Problem 23 - Calculus: Early Transcendentals

Question

In a seasonal-growth model, a period function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for examples, be caused by seasonal changes in the availability of food. (a) Find the solution of the seasonal-growth model \( \frac {dP}{dt} = kP \cos(rt - \phi) P(0) = P_0 \) where \( k, r, \) and \( \phi \) are positive constants. (b) By graphing the solution for several values of \( k, r, \) and \( \phi, \) explain how to the values of \( k, r, \) and \( \phi \) affect the solution. What can you say about lim \( _{t \to \infty} P(t)? \)

Step-by-Step Solution

Answer

The solution is \( P(t) = P_0 e^{\frac{k}{r} \sin(\phi) + \frac{k}{r} \sin(rt - \phi)} \), and \( P(t) \) oscillates indefinitely.

Step 1: Write down the differential equation

We are given the differential equation \( \frac{dP}{dt} = kP \cos(rt - \phi) \). This equation models population growth with seasonal variation influenced by the cosine term.

Step 2: Solve the differential equation

The equation is separable. Rearrange it as:\[\frac{1}{P} dP = k \cos(rt - \phi) dt\]Integrate both sides:\[\int \frac{1}{P} dP = \int k \cos(rt - \phi) dt\]This results in:\[\ln|P| = k \int \cos(rt - \phi) dt + C_1\]

Step 3: Integrate the right side

The integral on the right side is\[k \int \cos(rt - \phi) dt = \frac{k}{r} \sin(rt - \phi) + C_2\]Substitute back to get:\[\ln|P| = \frac{k}{r} \sin(rt - \phi) + C\]

Step 4: Solve for P

Exponentiate both sides to solve for \(P\):\[P(t) = Ce^{\frac{k}{r} \sin(rt - \phi)}\]Since we have the initial condition \(P(0) = P_0\), we determine \(C\) by substituting \(t = 0\):\[P_0 = Ce^{\frac{k}{r} \sin(-\phi)}\]Therefore,\[C = P_0 e^{-\frac{k}{r} \sin(-\phi)} = P_0 e^{\frac{k}{r} \sin(\phi)}\]Finally, the solution is:\[P(t) = P_0 e^{\frac{k}{r} \sin(\phi) + \frac{k}{r} \sin(rt - \phi)}\]

Step 5: Analyze the effect of parameters

1. **Effect of \(k\):** Affects the rate of maximum seasonal growth variation. Larger \(k\) leads to more pronounced seasonal effects.2. **Effect of \(r\):** Determines frequency of seasonal changes. A larger \(r\) increases oscillation frequency.3. **Effect of \(\phi\):** Shifts the phase of the seasonal variation.Graphing the solution with different values of these parameters shows these effects clearly.

Step 6: Evaluate the limit as t approaches infinity

Since \( \sin(rt - \phi) \) is bounded between -1 and 1, the exponential expression in \(P(t)\) can dominate the solution behavior due to \(e^{\frac{k}{r} \sin(rt - \phi)}\).However, since \( \sin \) is bounded and periodic, \( P(t) \) will oscillate between exponential bounds without converging to a finite limit unless \(k=0\). Therefore:\[\lim_{t \to \infty} P(t)\]does not exist in a traditional sense but is indefinitely oscillating.

Key Concepts

Differential Equation

In mathematics, a differential equation is an equation that relates a function with its derivatives. In the context of the seasonal-growth model, the differential equation given is \( \frac{dP}{dt} = kP \cos(rt - \phi) \). This equation represents how a population \( P \) changes over time \( t \), where the change is influenced by the growth constant \( k \) and periodic factors represented by \( \cos(rt - \phi) \). The cosine term introduces oscillations to mimic real-world seasonal effects, such as changes in food availability or environment.

Differential equations like these are powerful tools in modeling continuous processes that change over time. When solving them, one typically needs to find a function \( P(t) \) that satisfies the equation for given initial conditions.

Population Growth

Population growth in the seasonal-growth model is unique compared to constant growth models. It considers not only the natural growth of a population due to reproduction (captured by the \( kP \) term) but also seasonal variance (influenced by \( \cos(rt - \phi) \)).

Such seasonal variations can arise from factors like

• fluctuations in resource availability
• environmental changes
• temperature extremes
  

In a seasonal-growth model, these factors periodically accelerate or decelerate growth. The model ultimately predicts that population size will fluctuate over time within an oscillatory bound, rather than moving toward a fixed exponential growth trend.

Separable Differential Equation

The differential equation presented in this exercise is a separable differential equation. This type of equation allows us to separate the variables on either side of the equation before solving. The given equation \( \frac{dP}{dt} = kP \cos(rt - \phi) \) can be rewritten as \( \frac{1}{P} dP = k \cos(rt - \phi) dt \), which conveniently separates variables \( P \) and \( t \).

Once separated, both sides of the equation can be integrated independently:

• The left side gives \( \int \frac{1}{P} dP = \ln|P| + C_1 \)
• The right side integrates to \( \frac{k}{r} \sin(rt - \phi) + C_2 \)

This solution method is advantageous because it turns complex relationships of change into more tractable calculations. With initial conditions, we can determine the constant \( C \) and derive \( P(t) \), providing insights into the system's behavior.

Oscillation Frequency

Oscillation frequency plays a vital role in the seasonal-growth model. It is denoted by the parameter \( r \) in the cosine term \( \cos(rt - \phi) \). This frequency determines how often the population experiences peaks and troughs over time.

When \( r \) is larger, the population oscillates more frequently, implying quick changes back and forth in growth. Conversely, a smaller \( r \) results in slower oscillations, indicating longer periods between growth peaks and decreases.
This model exhibits that, rather than settling into a steady state, the population continually experiences cycles of growth and decline influenced by the seasonal model's parameters. As such, understanding \( r \) helps in predicting population behavior over various seasonal cycles.