Problem 21 - Calculus for Biology and Medicine

Question

Forest Disturbances Disturbances in forests (wind, fire, etc.) create gaps by killing trees. These gaps are eventually filled by new trees. We will model this process by a two-compartment model. 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 \(\left.\frac{d}{d t}\left(x_{1}+x_{2}\right)=0 .\right]\) (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 (11.51) by \(20-x_{1}\), and show that doing so reduces the system (11.51) and (11.52) 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 (11.51) and (11.52), 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

Adult trees will occupy two-thirds of the forest as \(t \rightarrow \infty\).

Step 1: Create a Compartment Diagram

To create the compartment diagram, visualize two compartments: one for gaps, denoted as \(x_1\), and another for adult trees, denoted as \(x_2\). The flow between the compartments is shown by arrows: \(0.2 x_1\) is the flow from gaps to adult trees, and \(0.1 x_2\) is the flow from adult trees to gaps. No external input or output is included, indicating a closed system.

Step 2: Show \(x_1(t) + x_2(t)\) is Constant

Compute the derivative \(\frac{d}{dt}(x_1+x_2)\) using the given equations: \[\frac{d}{dt}(x_1 + x_2) = \frac{d x_1}{dt} + \frac{d x_2}{dt} = (-0.2 x_1 + 0.1 x_2) + (0.2 x_1 - 0.1 x_2) = 0.\] Since the derivative is zero, \(x_1(t) + x_2(t)\) is constant for all \(t\). This constant, \(A\), represents the total area of the forest that is unchanged over time.

Step 3: Explain the Initial Condition's Implication

Given \(x_1(0) + x_2(0) = 20\) and knowing from Step 2 that \(x_1(t) + x_2(t) = A\) is constant, we see that \(A = 20\). Therefore, for all \(t > 0\), \(x_1(t) + x_2(t) = 20\). Thus the total area of the forest remains consistent over time.

Step 4: Substitute \(x_2(t) = 20 - x_1(t)\) into the Equations

Replace \(x_2(t)\) in the first equation with \(20 - x_1(t)\), as per the result of Step 3: \[ \frac{dx_1}{dt} = -0.2x_1 + 0.1(20 - x_1) = 2 - 0.3x_1.\] The second equation is not needed since \(x_2\) is expressed in terms of \(x_1\), confirming the system reduces to: \[ \frac{dx_1}{dt} = 2 - 0.3x_1. \]

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

This is a first-order linear differential equation. First, find the general solution: \[ \frac{dx_1}{dt} = 2 - 0.3x_1 \] which can be rewritten and solved as: \[ \frac{dx_1}{dt} + 0.3x_1 = 2. \] The integrating factor is \(e^{0.3t}\). Multiply the equation by the integrating factor and integrate both sides: \[ e^{0.3t}x_1 = \frac{2}{0.3}e^{0.3t} + C. \] Simplifying further, \[ x_1(t) = \frac{20}{3} + Ce^{-0.3t}. \] Using \(x_1(0) = 2\), solve for \(C\): \[ 2 = \frac{20}{3} + C \Rightarrow C = 2 - \frac{20}{3} = -\frac{14}{3}. \] Thus, \[ x_1(t) = \frac{20}{3} - \frac{14}{3}e^{-0.3t}. \]

Step 6: Determine \(x_2(t)\) and Fraction of Forest with Adult Trees

Since \(x_2(t) = 20 - x_1(t)\), substitute \(x_1(t)\) from Step 5: \[ x_2(t) = 20 - \left(\frac{20}{3} - \frac{14}{3}e^{-0.3t}\right) = \frac{40}{3} + \frac{14}{3}e^{-0.3t}. \] Initially, \(x_2(0) = 18\), confirming \(x_1(0) = 2\) and \(x_1(0) + x_2(0) = 20\). As \(t \rightarrow \infty\), \(e^{-0.3t} \rightarrow 0\), so \(x_1(t)\) approaches \(\frac{20}{3}\) and \(x_2(t)\) approaches \(\frac{40}{3}\). The fraction of the forest with adult trees stabilizes at \(\frac{2}{3}\).

Key Concepts

Two-Compartment Model

In mathematical modeling, the two-compartment model is a simple yet powerful way to understand and visualize how different parts of a system interact. In this context, we are modeling a forest ecosystem with two compartments: gaps and adult trees. The first compartment, gaps, contains open areas created by natural disturbances like wind or fire, denoted by \(x_1(t)\). The second compartment consists of spaces filled by adult, fully-grown trees, denoted by \(x_2(t)\).

The movement or the flow between these compartments is described by differential equations. Here, gaps grow into areas with adult trees at a rate of \(0.2x_1\), while the reverse process happens, albeit at a slower rate of \(0.1x_2\), as trees die and create gaps. This setup forms a closed system where the total area is constant. Visualizing these flows with arrows helps understand the system dynamics intuitively, as each arrow represents a transfer of area between the compartments. This model is particularly useful in ecological studies to simulate forest regeneration and manage forest resources efficiently.

First-order Linear Differential Equation

First-order linear differential equations are among the simplest types of differential equations. They are crucial for modeling processes that change over time in various scientific fields. In our forest ecosystem example, we encounter such an equation when we simplify the system with the substitution from given problems: \( \frac{dx_1}{dt} = 2 - 0.3x_1 \). This equation describes how the area occupied by gaps changes with time.

A first-order linear differential equation is of the standard form \( \frac{dy}{dt} + py = q \), where \(p\) and \(q\) are constants or functions of \(t\). In this exercise, we solve it using an integrating factor method. By multiplying the differential equation by an integrating factor, we can convert it into a form that allows straightforward integration to find the solution. In this case, the integrating factor is \(e^{0.3t}\), leading to a solution \(x_1(t) = \frac{20}{3} - \frac{14}{3}e^{-0.3t}\).

This result tells us how the area of gaps evolves, starting from an initial condition, and approaching a steady state over time as \(t\) approaches infinity.

Forest Ecosystem Modeling

Forest ecosystem modeling is a way to understand and predict changes in forest areas over time. By creating mathematical models like the two-compartment system, scientists can simulate the effects of natural disturbances and regeneration processes. This is critical for monitoring forest health and making informed conservation decisions.

In our specific model, the forest is seen as a closed system where the total area – the sum of gaps \(x_1(t)\) and adult trees \(x_2(t)\) – remains constant at 20. By setting up the differential equations based on real-world rates of tree growth and decay, we gain insights into how quickly a forest can recover from disturbances, what the stable configuration looks like, and how these dynamics influence biodiversity.

Furthermore, the model shows that as time goes to infinity, the fraction of the forest covered by adult trees approaches \(\frac{2}{3}\), indicating the long-term distribution between gaps and trees. Ecosystem modeling like this helps forest managers and ecologists understand the resilience and sustainability of forests, which is vital for planning conservation and management strategies.