Problem 54 - Calculus Single Variable

Question

A population \(P(t)\) satisfies the logistic growth equation \(P^{\prime}(t)=\alpha \cdot P(t)-\beta \cdot P^{2}(t)\) for positive constants \(\alpha\) and \(\beta\) What is the carrying capacity?

Step-by-Step Solution

Answer

The carrying capacity is \( \frac{\alpha}{\beta} \).

Step 1: Identify the Logistic Equation

The logistic growth model is represented by the differential equation \( P'(t) = \alpha P(t) - \beta P(t)^2 \). This equation describes how a population grows initially exponentially then stabilizes over time.

Step 2: Recognize the Carrying Capacity

The carrying capacity, \( K \), is the maximum population size that the environment can sustain indefinitely. In the logistic equation, it can be determined by analyzing the terms, specifically where the population growth rate \( P'(t) \) becomes zero.

Step 3: Set the Growth Rate to Zero

Set \( P'(t) = 0 \) in the equation: \( \alpha P(t) - \beta P(t)^2 = 0 \). This helps identify the points where the population size stabilizes.

Step 4: Solve for Carrying Capacity

Factor the equation: \( P(t) (\alpha - \beta P(t)) = 0 \). This gives two solutions: \( P(t) = 0 \), and \( \alpha - \beta P(t) = 0 \), the latter indicating the non-zero stable population size.

Step 5: Isolate for \( P(t) \)

From \( \alpha - \beta P(t) = 0 \), solve for \( P(t) \) to find the carrying capacity. Rearranging gives \( \beta P(t) = \alpha \), thus \( P(t) = \frac{\alpha}{\beta} \).

Step 6: Conclusion for Carrying Capacity

Therefore, the carrying capacity \( K \) is \( \frac{\alpha}{\beta} \), which is the maximum sustainable population size dictated by the logistic equation's parameters.

Key Concepts

Differential Equations

Differential equations are equations that relate a function with its derivatives. They are fundamental in modeling real-world scenarios where change is continuous. Specifically, a differential equation is used to describe how a quantity changes over time or space. In the context of the logistic growth model, the differential equation is \( P'(t) = \alpha \cdot P(t) - \beta \cdot P(t)^2 \).- **\( P'(t) \):** Represents the rate of change of the population over time.- **\(\alpha \cdot P(t)\):** Describes the exponential growth term, showing how the population would grow if unlimited resources were available.- **\(- \beta \cdot P(t)^2\):** Accounts for the slowing growth as competition for resources increases.This equation captures the essence of logistic growth, where populations grow rapidly, slow down, and eventually stabilize, creating an S-shaped curve. Understanding how to solve and interpret such differential equations helps in predicting population trends and understanding interactions between species and their environments. Differential equations like this are not only mathematical expressions but tools for scientists to predict and analyze changes in numerous fields, from biology to economics.

Carrying Capacity

Carrying capacity, denoted by \( K \), is the maximum population size that an environment can sustain over a long period. It is determined by factors such as the availability of resources, space, and the interaction with other species. In the logistic growth equation, the carrying capacity can be found where the population growth rate equals zero.- **Setting \( P'(t) = 0 \):** This condition helps determine when the population's growth stops and stabilizes.- **Equation Result:** By solving \( \alpha P(t) - \beta P(t)^2 = 0 \), we find that \( P(t) = 0 \) (extinction scenario) or \( P(t) = \frac{\alpha}{\beta} \). The latter is the carrying capacity, the point \( K \) where the population remains stable.Thus, the formula \( K = \frac{\alpha}{\beta} \) provides a quantitative measure of the environmental limits. Carrying capacity is integral in ecology and resource management, helping to forecast changes in populations under varying environmental conditions.

Population Dynamics

Population dynamics explores the changes in population size and composition over time. It involves understanding various factors that influence these changes, such as birth rates, death rates, and migration patterns. In the context of the logistic growth model:- **Early Growth Phase:** Populations grow exponentially when resources are abundant and competition is minimal.- **Decelerating Growth:** As the population size increases, individuals begin to compete for diminishing resources.- **Stable Phase:** Eventually, the population stabilizes at the carrying capacity \( K \), representing a balance between growth and resource limits.Population dynamics is crucial for:- **Conservation Efforts:** Ensuring species sustainability by maintaining population sizes.- **Resource Management:** Allocating and managing resources efficiently in agriculture and wildlife management.By using models like the logistic growth equation, researchers can analyze trends, predict future changes, and create strategies to deal with issues like overpopulation and endangered species. Understanding population dynamics offers insights into the delicate balance between organisms and their environments.