STATEMENT OF PROBLEM

Consider the following table describing a population of protozoa.

Time Number of Time Number of
(in days) Protozoa (in days) Protozoa
0 40 11 470
1 53 12 526
2 70 13 577
3 91 14 622
4 118 15 660
5 152 16 691
6 193 17 716
7 240 18 736
8 293 19 752
9 351 20 764
10 411

One reliable model for the growth of protozoa is the Logistic Equation, a differential or rate equation which suggests that the rate of growth of the population depends on the size of the population, a natural species growth rate, and the carrying capacity or limiting number , i.e. the maximum number of the protozoa which the environment can support.

The differential equation model is

dp/dt = p'(t) = r p(t) (K - p(t))/K or

dp/dt = r p (K - p)/K = r p - r/K p^2 (where p = p(t)).

r is called the growth rate and K is the carrying capacity.

(1) What values of the population p would make the growth rate (dp/dt) zero? Which of these values is "interesting"? And what insight does this give to you concerning the term carrying capacity or limiting number for K?

(2) What would the population tend to do if the initial population were less than K? Hint: Examine the sign of dp/dt. What would the population tend to do if the initial population were greater than K? Again what insight does this give to you concerning the term carrying capacity or limiting number for K?

(3) What value of p would give the maximum value of dp/dt? I.e. if we could keep the population at one level, say by harvesting of the growth, so that our growth would be greatest what value of p would we choose?

(4) Ecologists who study populations of species are interested in determining values of r and K from collected data. In fact, the values of r and K often are used to characterize species. If a species has a large r value it is said to be r-selected and uses a high breeding rate for survival. If a species has a large K value it is said to be K-selected and uses its environment efficiently, i.e., for a fixed environment it has a higher carrying capacity or can fit more of its own kind into the fixed environment size.

We shall consider a set (see above) of data obtained on the population of a protozoan colony in a fixed or limited environment. Our data consists of (time, population size) observations for this protozoan colony. On the assumption that the logistic equation posed above is a viable model for population growth the task is to determine good values of r and K so that the data fits thismodel well.

Produce several ways to determine r and K. Compare the various results you get and suggest which approach you would prefer to use and why. Consider the following approaches:

(a) Estimate the rate of change (dp/dt) from the data and fit a quadratic function, a p - b p^2, of p to these estimates to obtain values of r and K.

(b) Estimate the rate of change (dp/dt) divided by p from the data and fit a linear function, a - b p, of p to these estimates to obtain values of r and K.

(c) Solve the differential equation with parameters as constants. Use initial condition to obtain constant of integration. Directly use a least non-linear squares technique on fitting the function to the data.