Problem 62 - Calculus for Biology and Medicine

Question

62\. You are an ecologist measuring the size of the population of plants living on an island. You measure the size of the population each year at the beginning of the breeding season. Denote the population size after \(t\) years by \(N_{t}\). The first year that you measure the population size there are 16 birds, that is \(N_{1}=16\). Each year you find \(50 \%\) more birds than the year before. (a) Write down the recursion for \(N_{t}\) and solve it. (b) According to your formula, when will the size of the population first exceed 100 birds? (c) When will the population size first exceed 1000 birds? (d) Use your formula for \(N_{t}\) to predict when the population size will first exceed one million birds. Should you believe this prediction?

Step-by-Step Solution

Answer

(a) \(N_t = 16 \times (1.5)^{t-1}\); (b) after 8 years; (c) after 15 years; (d) after 30 years; prediction is unreliable.

Step 1: Understand the Problem

We start with a population of 16 birds and each year the population increases by 50%. We need to find a recursive formula for the population and predict when it surpasses different thresholds.

Step 2: Establish the Recursion Formula

The population increases by 50% each year, which means the population at year \( t+1 \) is 1.5 times the population at year \( t \). The recurrence relation is \( N_{t+1} = 1.5 \times N_t \). Since \( N_1 = 16 \), this represents the initial condition.

Step 3: Solve the Recursion Formula

The solution to the recurrence relation can be represented as \( N_t = N_1 \times (1.5)^{t-1} \). Thus, \( N_t = 16 \times (1.5)^{t-1} \).

Step 4: Find When Population Exceeds 100 Birds

To find when the population exceeds 100, solve \( 16 \times (1.5)^{t-1} > 100 \). Divide both sides by 16 giving \( (1.5)^{t-1} > 6.25 \). Taking the logarithm base 10 of both sides: \( (t-1) \log_{10}(1.5) > \log_{10}(6.25) \). Solving gives \( t > \frac{\log_{10}(6.25)}{\log_{10}(1.5)} + 1 \). The solution results in \( t \approx 7.21 \). Hence, the population exceeds 100 birds after 8 years.

Step 5: Find When Population Exceeds 1000 Birds

Using a similar approach, set \( 16 \times (1.5)^{t-1} > 1000 \), leading to \( (1.5)^{t-1} > 62.5 \). Logarithmizing gives \( (t-1) > \frac{\log_{10}(62.5)}{\log_{10}(1.5)} \). Solving yields \( t > 14.44 \). Therefore, it takes 15 years for the population to exceed 1000 birds.

Step 6: Predict When Population Exceeds 1 Million Birds

For 1 million birds: \( 16 \times (1.5)^{t-1} > 1000000 \), yielding \( (1.5)^{t-1} > 62500 \). After logarithmizing, \( t-1 > \frac{\log_{10}(62500)}{\log_{10}(1.5)} \). Solving gives \( t > 29.89 \). Thus, the population exceeds 1 million after 30 years.

Step 7: Assess Prediction Validity

While the mathematical model predicts 1 million birds after 30 years, such exponential growth is usually unsustainable in real ecosystems due to resource limits, suggesting the prediction is unreliable.

Key Concepts

Population Dynamics

Population dynamics is the study of how populations of living organisms change over time and space. In this scenario, we are considering a population of birds on an island and how their numbers increase annually. Understanding these dynamics involves examining factors such as birth rates, death rates, immigration, and emigration. In the exercise, we assume a simplistic model where the only factor affecting the population is a consistent annual growth rate, representing 50% more birds each year. This linear factor influences the size of the population without considering ecological constraints or changes.

In real-world ecology, many factors influence population size, including availability of food, predation, disease, and habitat space. While the model gives a clear mathematical prediction, actual population dynamics are more complex. Ecologists must consider these variables when making predictions about future population sizes.

Logarithmic Calculations

Logarithms are extremely useful for solving equations involving exponential growth or decay, as they convert multiplicative relationships into additive ones. In the context of this exercise, logarithmic calculations help us determine when the bird population will surpass certain numbers.

Let's break it down: to find when the population exceeds a specific number of birds, we transformed exponential inequalities into linear ones using logarithms. For instance, to identify when the population surpasses 100 birds, we have the inequality \( (1.5)^{t-1} > 6.25 \). Applying a logarithm allows us to reframe this as \( (t-1) \log_{10}(1.5) > \log_{10}(6.25) \). Through this method, we simplify the process of finding the required time \(t\) by working with logarithms to isolate \(t\) as the subject.

Learning how to use logarithms to solve these types of problems will be beneficial, as it provides a systematic approach to solve exponential equations common in various scientific and practical fields.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. In other words, the larger the population, the quicker it grows. The exercise highlights this concept by demonstrating that each year, the bird population on the island grows by a constant factor of 1.5. Hence, the population's size is described by the formula \( N_t = 16 \times (1.5)^{t-1} \).

Exponential growth is common in nature and technology, from bacterial reproduction to computer processing power increases. However, in natural ecosystems, exponential growth is frequently unsustainable longer term due to limits in resources, space, and other factors. Once resources become limited, populations often transition to logistic growth, where the growth rate slows as the population approaches the environment's carrying capacity.

Understanding exponential growth helps one grasp why populations might seemingly explode in growth at first, before potentially crashing or stabilizing as external constraints manifest. It's a fundamental concept in ecology, economics, and many scientific disciplines.