Question

Let the rate of growth \(d N / d t\) of a colony of bacteria be proportional to the square root of the number present at any time. If there are no bacteria present at \(t=0\) how many are there at a later time? Observe here that the routine separation of variables solution gives an unreasonable answer, and the correct answer, \(N \equiv 0\), is not obtainable from the routine solution. (You have to think, not just follow rules!)

Short answer

The number of bacteria remains zero for all time, \(N(t) = 0\).

Step-by-step solution

Identify the given information

The rate of growth of the bacteria colony, \(\frac{dN}{dt}\), is proportional to the square root of the number of bacteria, \(N\), present at any time. We are given that initially, there are no bacteria, i.e., \(N(0) = 0\).

Write the differential equation

Since the rate of growth is proportional to the square root of \(N\), we can write the differential equation as: \[ \frac{dN}{dt} = k \sqrt{N} \]where \(k\) is the constant of proportionality.

Use the initial condition

We are given that at time \(t=0\), the number of bacteria present is zero, i.e., \(N(0) = 0\). So initially: \[ \frac{dN}{dt} = k \sqrt{0} = 0 \]

Analyze the differential equation

The differential equation implies that if the initial number of bacteria is zero, the rate of growth is also zero. No bacteria present initially means no growth will occur at any moment in time. Thus, the number of bacteria remains zero for all time.

Conclude the solution

Since the bacteria do not start multiplying in the absence of an initial population, the correct solution to the problem is \(N(t) = 0\) for all \(t \). The routine separation of variables might give different solutions, but they are not applicable since the initial condition is crucial here.

Key concepts

Bacterial Growth

Bacterial growth refers to how bacteria multiply over time. Understanding this concept involves recognizing that bacteria reproduce by dividing, meaning one cell splits to become two, then four, and so on. Typically, this process happens exponentially under favorable conditions.
The specific exercise deals with a scenario where the rate of bacterial growth is proportional to the square root of the number of bacteria present. This differs from the common exponential growth model,
making it an interesting case for studying mathematical biology.

When bacteria are cultured, they usually go through several growth phases:

• Lag Phase: Bacteria are adjusting to their environment.
• Exponential Phase: Rapid cell division occurs.
• Stationary Phase: Nutrient depletion slows growth.
• Death Phase: Nutrient exhaustion and waste accumulation lead to a decline in bacteria.

Observing how differential equations model these phases can help us understand and predict bacterial behavior in various environments.

Initial Conditions

Initial conditions in differential equations are the starting values that help determine the specific solution to a problem. They are critical because they provide context for the model.
In our problem, the initial condition is that there are no bacteria present at time zero, i.e., when \( N(0) = 0 \).

Initial conditions allow us to solve for unknown constants in differential equations. They essentially 'anchor' our solution. For example, the equation \( \frac{dN}{dt} = k \sqrt{N} \) would have numerous possible solutions otherwise. However, knowing \( N(0) = 0 \) helps us deduce that \( N(t) = 0 \) for all time points \( t \).
In life sciences, using correct initial conditions ensures models accurately predict real-world scenarios, such as the spread of diseases, population dynamics, or even chemical reactions.

Rate of Growth

The rate of growth in our context refers to how quickly the number of bacteria increases over time. It is described by the differential equation \( \frac{dN}{dt} = k \sqrt{N} \).

The term 'rate of growth' generally means the speed at which a variable transforms.
In mathematical terms, it is the derivative of the quantity with respect to time. A positive rate of growth suggests an increase over time, whereas a negative rate would indicate a decrease.

Understanding the rate of growth is essential for predicting future behavior. If, for instance, the rate were proportional to the number of bacteria, we'd expect exponential growth. However, our problem uses the square root of \( N \), indicating a more complex relationship.
In practical applications, knowing the rate of growth helps in areas like:

• Public health: Predicting disease outbreaks.
• Agriculture: Understanding crop yields.
• Ecology: Monitoring species populations.
• Pharmaceuticals: Drug response in organisms.

Through differential equations, we can model and predict how growth processes will unfold under different conditions.