Question

Leaves accumulate on the forest floor at a rate of \(4 \mathrm{~g} / \mathrm{cm}^{2} / \mathrm{yr}\). These leaves decompose at a rate of \(10 \%\) per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?

Short answer

The steady state is 40 grams of leaf litter per square centimeter.

Step-by-step solution

Define the Terms

Let \( L(t) \) represent the number of grams of leaf litter per square centimeter of the forest floor at time \( t \). The problem asks us to find a differential equation involving \( L(t) \).

Set Up the Rate of Change

The rate of change of \( L(t) \) can be defined as the difference between the accumulation rate and the decomposition rate. Therefore, \( \frac{dL}{dt} = 4 - 0.1L \), where 4 is the accumulation rate \( \frac{4 \mathrm{~g}}{\mathrm{cm}^2 \mathrm{~yr}} \) and \(-0.1L\) is the decomposition rate per year based on \( L(t) \).

Determine Steady State Value

To find the steady value of leaf litter, we set \( \frac{dL}{dt} = 0 \), which signifies no change in \( L(t) \). Solving the equation \( 0 = 4 - 0.1L \), we find \( L = 40 \). This means that when \( L = 40 \), the amount of leaf litter does not change over time.

Confirm Steady State Interpretation

The steady state value of \( L = 40 \) indicates that the litter accumulation and decomposition rates are balanced, leading to zero net change in the amount of litter over time.

Key concepts

Rate of Change

Understanding the rate of change is crucial for analyzing how quickly or slowly quantities evolve over time. It is often represented by a derivative in differential equations.
In our exercise, the rate of change describes how the amount of leaf litter on the forest floor changes every year. This rate of change is given by the accumulation of leaves and the decomposition process, which removes leaf material.
This is represented by the differential equation:

• Accumulation Rate: The forest floor receives leaves at a rate of \(4 \, \mathrm{g/cm}^2/\mathrm{yr}\), continuously adding material.
• Decomposition Rate: This is summarized as \(-0.1L\), indicating that 10% of the existing leaf litter is decomposed each year.

The change in litter mass over time is expressed as \(\frac{dL}{dt} = 4 - 0.1L\), where \(\frac{dL}{dt}\) captures the net change.

Steady State

The steady state in this context refers to a point where the amount of leaf litter remains constant over time. It occurs when the rate of change is zero.
Once the differential equation is set to \(\frac{dL}{dt} = 0\), it means there is no net change happening in the system. The equation simplifies to \(0 = 4 - 0.1L\).
This can be solved:

• Setting the equation to zero allows solving for \(L\), leading to \(L = 40\).
• In the steady state, the litter accumulation from new leaves perfectly balances the decomposition rate.

This steady value of 40 grams per square centimeter indicates the long-term balance of leaf accumulation and decomposition on the forest floor.

Decomposition Rate

The decomposition rate is a vital factor in determining the dynamic changes in leaf litter over time. It essentially governs how quickly the forest floor clears the accumulated leaves.

• In the differential equation, it is expressed as \(-0.1L\), meaning 10% of the current leaf amount decomposes each year.
• This process continuously extracts leaf material, influencing the net rate of change in the leaf litter.

Given its influential role, the decomposition rate directly impacts the time it takes for the system to reach the steady state. The faster the decomposition, the quicker the approach to a stable value. Understanding this rate helps in predicting how external factors, like climate conditions, might alter decomposition speed, affecting overall litter dynamics.