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The Blind Men and the Elephant

by John Godfrey Saxe (1816-1887)

It was six men of Indostan To learning much inclined, Who went to see the Elephant (Though all of them were blind), That each by observation Might satisfy his mind. The First approached the Elephant, And happening to fall Against his broad and sturdy side, At once began to bawl: "God bless me! but the Elephant Is very like a WALL!" The Second, feeling of the tusk, Cried, "Ho, what have we here, So very round and smooth and sharp? To me 'tis mighty clear This wonder of an Elephant Is very like a SPEAR!" The Third approached the animal, And happening to take The squirming trunk within his hands, Thus boldly up and spake: "I see," quoth he, "the Elephant Is very like a SNAKE!" The Fourth reached out an eager hand, And felt about the knee "What most this wondrous beast is like Is mighty plain," quoth he: "'Tis clear enough the Elephant Is very like a TREE!" The Fifth, who chanced to touch the ear, Said: "E'en the blindest man Can tell what this resembles most; Deny the fact who can, This marvel of an Elephant Is very like a FAN!" The Sixth no sooner had begun About the beast to grope, Than seizing on the swinging tail That fell within his scope, "I see," quoth he, "the Elephant Is very like a ROPE!" And so these men of Indostan Disputed loud and long, Each in his own opinion Exceeding stiff and strong, Though each was partly in the right, And all were in the wrong!
The Blind Men and the Elephant

The Blind Men and the Elephant

Models as lenses

Metaphors are a core component of human thought, but one that seldom get's much attention in scientific study.

The metaphors we use frame not just how we interpret the world, but how we organize it.

For example, we usually view sport leagues hierarchically -- teams are ranked in a linear order from best to worst. In the NCAA college basketball tournament, the teams are organized in a bracket allegedly to efficiently determine which team is the "best" team. This is so common, we barely even notice what we are doing. However, there is an alternative metaphor that is equally reasonable. We could view sporting competitions as generalized rock-paper-scissors games, where there is no "best" team. From this perspective, the NCAA tournament is a rather silly exercise because it never really establlishess that the winning team is the best. The most one can reasonably conclude is that the winning team is better than the 6 teams they played.

How do you discuss something without a metaphor for it?

Modelling crime metaphors

The results of Thibodeau and Boroditsky and other researchers suggest that metaphors are powerful communications tools, providing a compact way of communicating a significantly more complex phenomena. However, the specific meanings of metaphors can be elusive. We all have some visceral feel of what it means for a predator to be lurking in the dark waiting to pounce on us, and how it feels during flu season, but it takes a fair amount of work to unpack those feelings and translate them into a specific and tangible way. Let \(N\) be the count regular city residents, \(V\) be the count of crime victims.

Predator-crime model

For the predator metaphor, we let \(P\) be the count of criminals who prey upon the city.

\[\begin{align} \dot{N} &= -\kappa P N, \\ \dot{V} &= \kappa P N, \\ \dot{P} &= \eta - \gamma P. \end{align}\]

The effective interventions will reduce opportunities for crime (reduce \(c\)), reduce the number of predators by slowing immigration (reduce \(\eta\)), or accelerating incarceration (increase \(\gamma\)).

Disease-crime model

For the disease metaphor, let \(I\) be the city residents infected with a need to commit crimes.

\[\begin{align} \dot{N} &= -\beta I N - \alpha N + \rho I, \\ \dot{V} &= \beta I N, \\ \dot{I} &= \alpha N - \rho I . \end{align}\]

The effective interventions will reduce opportunities for crime (reduce \(\beta\)), reduce the number of criminals by improving well-being and reducing the motivations for criminal activity (reduce \(a\), increasing \(r\)) .

These two models do capture some of the differences implicit in the predator and disease metaphors used to describe crime. Which metaphor is correct? The simple data we have on the rates of crimes does not help us distinguish between these two models. Both can be made to fit the crime rates equally well. We all have different personal experiences that may give us reasons to see the world through one or the other of these metaphors. And each metaphor leads to different ideas and conclusions. But crime is a real phenomena. We have to talk about it and confront it.

When we only can talk in terms of these metaphors, this is a hard question. But once we have our models of these metaphors, we can now imagine not just how these models contrast each other, but also how they may fit together as puzzle pieces. We can find a third perspective by building a single combined model that incorporates the important aspects of both metaphors.

\[\begin{align} \dot{N} &= -\beta I N - \kappa P N - \alpha N + \rho I, \\ \dot{I} &= \alpha N - \rho I - \delta I,\\ \dot{P} &= \eta - \gamma P, \\ \dot{V} &= \beta I N + \kappa P N. \end{align}\]

Within our combined model, we can now drill down and seek out data that will quantify the rates and impacts of different components. We can go a step further and try to model the costs and effectivenesses of different interventions, as well. Not only that. The combined model gives us a new frame in which we can imagine the impacts of ideas that we hadn't considered before like the potential benefits of social safety nets, the secondary effects of incarceration, and the potential for recidivism.

Fire and qualitative analysis

At this point in our studies, we have introduced differential equations as a modelling tool and shown how we can use numerical methods to solve these equations. However, numerical solutions provide us a (literally) narrow slice of information for specific initial conditions and parameter values. If we change the initial condition or parameter values, the solution also changes. Rather than just predicting one solution, we sometimes would sometimes prefer an algorithm that can efficiently predict the solution of the system at any time in the future for any initial conditions and parameter values. Unfortunately, no such general algorithm can exist. Only in rare cases can we find efficient closed-form solutions expressed with standard functions.

Although closed-form solutions are rare, there are still a myriad of methods we can use to gain intuition about the behavior of solutions of differential equations without actually solving them. The study of these methods was pioneered by Henri Poincare and Aleksandr Lyapunov in the late 1800's and is now generally referred to as the qualitative theory of differential equations. To give you a taste of the ideas and methods applied in qualitative differential equation theory, let us consider a very old human activity -- lighting a fire.

A model of fire-starting

Lighting a wood fire with a match requires care. It is not simply a matter of throwing a lite match onto a pile of wood. That wood might be great fuel for a roaring fire, but it won't just burst into flame because a match is thrown onto it. A likely outcome of such a scenario is that the match will burn out without starting the fire. For best results, care should be taken in arranging the fuel and applying the match's flame to those pieces of tinder that are easiest to ignite. More kindling and wood is then added to the growing flame at a steady rate until the tinder and kindling are all burned away but the biggest logs have caught and the fire is roaring. If fuel is added too quickly or recklessly, and nascent fire may be smothered, but if it is added too slowly, it may burn out.

This is a familiar process for many, but it is rather odd. Why is it that a roaring fire must be built up with tinder and kindling rather than bursting forth immediately when the match is thrown in? We know all the needed fuel is present, but for some reason, that one little match is not enough. Shouldn't that match start a little fire, which starts feeds a bigger and bigger one no matter how the fuel is arranged?

Fire is an example of an exothermic reaction system -- a system of reactions that release heat as a product. Wood is made of a carbon-based polymer called cellulose that creates \(C O _ 2\) and heat when burned with oxygen. The rate of the reactions in this system obeys Arrhenius's law -- the negative logarithm of the reaction rates is inversely proportional to the temperature and the higher the temperature, the faster the reactions go. For Arrhenius's law to apply, temperature must be measured on an absolute scale such as degrees Kelvin. The heat released by these reactions raises the temperature in the system. This increase in temperature is counteracted by the lose of heat to the surrounding environment which has not been heated up yet by the reactions.

A quantitative conceptual model of a wood fire is given as follows. Let \(\theta(t)\) be the temperature of the wood at time \(t\). The rate of change in temperature is driven by two components -- heat input from the burning and heat lose to the environment. \[\frac{d\theta}{dt} = I - L\] Assume that the increase in temperature is proportional to the reaction rate obeying Arrhenius's law \[I = a e^{-r/\theta}\] where \(a\) is a proportionality constants and \(r\) is the Arrhenius constant, while the temperature decrease obeys Newton's law of cooling -- the rate of heat lose is proportional to the difference between the current temperature and the background temperature \[L = b (\theta - h)\] where \(b\) is a proportionality constants and \(h\) is the background temperature. Thus, the rate of change in temperature satisfies \[\frac{d\theta}{dt} = a e^{-r/\theta} - b (\theta - h).\] Let's call this the fire equation.

Dimensional Analysis

The fire equation has one independent variable (time), one dependent variable (temperature), and four parameters (\(a\), \(b\), \(r\), and \(h\)). Each of the parameters has its own units. The heating rate \(a\) has units degrees per time. The Arrhenius constant \(r\) has units of degrees. The cooling rate constant \(b\) has units of per-time. And the background temperature \(h\) has units of degrees. By choosing the scales with which we measure temperature \(\theta = r \hat{\theta}\) and time \(t= (a/r) \hat{t}\) according to dimensional analysis, we can show that all solutions of our original model are the same as solutions of the simplified equation \[\frac{d\hat{\theta}}{d\hat{t}} = e^{-1/\hat{\theta}} - \hat{b} (\hat{\theta} - \hat{h})\] for the dimensionless cooling rate \(\hat{b} = br/a\) and the dimensionless background temperature \(\hat{h}=h/r\).

The hat-notation introduced above is a very convenient way of describing non-dimensionalization. However, it somewhat clunky to continue using it for all the variables. Instead, we adopt an alternative convention. For the rest of this chapter, we drop the hat notation and assuming \(a=r=1\), so the fire equation is written \[\frac{d{\theta}}{d{t}} = e^{-1/{\theta}} - {b} ({\theta} - {h}).\] If needed, the results we obtain can be converted back to dimensional variables by substituting for the hat-variables in the change-of-variables formulas above. Keep in mind for all of the discussion to follow that whenever we say background temperature \(h\) is large (\(h \gg 1\)), we really mean that it is much larger than the Arrhenius constant (\(h \gg r\)), and vice-versa. Similarly, when we say the cooling rate \(b\) is small (\(b \ll 1\)), we really mean the cooling rate is small relative to ratio of the heating rate and Arrhenius constant (\(b \ll a/r\)).

Steady-state Analysis

The fire equation does not have a simple answer, but we can characterize its dynamics very clearly using a geometric method. The core ideas are that when cooling removes less heat than is being created by burning, the system will heat up, but when cooling removes more heat than the burning is producing, the system cools down. If we plot the rate of change in temperature as a function of temperature itself, we can see when the temperature will increase and when the temperature will decrease. For example, if the cooling rate is large, then the heat created by the "fire" will be quickly lost to the environment. The temperature of the system will be drawn toward the background temperature -- if it is too hot, it will cool down, and if it is too cold, it will heat up until \(\theta \approx h\).

Temperature has a steady-state solution (right), determined by the balance of heating according to the Arrhenius law (left) and cooling according to the Newton law (middle).

Temperature has a steady-state solution (right), determined by the balance of heating according to the Arrhenius law (left) and cooling according to the Newton law (middle).

The temperature the system converges to is a steady-state solution. A steady-state solution is any solution that does not change in time. By definition, steady-states must solve the equation \(d\theta/dt = 0\). Steady-states are often denoted with an asterisk, as \(\theta^*\). For the fire equation, steady-states must be solutions of the transcendental equation \[0 = e^{-1/\theta^*} - b (\theta^* - h).\] This equation that can not be solved for \(\theta^*\) in terms of elementary functions. It can be solved in terms of a special function known as Lambert's W function, but for our purposes, this solution is a distraction that doesn't provide any illumination. We can, however, understand the steady-state solutions geometrically.

A steady-state is an intersection point between the sigmoid-curve created by the Arrhenius law and a straight line created by Newton's law of cooling. Because the sigmoid curve is positive and increasing, all steady-state solutions must be larger than the background temperature \(h\), though in some cases they may be very close to the background temperature. When the cooling rate is large, there is only one intersection point. But if this cooling rate is slow enough, there might be three intersection points between these two curves.

We can perform a little algebra to determine exactly when there will be 3 equilibria (see Exercises). We find that they only appear in a region of parameter space where both the cooling rate and background temperature are small. If \(b > 4 e^{-2} \approx 0.54\) or \(h > 1/4\), then there is a single steady-state solution \(\theta^{ * }\).

Number of steady-state solutions of the fire equation depending on the cooling rate b and the background temperature h

Number of steady-state solutions of the fire equation depending on the cooling rate \(b\) and the background temperature \(h\)

Examples where there are exactly two steady-state solutions. These are boundary-cases between 1 and 3 steady-state solutions -- a small perturbation in one direction creates 3 steady-state solutions, but a small perturbation in the other direction leaves only 1 steady-state solution. The final panel shows the unique triple-root situation at the cusp when b=4 e^{-2} and h=1/4.

Examples where there are exactly two steady-state solutions. These are boundary-cases between 1 and 3 steady-state solutions -- a small perturbation in one direction creates 3 steady-state solutions, but a small perturbation in the other direction leaves only 1 steady-state solution. The final panel shows the unique triple-root situation at the cusp when \(b=4 e^{-2}\) and \(h=1/4\).

These steady-states aren't all the same. Temperature drifts towards some of these steady-states and away from others. The local stability of a steady-state is how the solutions behave near to it. If solutions that start near enough to a steady-state always move closer to it over time, then we say the steady-state solution is asymptotically stable. If, on the other hand, some solutions that start near a steady-state move away from the steady-state over time, we say the steady-state solution is locally unstable. For 1-dimensional differential equations of the form \(dx/dt = f(x)\), the stability of a steady-state \(x^*\) where \(f(x^*)=0\) can usually be determined by the derivative \(f'(x^*)\). If \(f'(x^*) > 0\), the steady-state is locally unstable, but if \(f'(x^ * ) < 0\), then the steady-state is locally stable.

Let's now apply the local stability criteria to the fire equation. When there is a single steady-state solution, that steady-state is always locally stable. Whether the initial temperature is a little above or below that steady-state, it will relax back toward the steady state. In fact, the solution for every initial temperature will eventually converge to this steady-state -- it is what we call a globally attracting steady-state solution.

When there are three steady-state solutions to the fire equation, the middle one is locally unstable, and the other two are always locally stable. This is called bi-stability. Depending on the initial temperature, the solution may approach one or the other of these two steady-state solutions -- if the system is initially close to the background temperature, then the system temperature will converge to the lower stable steady-state. But if the initial temperature is hot enough, the solution will converge to the hotter steady-state. The unstable steady-state in the middle is the threshold between these two outcomes. Solutions with initial temperatures below the unstable steady-state will converge to the lower stable steady-state. Solutions with initial temperatures above the unstable steady-state will converge to the higher stable steady-state.

Solutions calculated numerically for various initial conditions in cases of a single steady-state (left) and three steady-states (right). In the case of a single steady-state, all solutions converge to the same steady-state value. In the case of three steady states, the solutions converge to either the highest or lowest steady-state.

Solutions calculated numerically for various initial conditions in cases of a single steady-state (left) and three steady-states (right). In the case of a single steady-state, all solutions converge to the same steady-state value. In the case of three steady states, the solutions converge to either the highest or lowest steady-state.

Interpretation

This dynamics described by the fire equation captures several features of fire-starting. Before we do anything, the system is at rest, corresponding to the lower stable steady-state. Then we try to light the fire. The heat of a single match alone isn't enough extra initial heat to get the fire going. But when we use tinder and kindling to build up more extra initial heat, the fire can take off on its own, and approach the higher stable steady-state solution.

The fire equation correctly predicts other effects. From the geometry of the triple-steady-state situation, For instance, according to the fire equation, increases in wind speed that increase the cooling rate \(b\) will make it harder to start a fire. This matches experience. And according to the fire equation, increases in the background temperature will reduce the temperature jump needed to start the fire, hence making it easier to start a fire. This matches the patterns with see with wild-fires, that are much more commonly started in the heat of summer than other times of year.

Despite its successes, the fire equation should not be interpreted too literally -- there are many things that have been left out. For one example, the fire equation assumes a constant availability of fuel, implying that the fire, once started, will burn forever -- that is not realistic. A fire runs out of fuel eventually if left unattended. For another example, our use of Newton's law of cooling is no more than a gross approximation of the actual processes of radiation, conduction, and convection by which energy is lost. And the effects of the cooling mechanism can be complicated by chemistry -- a stronger wind can increase the flow of oxygen to a fire, making it hotter, like in a blast furnace.

Discussion

Our analysis of fire-starting is a useful metaphor for many other problems. For example, consider the starting of a lawn-mower. It has two important states -- off and running. Both of these states are stable in some sense, and it can sometimes take allot of pulling to get the lawn-mower started. The same metaphor applies to starting an old car on a cold winter morning. Life itself can be thought of as an exothermic reaction with a collapse to a low-temperature steady-state corresponding to death. Even nations and governments might be described in this way.

Our study of the fire equation is an example of successful qualitative analysis of ordinary differential equations. By considering the geometric properties of the fire equation, we have developed a deep understanding of the behavior of its solutions, without actually needing to solve the equation. Our analysis relied on the consideration steady-states and their stability. In systems with more equations, more complex structures such as limit cycles and chaotic attractors are possible. We will encounter more examples in future chapters, but interested students may find Nonlinear Dynamics And Chaos by Steven Strogatz a good starting point for learning more.

Exercises

  1. In our non-dimensionalized fire model, the parameter-space boundary between 1 and 3 steady-state solutions is formed by the curve here there 1 steady-state is a double-root of the steady-state equation. This means \(f(\theta;b,h) =0\) and \(f'(\theta;b,h)=0\). Use these double-root conditions to derive parametric equations \(b(\theta)\) and \(h(\theta)\) for the boundary between 1 and 3 steady-states.

  2. An extension of the fire equation that explicitly incorporates fuel input and consumption is give below. Determine the steady-states, if any. \[\begin{align*} \dot{\theta} &= k u e^{\theta} - \theta, \\ \dot{u} &= - u e^{\theta} + m (1-u). \end{align*}\]

  3. A commonly used toy model for nuclear reactors uses the Frank-Kamenetskii exponoential approximation of the Arrhenius law to give \[\dot{\theta} = - a \theta + b e^{r \theta}\] when \(b = r = 1\).
    1. At steady-state (when temperature is not changing and \(\dot{\theta}=0\)), we should have \(a \theta = e^{\theta}\). Explain how the solutions of this equation change as the cooling rate \(a\) increases.
    2. Find the value of \(a\) for which there is a double root to the steady-state equation.
  4. Epidemic model reduced to single first-order autonomous equation \(\dot{R} = \gamma (n-ne^{-\beta R/\gamma} - R)\).
    1. Nondimensionalize the state and time variables this equation so that it depends on a single dimensionless parameter.
    2. Describe the dynamics of your dimensionless equation when your dimensionless variable is very large.
    3. Describe the dynamics of your dimensionless equation when your dimensionless variable is very small.
    4. Characterize any bifurcations in behavior as functions of your dimensionless parameter.

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