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Chapter 10: Problem 24

Suppose that the growth of a population \(y=y(t)\) is given by the logistic equation $$y=\frac{1000}{1+999 e^{-0.97}}$$ (a) What is the population at time \(t=0 ?\) (b) What is the carrying capacity \(L ?\) (c) What is the constant \(k ?\) (d) When does the population reach \(75 \%\) of the carrying capacity? (e) Find an initial-value problem whose solution is \(y(t)\)

Short Answer

Expert verified
(a) 1 (b) 1000 (c) 0.9 (d) 8.725 (e) \( \frac{dy}{dt} = 0.9y(1-\frac{y}{1000}), \ y(0) = 1 \)

Step by step solution

01

Determine Population at Time t=0

To find the population at time \( t = 0 \), substitute \( t = 0 \) into the logistic equation: \( y = \frac{1000}{1 + 999 e^{-0.9 \cdot 0}} \). Since \( e^0 = 1 \), the equation simplifies to \( y = \frac{1000}{1 + 999} = \frac{1000}{1000} = 1 \). Therefore, the population at \( t = 0 \) is 1.
02

Determine the Carrying Capacity (L)

In the logistic equation, the carrying capacity \( L \) is the maximum population limit as \( t \to \infty \). Looking at the logistic model \( y = \frac{1000}{1+999 e^{-0.9t}} \), the carrying capacity is \( L = 1000 \). This is because as \( e^{-0.9t} \to 0 \), the expression \( 999 e^{-0.9t} \to 0 \) and thus \( y \approx \frac{1000}{1} \).
03

Identify the Constant k

The logistic equation is usually expressed as \( y = \frac{L}{1 + Ce^{-kt}} \). By comparing \( y = \frac{1000}{1+999 e^{-0.9t}} \) with the standard form, we can identify the constant \( k = 0.9 \) which corresponds to the growth rate factor in the exponent.
04

Find When Population Reaches 75% of Carrying Capacity

To find when the population reaches 75% of the carrying capacity, set \( y = 0.75 \times 1000 = 750 \). Solve \( 750 = \frac{1000}{1+999 e^{-0.9t}} \). Rearranging gives \( 1 + 999 e^{-0.9t} = \frac{1000}{750} = \frac{4}{3} \). So, \( 999 e^{-0.9t} = \frac{1}{3} \) and \( e^{-0.9t} = \frac{1}{2997} \). Taking natural logs, \( -0.9t = \ln(\frac{1}{2997}) \) which simplifies to \( t \approx \frac{\ln(2997)}{0.9} \approx 8.725 \). So, it takes approximately 8.725 units of time to reach 75% of the carrying capacity.
05

Find an Initial-Value Problem

The logistic equation \( y = \frac{1000}{1+999 e^{-0.9t}} \) can be described by the differential equation \( \frac{dy}{dt} = 0.9y(1-\frac{y}{1000}) \). The initial condition based on Step 1 is \( y(0) = 1 \). So, the initial-value problem is \( \frac{dy}{dt} = 0.9y(1-\frac{y}{1000}), \ y(0) = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Dynamics
Population dynamics is the study of how populations change over time. They can grow, shrink, or remain stable, depending on various factors such as birth rates, death rates, and environmental conditions. Logistic growth is a common model used to describe how populations evolve. It considers both exponential growth and limiting factors that prevent indefinite growth. This model is more realistic than simple exponential models, as it incorporates a carrying capacity, which is the maximum population size that the environment can sustain indefinitely.
  • Logistic growth starts with rapid, exponential growth.
  • As resources become limited, growth slows down until it stabilizes.
  • Understanding these dynamics helps in forecasting population changes in ecosystems.
Differential Equations
Differential equations are mathematical expressions that describe how a function changes over time. They are vital in modeling natural phenomena, including population growth. In the context of population dynamics, a differential equation can predict changes in population size based on factors like growth rates and carrying capacity.
  • The logistic differential equation for population growth is: \( \frac{dy}{dt} = k y (1 - \frac{y}{L}) \).
  • Here, \( y \) is the population size, \( k \) is the growth rate, and \( L \) is the carrying capacity.
  • These equations help in understanding the changes in population over time, accounting for initial conditions and constraints.
Carrying Capacity
Carrying capacity is a fundamental concept when discussing population dynamics. It refers to the maximum number of individuals that an environment can sustainably support without degrading the resources needed for future generations. It鈥檚 a central component of logistic growth models.
  • Carrying capacity is denoted by \( L \) in the logistic equation.
  • Once a population reaches its carrying capacity, births and deaths should balance, stabilizing the population.
  • Understanding carrying capacity helps in planning resource use and conservation efforts.
Exponential Growth
Exponential growth describes a situation where the growth rate of a population is proportional to its current size, leading to faster growth as the population increases. In a limitless environment, populations would grow exponentially. However, in reality, exponential growth is often curtailed by environmental limits.
  • Initially, populations may grow exponentially, meaning they double at a consistent rate.
  • Exponential growth can lead to resource depletion if unchecked, making logistic growth a more realistic model.
  • Recognizing the signs of exponential growth can help in managing sustainable growth policies.
Initial-Value Problems
Initial-value problems involve solving differential equations with given starting conditions. They are fundamental in predicting how a system evolves over time from a known state. In population dynamics, knowing the initial population size helps forecast future population sizes.
  • In our logistic model, the initial-value problem is \( \frac{dy}{dt} = 0.9y(1-\frac{y}{1000}), \ y(0) = 1 \).
  • Initial conditions, such as \( y(0) = 1 \), specify the population size at the start.
  • This method is crucial for simulating population growth and assessing long-term viability.

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Most popular questions from this chapter

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Suppose that a population \(y(t)\) grows in accordance with the logistic model $$\frac{d y}{d t}=10(1-0.1 y) y$$ (a) What is the carrying capacity? (b) What is the value of \(k ?\) (c) For what value of \(y\) is the population growing most rapidly?
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