two fixed points.
Earlier, we found that period-one fixed points exist in all logistic systems, whether or not they approach only one point. Now, we will see that the condition x(n+2) = xn also has a solution for logistic systems with r < 3. This would seem to indicate that there are period-two fixed points as well as the period-one fixed point that we found as the limit to such systems in Eq. 5, so why do these systems not oscillate as well? The reason is clear if we graph the situation as we have in Image 6 for r = 2.9. We can see by the graph that, while both x(n+1) = xn and x(n+2) = xn have solutions, the two solutions are equal, so the system does not orbit to any higher-periodic fixed points.
Similarly, if we graph x(n+3) = xn for r = 3.4, as we have in Image 7, we do not find any new solutions that we did not see in Image 5. But for r = 3.54, shown as a system of equations in Image 8 and a web diagram in Image 9, we see that where there are solutions to higher-order iterations – that is, higher-periodic fixed points – the system will yield values corresponding to those solutions.
So what are we seeing here? There seems to be a pattern: logistic systems have limits located at their highest-periodic distinct fixed points. To form a general rule, we will bring back the term l(xn) from Eq. 9. Based on the pattern we have seen, at the highest value of k for which
has solutions, any solutions that do not appear at any lower k values are end behaviors of the logistic system in question. The logistic bifurcation diagram shows us that, as r values increase, the number of distinct higher-periodic fixed points also tends to increase. In fact, at r = 4, the number of distinct fixed points has increased to the point that it is infinite, and we observe chaos.
You can find the limits of any logistic system yourself, with no graphs or diagrams, by simply solving the system of equations produced by Eq. 12. Start at k = 1, and continue until you reach a k value that yields only solutions that appear at lower k values. The system of equations before that point contains the solutions you are looking for; all solutions to those equations that do not exist at lower k values are limits of the logistic system in question. Of course, if you find web diagrams more appealing, you will find the same results by directly diagramming the iterations of the logistic map.
r < 1
We have not paid much attention to those logistic systems in the range 0 < r < 1, because these systems all converge to xn = 0. The animation to the left should help illustrate why: no matter what x0 begins the system, the values inevitably move toward zero.To see the same result obtained mathematically,
r ≥ 4
As mentioned earlier, r = 4 marks the beginning of continuous chaos – any r value of 4 or greater yields a chaotic logistic system. To the left is a web diagram showing only 250 iterations of the logistic map at r = 4. It is clear that no pattern is emerging; if more iterations were added to the web diagram, it would begin to look completely "filled-in" with blue, but the iterations would continue to move to new points, distinct (by infinitesimal values) from any previous points.
If we were to attempt to analyze the system at r = 4 using a system of equations, as we did for previous systems, we would find that there is an infinite number of higher-order period functions that have intersections with the line x(n+1) = xn. We found above that the logistic map settles on its highest-order periodic oscillation with distinct solutions. But there is no such highest-order oscillation for a system with r ≥ 4, so such systems are non-periodic; they never repeat, instead moving chaotically.
x0 at a Lower-Periodic Fixed Point
Thus far, we have operated on the claim laid forth in the "Basic Description" that the r value has a much greater impact on the outcome of the logistic map than the x0 value does. In general, this is true. We can see the validity of the claim in the web diagrams; given an r value, almost any valid (that is, 0 < xn < 1) starting point on the xn axis will move to the same point, oscillation, or chaotic motion, determined by that r value. The only exception is for x0 equal to the one of the lower-periodic fixed points of the system. In that case, as shown on the left, the system will never yield any value other than the one or ones represented by the fixed point or points. This is the only case in which the value of x0 impacts the values generated by the logistic map.
Why It's Interesting
The Issue of Real-World Applications
Does the type of chaotic growth predicted by the logistic map ever actually occur in the natural world? Though the logistic map was created to mimic real population development, this question is still highly controversial among biologists. Several laboratory experiments on microscopic organisms and insects seem to indicate that chaotic population patterns are possible, but none have been conclusively proven to arise spontaneously, outside the lab.
This is in large part due to the high number of variables, many of them difficult to measure, that contribute to population changes when the population in question is not in a controlled setting. These make it almost impossible to determine whether populations that appear to behave chaotically are doing so because of fecundity rates predicted to cause chaos or because of other factors such as disease, drought, or famine. Coral reefs, for example, seem to exhibit chaos in many of the populations they support, but they are also very delicate systems. It is unclear whether this delicacy is a sign of logistic chaos or simply a complex web of other variables making stability look like chaos.
Many scientists argue that, if chaos is possible in nature, it must be incredibly rare and short-lived. Recall that "chaos" in this case means that the population size moves, at some point, to every possible value, and does this in a manner that has no recognizable pattern. A population that has such frequent and erratic dips to low sizes would be particularly susceptible to any natural disasters or drops in levels of resources, most likely quickly leading to its extinction. In this way, some scientists say, natural selection would eliminate any traits that lead to levels of fecundity that cause chaos. This, however, is only a theory; the debate is ongoing.
The logistic system, like many other dynamic systems, is created by applying a single process over and over to one initial condition. Because of this, any one step in a logistic system, no matter how far it is from x0, will look the same, mathematically, as any other single step. In this way, logistic systems share an important characteristic with fractals, and visual representations of them often have fractal dimensions of self-similarity. The logistic bifurcation diagram is one such representation; many parts of the diagram, taken by themselves, resemble the whole. You can use the applet below to explore the diagram by zooming in on different sections. Can you find the sections that exhibit self-similarity?
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