Problem 21 - Calculus for Biology and Medicine

Question

A very simple two-compartment model for gap dynamics in a forest assumes that gaps are created by disturbances (wind, fire, etc.) and that gaps revert to forest as trees grow in the gaps. We denote by \(x_{1}(t)\) the area occupied by gaps and by \(x_{2}(t)\) the area occupied by adult trees. We assume that the dynamics are given by $$ \begin{array}{l} \frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2} \\ \frac{d x_{2}}{d t}=0.2 x_{1}-0.1 x_{2} \end{array} $$ (a) Find the corresponding compartment diagram. (b) Show that \(x_{1}(t)+x_{2}(t)\) is a constant. Denote the constant by \(A\) and give its meaning. [Hint: Show that \(\frac{d}{d t}\left(x_{1}+x_{2}\right)=0 .\) ] (c) Let \(x_{1}(0)+x_{2}(0)=20\). Use your answer in (b) to explain why this equation implies that \(x_{1}(t)+x_{2}(t)=20\) for all \(t>0\). (d) Use your result in (c) to replace \(x_{2}\) in (11.44) by \(20-x_{1}\), and show that doing so reduces the system \((11.44)\) and \((11.45)\) to $$ \frac{d x_{1}}{d t}=2-0.3 x_{1} $$ with \(x_{1}(t)+x_{2}(t)=20\) for all \(t \geq 0\). (e) Solve the system (11.44) and (11.45), and determine what fraction of the forest is occupied by adult trees at time \(t\) when \(x_{1}(0)=2\) and \(x_{2}(0)=18\). What happens as \(t \rightarrow \infty\) ?

Step-by-Step Solution

Answer

(b) Yes, \(x_1(t) + x_2(t) = A = 20 \) (c) \( x_1(t) + x_2(t) = 20 \) due to constant sum (e) \(\frac{2}{3}\) of the forest is adult trees as \(t \to \infty\).

Step 1: Draw the Compartment Diagram

The compartment diagram represents the areas as compartments with arrows showing the flow between them. We have two compartments: one for gaps \(x_1(t)\) and one for adult trees \(x_2(t)\). The arrows between them show the flow rates: \(-0.2x_1\) from gaps returning to forest and \(0.1x_2\) from forest becoming gaps.

Step 2: Show Constant Sum of Areas

To show that \(x_1(t) + x_2(t)\) is constant, differentiate \(S(t) = x_1(t) + x_2(t)\) with respect to time \(t\): \[ \frac{dS}{dt} = \frac{d}{dt}(x_1 + x_2) = \frac{dx_1}{dt} + \frac{dx_2}{dt}. \] Substitute the given equations, \( \frac{dx_1}{dt} = -0.2x_1 + 0.1x_2 \) and \( \frac{dx_2}{dt} = 0.2x_1 - 0.1x_2 \), to get \[ \frac{dS}{dt} = (-0.2x_1 + 0.1x_2) + (0.2x_1 - 0.1x_2) = 0. \] Thus, \(S(t) = x_1(t) + x_2(t)\) is constant.

Step 3: Initial Condition and Constant Sum

Given \(x_1(0) + x_2(0) = 20\), it implies \(S(t) = 20\) for all \(t > 0\) because we showed \(S(t)\) is constant in Step 2. Therefore, \(x_1(t) + x_2(t) = 20\) always.

Step 4: Replace \(x_2\) and Simplify System

Substituting \(x_2 = 20 - x_1\) into \( \frac{dx_1}{dt} = -0.2x_1 + 0.1x_2 \), we have \[ \frac{dx_1}{dt} = -0.2x_1 + 0.1(20 - x_1) = 2 - 0.3x_1. \] We now have a simplified system \( \frac{dx_1}{dt} = 2 - 0.3x_1 \) with \( x_1(t) + x_2(t) = 20 \) for all \( t \geq 0 \).

Step 5: Solve the Differential Equation for \(x_1(t)\)

To solve \( \frac{dx_1}{dt} = 2 - 0.3x_1 \), separate variables and integrate: \[ \int \frac{1}{2 - 0.3x_1} dx_1 = \int dt. \] Solving this gives the solution for \(x_1(t)\): \[ x_1(t) = \frac{20}{3} + Ce^{-0.3t} \] where \( C \) is a constant determined by initial condition \(x_1(0) = 2\). Solving for \(C\), we have \( 2 = \frac{20}{3} + C \), giving \( C = 2 - \frac{20}{3} = -\frac{14}{3} \). Thus, \( x_1(t) = \frac{20}{3} - \frac{14}{3}e^{-0.3t} \).

Step 6: Find \(x_2(t)\) and Fractions

Using \(x_2(t) = 20 - x_1(t)\), substitute the solution from Step 5: \( x_2(t) = 20 - \left( \frac{20}{3} - \frac{14}{3}e^{-0.3t} \right) = \frac{40}{3} + \frac{14}{3}e^{-0.3t} \). The fraction of forest area occupied by adult trees is \(\frac{x_2(t)}{20}\). As \(t \to \infty\), \( e^{-0.3t} \to 0\), so \( x_1(t) \to \frac{20}{3} \) and \( x_2(t) \to \frac{40}{3} \), resulting in \(\frac{2}{3}\) of the forest area being adult trees indefinitely.

Key Concepts

Compartment Models

Differential equations are often used to model dynamic systems, and one powerful approach is through compartment models. In these models, the system is divided into interconnected compartments, each representing quantities of interest. For instance, in a forest dynamics context, compartments can represent areas of different land cover such as gaps and mature trees.

To understand a compartment model, envision it as a diagram with boxes and arrows:

• Each box is a compartment (e.g., gaps or trees in a forest).
• Arrows between boxes indicate the rate at which quantities move between compartments.

The given exercise presents a two-compartment model for forest dynamics, with two equations expressing the change in the area of gaps (\(x_1(t)\)) and adult trees (\(x_2(t)\)). The arrows indicate -0.2x_1 as the rate of gaps turning into forest, and 0.1x_2 as the rate of forest areas becoming gaps.

This model helps to understand how these areas change over time due to natural disturbances and regrowth.

Forest Dynamics

The dynamics of a forest can be complex, influenced by many factors such as disturbances and natural growth cycles. Understanding these dynamics is crucial for effective forest management and conservation. In the forest dynamics model described in the exercise, the key variables are:

• Gap area (\(x_1(t)\)): Represents the regions of the forest affected by disturbances.
• Adult tree area (\(x_2(t)\)): Represents the healthy, mature forest.

The system of differential equations captures the rate of change for these areas. The equations imply that:

• Gaps decrease as trees grow back (-0.2x_1).
• Forested areas can become gaps due to disturbances (0.1x_2).

By solving these equations, we can predict how much of the forest will remain intact over time and how quickly gaps are filled. This understanding is important when forecasting forest regeneration and planning resource management.

Constant Solutions

In the presented model, one interesting outcome is the concept of constant solutions, indicating that certain properties of the system do not change over time. Specifically, the sum of the areas for gaps and forest (\(x_1(t) + x_2(t)\)) is constant.

• This stems from natural balance: the area covered by forests and gaps together does not change due to the constraints of the environment.

To verify this:1. Differentiate the total area (\(S(t) = x_1(t) + x_2(t)\)) with respect to time.2. Substitute the given differential equations, showing that the rate of change (\(rac{dS}{dt}\)) is zero.

Thus, this sum remains constant, symbolized as \(S(t) = 20\) when initial conditions specify it. Constant solutions, like these, are vital because they simplify dynamics analysis and help in predicting long-term behavior, even as individual compartments change.