Question

Leaves accumulate on the forest floor at a rate of \(2 \mathrm{~g} / \mathrm{cm}^{2} / \mathrm{yr}\) and also decompose at a rate of \(90 \%\) per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?

Short answer

The leaf litter approaches a steady value of approximately 2.22 g/cm².

Step-by-step solution

Define Variables and Constants

Let \( L(t) \) represent the grams of leaf litter per square centimeter of forest floor at time \( t \). The leaves accumulate at a rate of 2 g/cm²/yr, so this is a constant source term. The decay rate is 90% per year, which means the leaves decompose at a rate of 0.9\( L(t) \).

Formulate the Differential Equation

The rate of change of leaf litter is the accumulation rate minus the decay rate. Thus, we can write the differential equation as:\[\frac{dL}{dt} = 2 - 0.9L\]

Analyze the Differential Equation for Steady State

To find if the litter approaches a steady state, set \( \frac{dL}{dt} = 0 \). Solving \( 2 - 0.9L = 0 \) gives \( L = \frac{2}{0.9} \).

Calculate the Steady State Value

Solving \( L = \frac{2}{0.9} \), we find \( L = \frac{20}{9} \approx 2.22 \) g/cm². This is the steady-state value of leaf litter.

Key concepts

Leaf Litter

Leaf litter refers to the dead leaves and organic matter that fall and accumulate on the forest floor. This natural process plays a crucial role in nutrient cycling within forest ecosystems. As leaves fall to the ground, they begin to contribute to the top layer of soil, enriching it with essential organic material.

• The leaf litter serves as a habitat for numerous organisms, such as insects and fungi, that help decompose the material.
• While the accumulation of leaf litter depends on several factors, in this scenario, it occurs at a defined rate of \(2 \text{ grams/cm}^2/ ext{yr}\).
• This example assumes an idealized condition where leaf fall is constant and unaffected by seasonal variations.

Understanding how leaf litter accumulates helps in constructing models, like differential equations, to evaluate its environmental impact.

Steady State

In the context of differential equations, a steady state implies a condition where a system remains constant over time despite continuous inputs or changes. For leaf litter, the system reaches a steady state when the rate of leaf accumulation equals the rate of decomposition.

• At steady state, the change in leaf litter over time is zero.
• This is found by setting the derivative \( \frac{dL}{dt} \) of the differential equation to zero.
• When solved for the given exercise, it means that the leaf litter approaches and remains at \( \approx 2.22 \text{ grams/cm}^2 \).

This steadiness is crucial, as it reflects a balance in the ecosystem, indicating neither accumulation nor depletion of resources on the forest floor.

Decay Rate

The decay rate in this scenario refers to the rate at which leaf litter decomposes or breaks down organically. It is often represented as a percentage of the existing material that decays over a given period. Here, it is specified as 90% per year.

• This means that each year, 90% of the existing leaf litter is decomposed.
• The decay rate is a crucial factor that helps determine how quickly or slowly the leaf litter diminishes.
• In the differential equation, the decay is represented by the term \(0.9L(t)\), where \(L(t)\) is the amount of leaf litter at time \(t\).

Understanding the decay rate is essential when calculating the long-term equilibrium, or steady state, of the leaf litter.

Accumulation Rate

The accumulation rate is the speed at which leaf litter is added to the forest floor over time. For this problem, the rate is constant at \(2 \text{ grams/cm}^2/ ext{yr}\). This constant rate serves as an input in our differential equation to determine the overall dynamics of leaf litter.

• The accumulation rate is consistent, assuming the same environmental conditions year-round.
• This constant influx of leaf material counteracts the loss through decomposition.
• In the equation \( \frac{dL}{dt} = 2 - 0.9L \), the accumulation term is represented by the number 2.

By understanding and determining the accumulation rate, it is possible to predict how leaf litter levels will change over time, ultimately leading to a steady state.