### 4.4.1 Revised Assumptions

The obvious implication for the food supply of any species allowed to grow unchecked is that the supply will simply run out or at best be diminished to the point where it can no longer support the population. When this happens, the assumptions made about the birth and death rates will no longer be true. You might well argue that the assumptions made were unrealistic anyway, now consider what will happen to the birth and death rates.

The Birth-rate

As the food supply runs out it is reasonable to suggest that the individual members of the population will weaken. This alone is sufficient to suggest that the birth-rate will not be constant, in fact, it can be expected to fall.

The Death-rate

As the food supply runs out it is suggested that tte individual members of the population will weaken to the point where they cannot search for and catch food. Given that they cannot catch food, they will die before the end of their natural life cycle. This suggest s that the death-rate will not be constant, in fact, it can be expected to increase.

We have now decided that the birth and death rates should be regarded as variables rather than constants. The implication of this decision is that we must now make further assumptions regarding the way in which the two rates vary.

What is the simplest type of variation that we can consider? We can choose to consider any type of variation that we like, for example

1. exponential variation;
2. quadratic variation;
3. linear variation.

Clearly, the assumption resulting in the greatest simplification is the assumption that the variation is linear. Assuming that this is a reasonable place to start we must now ask, "linear with what, time, food supply, population size"? You may be able to think of other possibilities, but we will consider the case where the variation in both birth-rate and death-rate is linear with the size of the population.