Problem Description:

Consider the fish population, P(t), in a fresh-water man-made pond. Assume that the initial population is set at 100 fish by stocking the pond with a combination of mature male and female fresh-water trout. Based on various environmental factors (food supply, water conditions, etc.), a reasonable approximation to the population dynamics for low population sizes suggests that the normalized fish birth and death rates are both inversely proportional to the square root of the population present at time t.

Based on the above description, derive a mathematical model for the fish population and find a general solution for P(t). Given that the initial population is 100 fish and that the measured population after six months is 169 fish, estimate how many fish will be in the pond after the first year. Using the same dynamics model, what will be the population size after five years? Based on your results, what can you say about the accuracy of the mathematical model for large population sizes?

Reference: This problem was derived from Prob. 2 on page 76 of the Differential Equations text by Edwards and Penny (Prentice Hall Inc., 1996).

Problem Solution:

where B and D are the normalized birth and death rates, respectively.

For the mathematical model described above, the birth and death rates are inversely proportional to the square root of the population size, or

Therefore, the population balance equation becomes

where k is constant. Thus, the mathematical model for this system is simply .

This is a separable equation whose solution can be developed as follows. We first rewrite the balance equation as

Integrating both sides gives

and, after some simple algebra, the general solution for the fish population becomes

Now, applying the initial condition gives ; therefore and the unique solution is

The growth rate constant, k, can be determined from the data given in the problem specification. We know that fish and after 6 months, fish. Evaluating eqn. (5) at t = 6 months gives

and solving this for k gives k = 1(note that the odd units here are needed to give consistency in the original balance equation -- since must have units of fish per month).

With the growth rate constant known, we can use the unique solution in eqn. (5) as a predictive model. Evaluating this expression at t = 1 year (12 months) and again at 5 years (60 months) gives

Thus, we see that the fish population grows rather rapidly, especially at larger values of P (since the rate of change of P is proportional to ). This model, however, predicts unbounded growth as time becomes large, and this is physically not possible for a pond of finite size and finite food supplies. The model may be adequate over a period of several years, but eventually its unbounded nature would lead to large errors in a real limited environment. Periodic comparison of the model’s predictive capability and the actual fish population would alert the user of the need to modify the mathematical model for this ecosystem.

This problem was solved rather easily using standard analytical techniques. It can also be solved numerically in Matlab, which we illustrate here. For use with a standard numerical ODE solver (like ODE23 in Matlab), one must have numerical values for all the constants within the mathematical model. In this problem, the only constant is the growth rate factor, k, but it is initially an unknown parameter. The usual way to handle this situation in numerical solutions is to simply guess a range of k values, solve the given IVP for each value of k, and then determine which value fits all the data given for the problem of interest.

This example is typical of many real problems where all the model coefficients are not known prior to solution of the defining balance equations. Instead, they are determined from actual measured data that are obtained from the physical system under study. In these situations, one must search for the unknown parameters by systematically guessing values for the unknown coefficients until the solution of the balance equations matches the known measured data. When this occurs, a proper set of coefficients has been identified, and a predictive model is now available. For problems with one unknown parameter, this search procedure is very straightforward (as illustrated in this example). For problems with higher dimensionality, the basic concept is similar, but actual implementation is more complicated because of the number of combinations that are available (formal optimization methods and search algorithms are available for these cases).