Differential Equations
Differential Equations
PARTIAL DIFFERENTIAL EQUATIONS
STOCHASTIC DIFFERENTIAL EQUATIONS
Differential equations are models of real systems that are believed to change their states continuously, or, to put it more precisely, at infinitesimally short intervals in time. Differential equations, or rather systems of differential equations, connect a change in the state of a system to its current state, or even the change in a change of the state of the same system, in a way that is comparable to the way difference equations allow the calculation of future states of a system from its current state. But unlike difference equations, the application of differential equations supposes that the processes within the system modeled by these equations are continuous in time, whereas with difference equations, processes are discrete in time.
For a number of real systems, the approach of differential equations seems appropriate, for instance in the case of the movement of an arrow through the air or of the local concentration of some pollutant in a lake. Here, one is only (or at least mainly) interested in the current value of some continuously measurable variable that is seen as varying continuously over time. More generally speaking, t is the parameter of a process {xt, tε T } where T is a continuous set with the same cardinality as that of the set of real numbers; the general form of a (first-order ordinary) difference equation is
dx/dt = ẋ = f (x )
Here, in a more symbolic way, dx is the change that occurs to the state variable x of the system in question during the infinitesimally short time interval dt at any time t. Differential equations of higher order are also possible; a second-order differential equation has the general form
d2x/dt2 = x∙∙ = f (x )
and is often transformed into a system of differential equations
ẏ = x∙∙ = f (x ) = g (y )
y = x̉ = h (x )
Strictly speaking, in the realm of the social and economic sciences, applications of differential equations and systems of them are only approximations, because the state variables of social and economic systems cannot undergo continuous changes. In demography, for example, we can only talk about the birth and death of an integer number of people, and in economics we can only calculate with a fixed number of products sold to the customer (not even with the exception of fluid, gaseous goods, or energy, which can be physically split down to molecules and energy quanta). In social psychology, it is still an open question whether attitudes change continuously (they are usually measured on four- or seven-point scales). And even if all these variables were continuous, the question remains whether these changes occur in a continuous manner: Children are born at a certain point in time, prices are paid at a certain point in time, and until the next payment arrives in one’s bank account, the balance is constant.
On the other hand, with a large number of demographic events or financial transactions, one could argue that a differential equation is a sufficiently good approximation that is, in most cases, more easily treatable than the discrete event formalization of the real process (this even applies when the alternative is a deterministic difference equation). Differential equations can also treat probabilistic problems (then we have stochastic differential equations) and can describe processes in time and space, for instance in diffusion processes where the distribution of local concentrations or frequencies changes over time.
UNRESTRICTED GROWTH
Linear differential equations of the type x̉ = λx and systems of such equations can always be solved, that is, it is always possible to write down the time-dependent function that obeys the differential equation (which is the exponential function x (t ) = Ae λt, where A and λ are two constants that depend on the initial condition and the proportionality constant between x̉ and x, respectively). If the proportionality constant is positive, this results in an infinite growth, whereas with a negative proportionality (“the higher the value of x, the higher its decrease”) the value of x approaches 0, though only in infinite time. This differential equation was first used in Thomas Malthus’s (1766-1834) theories of demographic and economic growth.
ARMS RACE
A system of linear differential equations has a vector-valued exponential function as its solution. One of the earlier applications of a very simple system of linear differential equations was Lewis Fry Richardson’s (1960) model of an arms race between two powers. The idea behind this model is that each block increases the armament budget both proportional to the current armament expenses of the other block and the budget available for other purposes. Thus, the change in the armament budget of block 1 is
x̉ = m (x max – x ) + ay = g + mx + ay with g = mx max
The same holds for the other block:
ỷ = bx + n (y max – y ) = h + bx + ny with h = ny max
The analytical solution for this system of two linear differential equations has the general form
q (t ) = θ 1q 1 exp (λ 1t ) θ2q2 exp (λ2t) + q3
where q and q • are vectors with elements x and y and elements x • and y • respectively, while θ 1, θ 2, λ 1, λ 2, q 1, q 2 and q 3 are constants that depend on a, b, g, h, m, and n. In a way, only λ 1 and λ 2 are of special interest, because they are—as multipliers in the arguments to the exponential functions in the analytical solution—responsible for the overall behavior of the system. They can be shown to be the eigen-values of the matrix formed of – m, a, b, and – n, and these eigen-values can be complex, which means that besides stationary solutions, periodic solutions are also possible, at least in principle (although not in this case, where m, a, b, and n are all positive). If both λ 1 and λ 2 are negative, q (t ) approaches q 3 as times goes by; if at least one of them is positive, q (t ) grows beyond all limits (which of course is impossible in the real world).
LOGISTIC GROWTH
One of the simplest cases of a differential equation in one variable—which also displays some interesting behavior— is the so-called logistic or Verhulst equation, which in its time-continuous version has the form
x̉ = rx (k – x )
One of the interpretations of this equation is that it describes a population in a habitat with carrying capacity K whose size changes continuously in such a way that the relative change (x •/x ) is proportional both to the current size x and to the difference between the current size and the carrying capacity (K – x, this difference is the proportion of the habitat that, in a way, is so far unused).
The equation has two stationary solutions, namely, xst 0 = 0 and xst 1 = K. The former is unstable: Even from the tiniest initial state, the population will grow until the carrying capacity is exactly exhausted. The time-dependent function x (t ), which obeys the differential equation, is a monotonically growing function whose graph is an S- shaped curve. This time-dependent function can be written as
x (t ) = Kx (0) exp (rt )/{K – x (0) [1 – exp (rt )]}
This differential equation is one of the simplest nonlinear ordinary differential equations.
THE LOTKA-VOLTERRA EQUATION
Another well-known system of nonlinear differential equations is the so-called Lotka-Volterra equation, which describes the interaction between predators and prey. It can also be applied to the interaction between a human population (predator) and its natural resources (prey). Here, the relative growth of the prey is a sum of a (positive) constant and a negative term that is proportional to the size of the predator population, whereas the relative growth of the predator population is a sum of a (negative) constant and a positive term that is proportional to the size of the prey population. In other words, in the absence of the predator population the prey would grow infinitely, whereas in the absence of the prey, the predator population would die out.
x̉ = x (a – by )
ỷ = y (–c + dx )
This system of differential equations does not have a closed solution, but it has a number of interesting features that show up no matter how detailed the model is for the interaction between predators and prey: The solution for this system of differential equations is a periodic function with constant amplitude that depends on the initial condition. There is only one stationary state of the system, which is defined by y = ab (this leads to x• = 0) and x= c/d (this leads to y• = 0); thus if both hold, then no change will happen to the state of the system. Otherwise the populations increase and decrease periodically without ever dying out.
PARTIAL DIFFERENTIAL EQUATIONS
In most applications of differential equations and their systems, the parameter variable will be time, as in the examples above. But it is also possible to treat changes both in time and space with the help of a special type of differential equation, namely, partial differential equations. They define the change of the value of some attribute at some point in space and time—for instance, the expected change K of the continuously modeled and measured attitude X of a person that has the value x at time t, where this change will be different for different x and perhaps also for different t —in terms of this point in time and space. Thus,
K (x, t ) = dx/dt = ∂V (x, t )/∂x
For an application, see the next paragraph. Partial differential equations are seldom used in the social sciences because, typically, continuous properties of individual human beings—if they exist at all in the focus of interest of social scientists and economists—are difficult to measure, and even more difficult to measure within time intervals that are short enough to estimate any parameters of functions such as K and V in the above equation.
STOCHASTIC DIFFERENTIAL EQUATIONS
Stochastic influences can also be inserted into the formulation of differential equations. The simplest case is the so-called Langevin equation, which describes the motion of a system in its state space when there is both a potential whose gradient it follows and some stochastic influence that prevents the system from following the gradient of this potential in a precise manner. This type of description can, for instance, be used to describe the attitudes of voters during an election campaign. Each voter’s attitude can be defined (and measured) in a continuous attitude space. Their motions through this attitude space (say, from left to right; see, e.g., Downs 1957, p. 117) are determined by a “potential” that is determined either by some parties that “attract” voters toward their own positions in the same attitude space or by the “political climate” defined by the frequency distribution over the attitude space. In the latter case, voters would give up their attitude if it is shared by only a few and change it into an attitude that is more frequent. Thus they follow a gradient toward more frequent attitudes; but while moving through the attitude space, they would also perform random changes in their attitudes, thus not obeying exactly the overall political climate. And by changing individual attitudes, the overall “climate” or potential is changed. The movement could be described as follows:
q̇ (t ) = – γ∂V (q, t )/∂q + εt
where
V (q, t ) = –Inf (q, t )
and f (q, t ) is the frequency distributions of voters over the attitude space at time t (V would be a polynomial up to some even order in q). One would typically find voters more or less normally distributed at the beginning of an election campaign, but the process described here would explain why and how polarization—a bimodal or multimodal frequency distribution—could occur toward the election date (Troitzsch 1990).
SEE ALSO Comparative Dynamics; Cumulative Causation; Difference Equations; Phase Diagrams; System Analysis; Taylor, Lance
BIBLIOGRAPHY
Downs, Anthony. 1957. An Economic Theory of Democracy. Boston: Addison-Wesley.
Lotka, Alfred J. 1925. Elements of Physical Biology. Baltimore, MD: Williams & Wilkins.
Richardson, Lewis Fry. 1960. Arms and Insecurity. A Mathematical Study of the Causes and Origins of War. Pittsburgh, PA: Boxwood.
Troitzsch, Klaus G. 1990. Self-Organization in Social Systems. In Computer Aided Sociological Research, eds. Johannes Gladitz and Klaus G. Troitzsch, 353-377. Berlin: Akademie-Verlag.
Verhulst, Pierre-François. 1847. Deuxième mémoire sur la loi d’accroissement de la population. Nouveaux mémoires de l’Academie Royale des Sciences et Belles-Lettres de Bruxelles 20: 1–32.
Volterra, Vito. 1926. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Atti della Accademia Nazional dei Lincei 6 (2): 31–113.
Klaus G. Troitzsch