Problem 74 - Calculus Early Transcendentals

Question

Steady states If a function \(f\) represents a system that varies in time, the existence of \(\lim f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value. The population of a colony of squirrels is given by \(p(t)=\frac{1500}{3+2 e^{-0.1 t}}\)

Step-by-Step Solution

Answer

Answer: Yes, a steady state exists for the population of squirrels, with a steady-state value of 500 squirrels.

Step 1: Identify the function

We are given the population of a colony of squirrels as a function of time: \(p(t) = \frac{1500}{3 + 2e^{-0.1 t}}\). Our goal is to evaluate the limit of this function as time approaches infinity: \(\lim_{t \to \infty} p(t)\). #Step 2: Evaluate the limit as t approaches infinity#

Step 2: Evaluate the limit

To evaluate the limit, we'll look at how the denominator \(3 + 2e^{-0.1t}\) behaves as \(t\) approaches infinity. As \(t\) goes to infinity, the term \(e^{-0.1t}\) goes to zero since the exponent is negative. Thus, the denominator approaches \(3\). So, we can rewrite the limit as \(\lim_{t \to \infty} p(t) = \lim_{t \to \infty} \frac{1500}{3+2 e^{-0.1 t}} = \frac{1500}{3}\). #Step 3: Determine if a steady state exists and find its value#

Step 3: Determine the steady state

Since the limit as \(t\) goes to infinity is a finite value, it means that a steady state exists. The steady state value is the limit we calculated: Steady state value = \(\frac{1500}{3} = 500\) Therefore, a steady state exists with a population of 500 squirrels.

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial in many areas of calculus. A limit describes the value that a function approaches as the input (or variable) approaches a certain value. In the case of steady states, we're interested in the limit of a population function as time goes to infinity. This can tell us what the population will stabilize at, known as its equilibrium state.

To find the limit of a function like the one given for the population of squirrels, we examine the behavior of the function as the input grows without bound. If the function approaches a specific value, that value is the limit. As in our example with the function \(p(t) = \frac{1500}{3 + 2e^{-0.1 t}}\), as \(t\) increases, the exponential term \(e^{-0.1t}\) decreases towards zero. The presence of \(e^{-0.1t}\) in the denominator means that for very large values of \(t\), its contribution to the value of \(p(t)\) becomes negligible, causing the function to approach a constant value. This constant is the 'steady state' value.

Exponential Decay

Exponential decay is a concept typically encountered in fields like physics, chemistry, and population dynamics. It's characterized by a decrease that occurs at a rate proportional to the value at any time, which results in a smooth and continuous decline. In mathematical terms, a function that models exponential decay can be written as \(Ae^{-kt}\), where \(^{ }\)

• \(A\) is the initial amount,
• \(e\) is the base of the natural logarithm,
• \(k\) is a positive constant that represents the rate of decay,
• \(t\) represents time.

In our particular exercise with the squirrel population, the term \(e^{-0.1t}\) showcases this decay applied to part of the population. Over time, as \(t\) increases, this term steadily reduces, illustrating that the rate of growth of the population is slowing down as it nears a steady state.

Calculus in Population Modeling

Calculus is an invaluable tool for modeling populations in biology and ecology. Differential equations, which involve calculus, are often used to describe how populations change over time. These models can include factors such as birth rates, death rates, predation, and migration.

For instance, to study how a population reaches a steady state, we can use equations to define the rate of change in the population over time. The steady state occurs when this rate of change becomes zero, meaning the population is no longer increasing or decreasing. In the squirrel population example, the function \(p(t)\) predicts the number of individuals at any given time \(t\). Over time, as birth and death rates equalize, the population approaches a steady state, a scenario where it becomes constant, neither growing nor shrinking. Calculus allows us to formalize these ideas and calculate precise population levels at this steady state.