Question

If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The population of a bacteria culture is given by \(p(t)=\frac{2500}{t+1}\).

Short answer

Answer: The steady-state value for this bacterial population is 0.

Step-by-step solution

Find the limit as \(t\) approaches infinity

First, we need to find the limit of the function \(p(t)=\frac{2500}{t+1}\) as \(t\) goes to infinity. We have $$\lim_{t \rightarrow \infty} p(t) = \lim_{t \rightarrow \infty} \frac{2500}{t+1}$$

Determine existence of the limit

In order to find the limit, we can use the fact that \(\lim_{t \rightarrow \infty} \frac{c}{t} = 0\) for any constant c. So we can rewrite the expression for this limit as: $$\lim_{t \rightarrow \infty} \frac{2500}{t+1} = \frac{2500}{\infty} = 0$$ Since the limit exists and is equal to 0, this means that the system does reach a steady state.

Provide the steady-state value

As we have found that the system reaches a steady state and the limit is equal to 0, the steady-state value of the population is 0. In conclusion, the bacterial population does reach a steady state, and the steady-state value is 0.

Key concepts

Steady State

In the context of dynamic systems, a steady state—or equilibrium—is reached when a system no longer changes as time progresses. Consider a situation where we are observing the population of bacteria over time. If the rate of change becomes zero, meaning the population stabilizes and no longer grows or declines, we can say that the system has reached a steady state.
To determine this mathematically, we examine the limit of the population function as time approaches infinity. If the limit exists, the system achieves a steady state.
In our problem, the function representing bacteria population is \( p(t)=\frac{2500}{t+1} \). As time \( t \) goes to infinity, the population function approaches a limit of \( 0 \). This signifies the system achieves a steady state with a population of zero. Thus, eventually, the bacteria population ceases to grow as resources become limited or other factors dominate the dynamics.

Limit at Infinity

Understanding the concept of limits at infinity is crucial for analyzing how functions behave as the input grows indefinitely large. Limits at infinity help us determine the long-term behavior of functions.
The concept is primarily concerned with what value a function approaches as its variable tends towards infinity. This utility makes it fundamental in fields such as calculus and mathematical modeling.

• When a function \( f(t) \) approaches a number \( L \) as \( t \) becomes infinitely large, we describe it with the notation \( \lim_{t \rightarrow \infty} f(t) = L \).
• For rational functions like \( f(t)=\frac{c}{t} \), where \( c \) is a constant, the limit as \( t \) approaches infinity is \( 0 \).

In our exercise, the given function representing bacteria population, \( p(t) = \frac{2500}{t+1} \), simplifies to \( \frac{2500}{\infty} \) as \( t \) grows larger, resulting in the limit of \( 0 \). This mathematical process illustrates how the bacteria population trends towards a steady state of zero over infinite time.

Bacteria Population Dynamics

Population dynamics in bacteria often follow patterns where populations grow rapidly until limited by environmental factors. Growth can be described by functions in which population size depends on time. Over time, factors like resource depletion, waste accumulation, and space constraints typically impede growth leading to a stable equilibrium.
In the given exercise, the bacterial population is modeled by \( p(t) = \frac{2500}{t+1} \). Initially, there is a high population growth. However, as \( t \) increases, the growth rate slows down and the population approaches zero.

• The inverse relationship between population size and time in the formula suggests an eventual stabilization.
• Factors such as nutrient limitation or increased competition within the environment contribute to reaching this steady state.

In essence, the function's long-run behavior demonstrates how real-world limitations impact growth, and why despite accelerated initial growth, a ceiling is often reached which corresponds to the concept of a steady state.