Question

Determine the population growth curve for 10 generations for an annual plant with a net reproductive rate of 6 and a starting density of 35 Assume a constant reproductive rate [Equation (1)].

Short answer

The population size at generation 10 is 2,116,316,160.

Step-by-step solution

Understanding the Variables

The net reproductive rate (R0) is given as 6, and the starting density (N0) is 35. We are tasked with determining the population size over 10 generations.

Population Growth Formula

The population size for each generation can be calculated using the formula: \( N_t = N_0 \times R_0^t \), where \( N_t \) is the population size at generation \( t \), \( N_0 \) is the initial population size, and \( R_0 \) is the net reproductive rate.

Calculate Population for Each Generation

Using the formula from Step 2, calculate the population size at each generation for 0 through 10.- Generation 0: \( N_0 = 35 \) (Initial population)- Generation 1: \( N_1 = 35 \times 6^1 = 210 \)- Generation 2: \( N_2 = 35 \times 6^2 = 1260 \)- Generation 3: \( N_3 = 35 \times 6^3 = 7560 \)- Generation 4: \( N_4 = 35 \times 6^4 = 45360 \)- Generation 5: \( N_5 = 35 \times 6^5 = 272160 \)- Generation 6: \( N_6 = 35 \times 6^6 = 1632960 \)- Generation 7: \( N_7 = 35 \times 6^7 = 9797760 \)- Generation 8: \( N_8 = 35 \times 6^8 = 58786560 \)- Generation 9: \( N_9 = 35 \times 6^9 = 352719360 \)- Generation 10: \( N_{10} = 35 \times 6^{10} = 2116316160 \)

Key concepts

Net Reproductive Rate

The net reproductive rate, often abbreviated as \(R_0\), is a crucial concept in understanding population dynamics. It refers to the average number of offspring that an individual in a population will produce over its lifetime. In simpler terms, it shows how many new individuals can be expected from one organism.

When \(R_0\) is greater than 1, the population is growing. A net reproductive rate of 1 means the population is stable, while a rate of less than 1 indicates a decline. In our exercise, the net reproductive rate is 6. That means each plant, on average, produces six offspring, leading to rapid population growth. This concept is especially relevant for understanding how quickly a population can expand under optimal conditions.

Being able to calculate \(R_0\) is key for predicting future population sizes and assessing the sustainability of populations, particularly in conservation and ecological studies. As such, one can see that a higher \(R_0\) often correlates with exponential population growth, assuming other factors remain constant.

Generational Analysis

Generational analysis involves studying a population from one generation to the next. It helps us determine the size and growth pattern of a population over time. This type of analysis is crucial for making predictions and understanding long-term trends.

In generational analysis, we follow an initial cohort, or a group of individuals, and calculate the changes in their population across successive generations. The calculation is based on the net reproductive rate. In our example, we started with a population density of 35 plants. By applying the formula \( N_t = N_0 \times R_0^t \), we can see how the population grows over 10 generations.

This analysis is particularly insightful in the ecological and conservation fields, where understanding long-term trends and the impact of various factors on population growth is essential. Through generational analysis, we can anticipate potential challenges like overpopulation or extinction risks. The exercise demonstrated is a simple model, but generational analysis can become complex when considering environmental pressures and resource limitations in real-world scenarios.

Exponential Growth

Exponential growth describes a situation where the population size grows by a constant rate over equal time periods. This type of growth results in a J-shaped curve when plotted on a graph. It occurs when resources are abundant and environmental conditions are optimal.

In the exercise given, exponential growth is illustrated by the rapid increase in the plant population across generations. Starting from 35, the population grows massively by the tenth generation due to the high net reproductive rate of 6. The exponential formula used was \( N_t = N_0 \times R_0^t \), where each subsequent generation number (\(t\)) results in a multiplication effect on the initial population.

Understanding exponential growth is important in fields like ecology and biology because it helps predict how quickly a population can expand under ideal conditions. This can warn scientists and policymakers about potential overpopulation issues. In real-world applications, though, resources are often limited, which eventually alters growth patterns from exponential to logistic, reflecting real-time resource constraints.