In this article, we will delve into the fascinating world of population modeling, specifically focusing on the fox population in a certain region. We are given that the fox population exhibits a continuous growth rate of 5 percent per year, and the estimated population in the year 2000 was 17,400. Our primary goal is to develop a mathematical function that accurately models the population's growth t years after the year 2000 (where t = 0 represents the year 2000). Understanding population dynamics is crucial in various fields, including ecology, conservation, and wildlife management. By creating a mathematical model, we can gain valuable insights into how the fox population is likely to change over time, allowing us to make informed decisions regarding conservation efforts and resource allocation. This model will enable us to predict future population sizes, assess the impact of environmental changes, and evaluate the effectiveness of different management strategies.
(a) Finding the Exponential Growth Function
To model the fox population growth, we'll utilize the exponential growth model, a fundamental concept in mathematical biology and population dynamics. The exponential growth model is particularly well-suited for describing populations that increase at a constant percentage rate over time. This model assumes that the growth rate is proportional to the current population size, meaning that larger populations tend to grow faster. The general form of the exponential growth function is given by:
N(t) = N₀ * e^(kt)
Where:
- N(t) represents the population size at time t.
- N₀ is the initial population size (the population at time t = 0).
- e is the base of the natural logarithm, approximately equal to 2.71828.
- k is the continuous growth rate (expressed as a decimal).
- t is the time elapsed since the initial time (in years in this case).
In our specific scenario, we are provided with the following information:
- The initial population size in the year 2000 (N₀) is 17,400.
- The continuous growth rate (k) is 5 percent per year, which translates to 0.05 when expressed as a decimal.
Substituting these values into the exponential growth function, we obtain the following model for the fox population:
N(t) = 17400 * e^(0.05t)
This function, N(t) = 17400 * e^(0.05t), is the mathematical representation of the fox population's growth over time. It allows us to estimate the population size at any given time t years after the year 2000. The exponential nature of the function highlights the fact that the population will grow at an accelerating rate, as the growth rate is applied to an ever-increasing population size. Understanding the implications of this exponential growth is crucial for long-term planning and conservation efforts.
Explanation of the Formula Components
- 17400 (N₀): This is the starting point, the foundation upon which the population growth is built. It represents the number of foxes present in the year 2000, our baseline for tracking population changes.
- e^(0.05t): This is the engine of growth. The exponential term, with the natural base e, captures the essence of continuous growth. The growth rate, 0.05 (or 5%), is embedded within this term, dictating how quickly the population expands. The variable t (time) acts as the accelerator, determining the duration over which the growth is applied. As t increases, the exponential term grows rapidly, leading to a substantial increase in population size.
- The interaction between these components is key to understanding the overall population dynamics. The initial population size sets the scale, while the exponential term governs the rate and pattern of growth. Together, they paint a picture of how the fox population is expected to evolve over time.
(b) Estimating the Fox Population in 2008
Now that we have our population model, N(t) = 17400 * e^(0.05t), we can use it to estimate the fox population in the year 2008. To do this, we need to determine the value of t that corresponds to the year 2008. Since t represents the number of years after 2000, we simply subtract 2000 from 2008:
t = 2008 - 2000 = 8
Therefore, t = 8 represents the year 2008. We now substitute this value of t into our population model:
N(8) = 17400 * e^(0.05 * 8)
To calculate this, we first multiply 0.05 by 8:
0. 05 * 8 = 0.4
Next, we evaluate the exponential term e^(0.4). Using a calculator, we find that:
e^(0.4) ≈ 1.49182
Finally, we multiply this value by the initial population size, 17,400:
N(8) ≈ 17400 * 1.49182 ≈ 25957.7
Since we are dealing with a population of foxes, we need to round this value to the nearest whole number. Therefore, our estimate for the fox population in the year 2008 is approximately 25,958 foxes.
Interpretation of the Result
This result, approximately 25,958 foxes, provides a snapshot of the population size eight years after our initial measurement in 2000. It demonstrates the power of the exponential growth model in predicting population changes over time. The increase from 17,400 foxes in 2000 to an estimated 25,958 foxes in 2008 highlights the impact of the 5% annual growth rate. This information can be valuable for wildlife managers and conservationists, helping them to assess the health of the fox population and make informed decisions about resource allocation and conservation strategies. By comparing this estimate with actual population counts or other data, we can further refine our model and improve its accuracy for future predictions.
(c) Determining the Time to Reach a Population of 30,000
Our next challenge is to determine the time it will take for the fox population to reach 30,000. To address this, we'll use our established population model, N(t) = 17400 * e^(0.05t), and solve for t when N(t) equals 30,000. This involves a bit of algebraic manipulation and the use of logarithms.
First, we set N(t) to 30,000:
30000 = 17400 * e^(0.05t)
Next, we isolate the exponential term by dividing both sides of the equation by 17,400:
30000 / 17400 = e^(0.05t)
Simplifying the left side, we get:
1. 724137931 ≈ e^(0.05t)
To solve for t, we need to eliminate the exponential function. We achieve this by taking the natural logarithm (ln) of both sides of the equation:
ln(1.724137931) = ln(e^(0.05t))
Using the property of logarithms that ln(e^x) = x, we simplify the right side:
ln(1.724137931) = 0.05t
Now, we calculate the natural logarithm of 1.724137931 using a calculator:
ln(1.724137931) ≈ 0.544865
So our equation becomes:
0. 544865 = 0.05t
Finally, we solve for t by dividing both sides by 0.05:
t = 0.544865 / 0.05 ≈ 10.8973
Rounding this value to two decimal places, we get t ≈ 10.90 years.
Interpretation and Context
This result, approximately 10.90 years, signifies the time it will take for the fox population to reach 30,000, starting from the year 2000. To determine the specific year, we add this value to 2000:
Year = 2000 + 10.90 ≈ 2010.90
Therefore, according to our model, the fox population is projected to reach 30,000 sometime during the year 2010. This prediction provides valuable insights for wildlife management and conservation planning. It allows us to anticipate potential challenges, such as resource scarcity or increased competition, and to implement proactive strategies to ensure the long-term health and sustainability of the fox population. By regularly monitoring the population and comparing actual data with our model's predictions, we can refine our understanding of the population dynamics and adapt our management approaches as needed.
In conclusion, we have successfully modeled the fox population growth using an exponential function, N(t) = 17400 * e^(0.05t). This model has allowed us to estimate the population size in 2008 (approximately 25,958 foxes) and to predict the time it will take for the population to reach 30,000 (approximately 10.90 years after 2000). Population modeling is a powerful tool for understanding and predicting population dynamics, providing valuable information for conservation efforts and wildlife management. By understanding the factors that influence population growth and decline, we can make informed decisions to ensure the long-term health and sustainability of animal populations in a changing world. The exponential growth model, while a simplification of real-world complexities, provides a valuable framework for understanding the potential for population increase under favorable conditions. However, it is crucial to acknowledge the limitations of the model and to consider other factors, such as carrying capacity, resource availability, and environmental changes, that may influence population growth in the long term.