97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


c. How fast (in fish per year) is the population growing at t=0? At t=5?

Verified step by step guidance

1. Identify the logistic growth function given in the problem: P(t) = \(\frac{400,000}{50 + 7950e^{-0.5t}\)}.
2. To find the rate of change of the population, calculate the derivative of P(t) with respect to t, denoted as P'(t). This derivative will give the rate of growth of the population at any time t.
3. Apply the quotient rule to differentiate the function P(t). The quotient rule states that if you have a function \(\frac{f(t)}{g(t)}\), its derivative is \(\frac{f'(t)g(t) - f(t)g'(t)}{(g(t))^2}\). Here, f(t) = 400,000 and g(t) = 50 + 7950e^{-0.5t}.
4. Calculate f'(t) and g'(t). Since f(t) = 400,000 is a constant, f'(t) = 0. For g(t), use the chain rule to find g'(t). The derivative of 7950e^{-0.5t} is -3975e^{-0.5t}.
5. Substitute f'(t), f(t), g(t), and g'(t) into the quotient rule formula to find P'(t). Evaluate P'(t) at t = 0 and t = 5 to find the rate of population growth at these times.

Key Concepts

Logistic Growth Function

The logistic growth function models population growth that is initially exponential but slows as the population approaches a maximum limit, known as the carrying capacity (K). The function is represented as P(t) = P₀K / (P₀ + (K - P₀)e^(-r₀t)), where P₀ is the initial population, r₀ is the growth rate, and t is time. This model is crucial for understanding how populations grow in environments with limited resources.

Carrying Capacity

Carrying capacity (K) refers to the maximum population size that an environment can sustain indefinitely without degrading the habitat. It is a critical concept in ecology and population dynamics, as it influences the growth rate and stability of populations. In the context of the logistic model, as the population approaches K, the growth rate decreases, leading to a leveling off of the population size.

Rate of Change

The rate of change in a population, often represented as dP/dt, indicates how quickly the population is increasing or decreasing at a given time. In the logistic growth model, this rate can be calculated by differentiating the logistic function with respect to time. Understanding the rate of change at specific time points, such as t=0 and t=5, is essential for predicting population dynamics and making informed management decisions.