Problem 71
Question
The population models \(P^{\prime}(t)=k P(t)\) and \(P^{\prime}(t)=k[P(t)]^{1.1}\) look very similar. The first is called exponential growth and is studied in detail in section \(7.1 .\) The second is sometimes called a doomsday model. Solve the general doomsday equation. Assuming that \(P(0)\) and \(k\) are positive, find the time at which the population becomes infinite.
Step-by-Step Solution
Verified Answer
The time at which the population becomes infinite in the Doomsday model is \(T= \frac{1}{0.1k[P(0)]^{0.1}}\).
1Step 1: Find the integral
We start by finding the integral for this differential equation. If we rearrange the equation and use Separation of Variables, we get \(\frac{1}{P^{1.1}} \cdot \frac{dP}{dt}=k\). We can now integrate both sides with respect to \(t\) which then gives us \(\frac{-1}{0.1P^{0.1}}=kt+C1\) where \(C1\) is the constant of integration.
2Step 2: Determine the constant of integration
To determine the constant \(C1\), we use the condition given in the exercise that the population is positive at time \(t=0\), meaning \(P(0)>0\). Substitute these initial conditions \(P(0)\) into the integrated equation. Doing so we get the constant of integration, \(C1= -\frac{1}{0.1[P(0)]^{0.1}}.\)
3Step 3: Find the infinite time
Lastly, to find the time (\(T\)) at which the population becomes infinite, we set \(P(T)=\infty\) and solve the equation for \(T\). Therefore, substitute \(P(T)\) with \(\infty\) and \(C1\) with its value. Solving the equation we get \(T= \frac{1}{0.1k[P(0)]^{0.1}}\). This gives the time at which the population becomes infinite in the Doomsday model.
Key Concepts
Doomsday ModelPopulation GrowthSeparation of Variables
Doomsday Model
The Doomsday Model is compelling in the study of population dynamics. It represents a situation where the population grows at an accelerated pace compared to standard exponential growth. This model is defined by the differential equation \(P'(t) = k[P(t)]^{1.1}\). The characteristic exponent greater than one (here, 1.1) suggests that as the population grows, the rate of growth itself accelerates.
This eventually leads to the population reaching infinity in a finite time. This phenomenon can often refer to theoretical discussions about how certain populations might expand without bound under specific environmental conditions.
Understanding the implications of the Doomsday Model helps highlight the importance of constraints in natural systems. In practical terms, similar dynamics can be linked to resource consumption or unregulated growth in certain ecological or economic environments.
This eventually leads to the population reaching infinity in a finite time. This phenomenon can often refer to theoretical discussions about how certain populations might expand without bound under specific environmental conditions.
Understanding the implications of the Doomsday Model helps highlight the importance of constraints in natural systems. In practical terms, similar dynamics can be linked to resource consumption or unregulated growth in certain ecological or economic environments.
Population Growth
Population growth is a fundamental concept in ecology and biology. It refers to the change in the number of individuals in a population over time. There are several models to describe population growth; one is the exponential model, while another related to this is the Doomsday model.
Exponential growth describes a situation where the population size increases at a consistent percentage over time. The mathematical model is \(P'(t) = kP(t)\), which leads to an exponentially increasing population size if \(k > 0\). In contrast, the Doomsday model, with its elevated exponent, presents an even more dramatic increase.
Population growth is influenced by various factors such as birth rates, death rates, immigration, and emigration. Understanding these different models helps in predicting trends and managing resources sustainably in real-world scenarios.
Exponential growth describes a situation where the population size increases at a consistent percentage over time. The mathematical model is \(P'(t) = kP(t)\), which leads to an exponentially increasing population size if \(k > 0\). In contrast, the Doomsday model, with its elevated exponent, presents an even more dramatic increase.
Population growth is influenced by various factors such as birth rates, death rates, immigration, and emigration. Understanding these different models helps in predicting trends and managing resources sustainably in real-world scenarios.
Separation of Variables
Separation of Variables is a powerful technique in solving differential equations, particularly useful for problems involving rates of change. This method works by manipulating the equation to isolate the different variables, allowing us to integrate and solve for the desired function.
In the context of the Doomsday Model, the equation \(P'(t) = k[P(t)]^{1.1}\) can be rearranged to \(\frac{1}{P^{1.1}} \cdot \frac{dP}{dt} = k\). Here, we separate variables \(P\) and \(t\). By integrating both sides, we obtain a general solution where the constant of integration can be determined using initial conditions.
This technique is not only applicable to population growth problems but extends to many areas in mathematics. Mastery of separation of variables opens up avenues to solve a variety of differential equations with ease.
In the context of the Doomsday Model, the equation \(P'(t) = k[P(t)]^{1.1}\) can be rearranged to \(\frac{1}{P^{1.1}} \cdot \frac{dP}{dt} = k\). Here, we separate variables \(P\) and \(t\). By integrating both sides, we obtain a general solution where the constant of integration can be determined using initial conditions.
This technique is not only applicable to population growth problems but extends to many areas in mathematics. Mastery of separation of variables opens up avenues to solve a variety of differential equations with ease.
Other exercises in this chapter
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