Growth Models, Part 5

Population Growth Models

When you click on the button below corresponding to your CAS, you will be able to download the file corresponding to Part 5 of this module.

Part 5: Logistic Growth with Harvesting

Up to this point, our population models have focused primarily on human populations. However, the underlying principles apply equally well to biological populations of any species, at least to populations that are large enough to make a differential equation model appropriate. Natural growth is likely to occur at a rate proportional to the population size, and crowding is likely to impose a logistic constraint.

For the remainder of this module we will study populations managed by humans, specifically, fisheries. Thus, in addition to the factors of reproductive rates and competition for resources, we have to consider the effect of fishing on the rate of growth. If we were studying forest management, the analogous factor would be logging. If it were farm management, we might be concerned with slaughtering cattle or harvesting wheat. The general term for removing members of a managed biological population is harvesting.

fishing boat After we study the effects of adding a harvesting component to the logistic model, we will study a logistic equation model for the population growth of lobsters along the Maine coast, based on actual catch/effort data for the years 1940 to 1976. This model will give insight into the condition of the lobster industry and will help us determine whether this resource is being over-exploited. We will consider both biological and economic factors in our analysis.

Is this an important problem? Click here to read an Associated Press news item from 1994 about ocean fisheries being "stressed to the limit."

5.1 Constant-Rate Harvesting
5.2 Proportional Harvesting
5.3 Solutions of the Proportional Harvesting Model
5.4 Maine Lobsters: Catch and Traps
5.5 Lobstering Effort
5.6 The Equilibrium Catch-Effort Curve:
Biological Overfishing?
5.7 The Equilibrium Revenue-Effort Curve:
Economic Overfishing?
5.8 Environmental Carrying Capacity

For acknowledgements and references, click here.

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