Modeling via Differential Equations
One of the most difficult problems that a scientist deals with in his everyday research is: “How do I translate a physical phenomenon into a set of equations which describes it?”
It is usually impossible to describe a phenomenon totally, so one usually strives for a set of equations which describes the physical system approximately and adequately.
In general, once we have built a set of equations, we compare the data generated by the equations with real data collected from the system (by measurement). If the two sets of data “agree” (or are close), then we gain confidence that the set of equations will lead to a good description of the real-world system. For example, we may use the equations to make predictions about the long-term behavior of the system. It is also important to keep in mind that the set of equations stays only “valid” as long as the two sets of data are close. If a prediction from the equations leads to some conclusions which are by no means close to the real-world future behavior, then we should modify and “correct” the underlying equations. As you can see, the problem of generating “good” equations is not an easy exercise.
Note that the set of equations is called a Model for the system.
How do we build a Model?
The basic steps in building a model are:
- Step 1: Clearly state the assumptions on which the model will be based. These assumptions should describe the relationships among the quantities to be studied.
- Step 2: Completely describe the parameters and variables to be used in the model.
- Step 3: Use the assumptions (from Step 1) to derive mathematical equations relating the parameters and variables
- The best example of mathematical modeling is the one related to population growth problems. Keep in mind that this problem has many ramifications ranging from population explosion to extinction phenomena.
Here are some natural questions related to population problems:
- What will the population of a certain country be in ten years?
- How are we protecting the resources from extinction?
More can be said about the problem but, in this little review we will not discuss them in detail. In order to illustrate the use of differential equations with regard to this problem we consider the easiest mathematical model offered to govern the population dynamics of a certain species. It is commonly called the exponential model, that is, the rate of change of the population is proportional to the existing population. In other words, if P(t) measures the population, we have
where the rate k is constant. It is fairly easy to see that if k > 0, we have growth, and if k <0, we have decay. This is a linear equation which solves into
where is the initial population, i.e. . Therefore, we conclude the following:
- if k>0, then the population grows and continues to expand to infinity, that is,
- if k<0, then the population will shrink and tend to 0. In other words we are facing extinction.
Clearly, the first case, k>0, is not adequate and the model can be dropped. The main argument for this has to do with environmental limitations. The complication is that population growth is eventually limited by some factor, usually one from among many essential resources. When a population is far from its limits of growth it can grow exponentially. However, when nearing its limits the population size can fluctuate, even chaotically. Another model was proposed to remedy this flaw in the exponential model. It is called the logistic model (also called Verhulst-Pearl model). The differential equation for this model is
where M is a limiting size for the population (also called the carrying capacity). Clearly, when P is small compared to M, the equation reduces to the exponential one. In order to solve this equation we recognize a nonlinear equation which is separable. The constant solutions are P=0 and P=M. The non-constant solutions may obtained by separating the variables
The partial fraction techniques gives
Easy algebraic manipulations give
where C is a constant. Solving for P, we get
If we consider the initial condition (assuming that is not equal to both 0 or M), we get
which, once substituted into the expression for P(t) and simplified, we find
It is easy to see that
However, this is still not satisfactory because this model does not tell us when a population is facing extinction since it never implies that. Even starting with a small population it will always tend to the carrying capacity M.