Problem 75 - Calculus for Biology and Medicine

Question

A bird population lives in a habitat where the number of nesting sites is a limiting factor in population growth. In which of the following cases would you expect that the growth of this bird population over the next few generations could be reasonably well approximated by exponential growth? (a) All nesting sites are occupied. (b) The bird population just invaded the habitat, and the population size is still much smaller than the available nesting sites. (c) In the previous year, a hurricane killed more than \(90 \%\) of the birds in this habitat.

Step-by-Step Solution

Answer

Exponential growth is expected in case (b), where the population is small relative to resources.

Step 1: Understanding Exponential Growth

Exponential growth occurs when the growth rate of a population is proportional to its current size. This generally means that the larger the population becomes, the faster it grows, as there are no significant limiting factors holding it back.

Step 2: Analyze Option (a)

The situation where all nesting sites are occupied indicates that the population is at its carrying capacity. Here, resources are limited due to a lack of nesting sites, meaning exponential growth is not expected as the population size is not free to increase past this capacity.

Step 3: Analyze Option (b)

In this case, the bird population has just invaded the habitat and is much smaller than the number of available nesting sites. This implies resources are not currently limiting, which allows for rapid growth of the bird population since there is an abundance of resources (nesting sites), thus indicating exponential growth is likely.

Step 4: Analyze Option (c)

With more than 90% of the population killed by a hurricane, the population is drastically reduced. Although the initial conditions after the severe reduction may allow for some growth, factors such as genetic diversity and recovery might limit exponential growth. However, with resources available and little competition, there might be a temporary period where exponential-like growth could occur as the population rebounds.

Step 5: Conclusion

Among the three cases presented, option (b) provides the best conditions for the bird population to exhibit exponential growth as it highlights a new population entering a habitat with abundant resources and minimal competition.

Key Concepts

Limiting Factors

Limiting factors are environmental conditions that restrict the growth, abundance, or distribution of a population within an ecosystem. They can be biotic, like food supply, or abiotic, such as space, weather, or availability of water. Limiting factors essentially cap a population's growth by creating obstacles the population must navigate to survive and thrive.

When a specific resource is low or limited, it directly impacts the population's ability to reproduce and increase in size. Common limiting factors include:

• Nesting sites for birds
• Food availability
• Predator presence
• Weather conditions
• Habitat destruction

These factors can prevent a population from following a natural exponential growth pattern, as seen in the options discussed, particularly in option (a) where all nesting sites are occupied.

Population Growth

Population growth refers to the change in the number of individuals in a population over time. Populations can grow in different patterns depending on environmental conditions and the presence of limiting factors. In favorable conditions with adequate resources and minimal competition, populations can exhibit exponential growth.

Exponential growth occurs when each generation of a population grows at a rate proportional to its size, contributing to rapid increases in population numbers. However, this type of growth is typically unsustainable over the long term due to limiting factors. For instance, in the case of a bird population just invading a habitat (option b), the initially small population can grow exponentially until nesting sites become fully occupied or other limiting factors present themselves. Thus, understanding the intricacies of population growth helps in predicting how a population might behave under different ecological circumstances.

Carrying Capacity

The concept of carrying capacity refers to the maximum number of individuals that an environment can sustainably support without degrading it. Once a population reaches this point, its growth rate slows down and stabilizes due to competition for limited resources.

In the context of the bird population exercise, when all nesting sites are occupied (option a), the population is at or near its carrying capacity. This means there is no room for new individuals to occupy without additional resources becoming available or existing birds dying.

• Population sizes fluctuate around the carrying capacity as they attempt to maintain equilibrium with the available resources.
• Carrying capacity can change due to alterations in the ecosystem, such as weather events, habitat destruction, or human intervention.

Understanding carrying capacity is crucial for effectively managing wildlife populations and ensuring that they do not exhaust their resources, leading to declines in the population and possibly to extinction.