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Chapter 1: Problem 18

Blood \(\mathrm{CO}_{2}\) and ventilation. We now consider the problem of blood \(\mathrm{CO}_{2}\) and the physiological control of ventilation. (a) As a first model, assume that the amount of \(\mathrm{CO}_{2}\) lost, \(\mathscr{L}\left(V_{n}, C_{n}\right)\), is simply proportional to the ventilation volume \(V_{n}\) with constant factor \(\beta\) (and does not depend on \(C_{n}\) ). Further assume that the ventilation at time \(n+1\) is directly proportional to \(C_{n}\) (with factor \(\alpha\) ), i.e., that \(\mathscr{F}\left(C_{.}\right)=\alpha C_{m-}\) (This may be physiologically unrealistic but for the moment it makes the model linear.) Write down the system of equations (49) and show that it corresponds to a single equation $$ C_{n+1}-C_{n}+\alpha \beta C_{n-1}=m \text {. } $$ (b) For \(m \neq 0\) the equation in part (a) is a nonhomogeneous problem. Use the steps outlined in problem (11) to solve it. (1) Show that \(C_{n}=m / \alpha \beta\) is a particular solution. (2) Find the general solution. (c) Now consider the nature of this solution in two stages. (1) First assume that \(4 \alpha \beta<1\). Interpret this inequality in terms of the biological process. Give evidence for the assertion that under this condition, a steady blood \(\mathrm{CO}_{2}\) level \(\mathrm{C}\) equal to \(m / \alpha \beta\) will eventually be established, regardless of the initial conditions. What will the steady ventilation rate then be? (Hint: Show that \(\left|\lambda_{i}\right|<1\).) (2) Now suppose \(4 \alpha \beta>1\). Show that the \(\mathrm{CO}_{2}\) level will undergo oscillations. If \(\alpha \beta\) is large enough, show that the oscillations may increase in magnitude. Find the frequency of the oscillations. Comment on the biological relevance of this solution. May (1978) refers to such situations as dynamical diseases. (d) Suggest how the model might be made more realistic. The equations you obtain may be nonlinear. Determine whether they admit steady-state solutions in which \(C_{n+1}=C_{n}\) and \(V_{n+1}=V_{n}\).

Short Answer

Expert verified
The equation \(C_{n+1} - C_{n} + \alpha \beta C_{n-1} = m\) was derived and the solution found. For \(4 \alpha \beta < 1\), \(C_{n}\) stabilizes to \(\frac{m}{\alpha \beta}\). For \(4 \alpha \beta > 1\), \(C_{n}\) oscillates.

Step by step solution

01

Understanding the Problem

Parse the two key assumptions: (1) The amount of CO2 lost, \mathscr{L}(V_n, C_n)\, is proportional to the ventilation volume \(V_n\) with constant \beta that is independent of \(C_n\). (2) Ventilation at time \(n+1\) is proportional to \(C_n\) with factor \alpha, i.e., \(\mathscr{F}(C_{n}) = \alpha C_{n}\).
02

Set Up the Equations

Based on step 1, we write down the system: $$ C_{n+1} = C_{n} - \beta V_{n} + m $$ and $$ V_{n+1} = \alpha C_{n} $$.
03

Combine the Equations

Substitute \(V_{n}\) from the second equation into the first equation: $$ V_{n} = \alpha C_{n-1} $$, then $$ C_{n+1} = C_{n} - \beta (\alpha C_{n-1}) + m $$, resulting in the equation: $$ C_{n+1} - C_{n} + \alpha \beta C_{n-1} = m $$.
04

Show Particular Solution

To solve the nonhomogeneous equation, first show that \(C_{n} = \frac{m}{\alpha \beta}\) is a particular solution. Substituting this value into the nonhomogeneous equation results in zero, confirming that it is a valid particular solution.
05

Find General Solution for Homogeneous Equation

Solve the corresponding homogeneous equation: $$ C_{n+1} - C_{n} + \alpha \beta C_{n-1} = 0 $$ yielding the general solution involving characteristic roots.
06

Analyze for 4αβ

When \(4 \alpha \beta < 1\), show that the roots of the characteristic equation are within the unit circle (\left| \lambda \right| < 1). Conclude that the steady-state solution is \(\frac{m}{\alpha \beta}\) and that the system stabilizes regardless of initial conditions.
07

Analyze for 4αβ > 1

When \(4 \alpha \beta > 1\), solve for the characteristic roots to show they are complex, resulting in oscillatory behavior of \(C_n\). If \(\alpha \beta\) is large enough, the oscillations may grow in amplitude, with frequency related to the imaginary part of the roots.
08

Discuss Realistic Model

Consider factors necessary to make the model more realistic, including nonlinear terms and other physiological factors. Check if these new equations allow for steady-state solutions where \(C_{n+1} = C_{n}\) and \(V_{n+1} = V_{n}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

physiological control
Physiological control refers to the body's ability to maintain stable internal conditions despite changes in the external environment. In the context of blood \(\text{CO}_2\) levels, the body uses a control system to regulate ventilation, the process of moving air in and out of the lungs. This system helps ensure that \(\text{CO}_2\) levels remain within a healthy range.
In our exercise, we consider a simplified model where the amount of \(\text{CO}_2\) lost is proportional to the ventilation volume \(V_n\) with a constant factor \(\beta\). Also, the ventilation at time \(n+1\) is directly proportional to the \(\text{CO}_2\) concentration \(C_n\) through another constant, \(\alpha\). These factors are part of the body's physiological control system's response to maintain stable \(\text{CO}_2\) levels.
ventilation volume
Ventilation volume is the amount of air moved in or out of the lungs during a given time period. It is a crucial factor in determining \(\text{CO}_2\) levels in the blood.
In our exercise, the relationship between ventilation volume \(V_n\) and \(\text{CO}_2\) concentration \(C_n\) is given by the equation \(V_{n+1} = \alpha C_n\). This means that higher \(\text{CO}_2\) levels lead to increased ventilation, helping to expel more \(\text{CO}_2\) from the body.
This relationship can be thought of as the body's natural response to rising \(\text{CO}_2\) levels—ventilating more to expel the excess \(\text{CO}_2\) and restore balance.
steady-state solutions
A steady-state solution is a condition where variables in a system remain constant over time. In our context, it would mean the \(\text{CO}_2\) concentration \(C_n\) and ventilation volume \(V_n\) do not change across time steps.
In the exercise, we explore if a steady state can be achieved. When \(4\alpha\beta < 1\), the roots of the characteristic equation are within the unit circle, implying that \(\text{CO}_2\) levels stabilize at \(\frac{m}{\alpha\beta}\).
At balance, even if initial conditions vary, the system will revert to this steady state, showing the effectiveness of the physiological control system in maintaining equilibrium.
nonlinear equations
Nonlinear equations feature variables that don't have a direct proportionality. In more realistic models of \(\text{CO}_2\) regulation, we might encounter nonlinear relationships.
Real-life \(\text{CO}_2\) dynamics include many interacting factors such as different feedback mechanisms, which can create nonlinear behaviors. For more accurate modeling, we could incorporate nonlinear terms that reflect the body's complex responses to changing \(\text{CO}_2\) levels.
Despite being more complex, one would still look for steady-state solutions to ensure that under certain conditions, the system can stabilize. Such refined models could better capture the nuances of physiological control in blood \(\text{CO}_2\) levels.

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Most popular questions from this chapter

(a) Solve the first order equation \(x_{n+1}=a x_{n}+b(a, b\) constants). (b) Consider the difference equation $$ a X_{n+2}+b X_{n+1}+c X_{n}=d $$ If \(d \neq 0\), the equation is called nonhomogeneous. Show that \(X_{n}=K\), where \(K\) is a particular constant, will solve this second order nonhomogeneous equation (provided \(\lambda=1\) is not an eigenvalue) and find the value of \(K\). (This is called a particular solution.) (c) Suppose the solution to the corresponding homogeneous equation \(a X_{n+2}+b X_{n+1}+c X_{n}=0\) is $$ X_{n}=c_{1} \lambda_{1}^{n}+c_{2} \lambda_{2} $$ where \(c_{1}\) and \(c_{2}\) are arbitrary constants. (This is called the complementary or homogeneous solution.) Show that $$ X_{n}=K+c_{1} \lambda_{1}^{n}+c_{2} \lambda_{2}^{n} $$ will be a solution to the nonhomogeneous problem (provided \(\lambda_{1} \neq \lambda_{2}\) and \(\lambda_{1} \neq 1\) ). This solution is called the general solution. Note: As in linear nonhomogeneous differential equations, one obtains the result that for nonhomogeneous difference equations: general solution = complementary solutions + particular solution.

The Rabbit problem. In 1202 Fibonacci posed and solved the following problem. Suppose that every pair of rabbits can reproduce only twice, when they are one and two months old, and that each time they produce exactly one new pair of rabbits. Assume that all rabbits survive. Starting with a single pair in the first generation, how many pairs will there be after \(n\) generations? (See figure.) (a) To solve this problem, define \(R_{n}^{0}=\) number of newborn pairs in generation \(n\), \(R_{n}^{1}=\) number of one-month-old pairs in generation \(n\), \(R_{n}^{2}=\) number of two-month-old pairs in generation \(n\). Show that \(R_{n}^{0}\) satisfies the equation $$ R_{n+1}^{0}=R_{n}^{0}+R_{n-1}^{0}. $$ (b) Suppose that Fibonacci initially had one pair of newborn rabbits, i.e., that \(R_{6}=1\) and \(R_{i}^{\varphi}=1\). Find the numbers of newborn, one- month-old, and two-month-old rabbits he had after \(n\) generations. (Historical note to problem 14: F.C. Gies quotes a different version of Fibonacci's question, reprinted from the Liber abaci in the 1981 edition of the Encyclopedia Britannica: "A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?")

(a) Consider a population with \(m\) age classes, and let \(p_{n}^{1}, p_{=1}^{2}, \ldots, p_{n}^{m}\) be the numbers of individuals within each class such that \(p_{n}^{0}\) is the number of newborns and \(p_{n}^{m}\) is the number of oldest individuals. Define $$ \begin{aligned} \alpha_{1}, \ldots, \alpha_{4}, \ldots, \alpha_{m}=& \text { number of births from individuals of a given } \\ & \text { age class, } \\ \sigma_{1}, \ldots, \sigma_{k}, \ldots, \sigma_{m-1}=& \text { fraction of } k \text { year olds that survive to be } \\ & k+1 \text { year olds. } \end{aligned} $$ Show that the system can be described by the following matrix equation: $$ \mathbf{P}_{n+1}=\mathbf{A} \mathbf{P}_{n} $$ where $$ \mathbf{P}=\left(\begin{array}{c} p^{1} \\ p^{2} \\ \vdots \\ p^{m} \end{array}\right), \quad A=\left(\begin{array}{cccc} \alpha_{1} & \alpha_{2} & \cdots & \alpha_{m} \\ \sigma_{1} & 0 & \cdots & 0 \\ 0 & \sigma_{2} & \cdots & \\ 0 & 0 & \cdots & \sigma_{m-1} 0 \end{array}\right). $$ A is called a Leslie matrix. (See references on demography for a summary of special properties of such matrices.) Note: For biological realism, we assume at least one \(\alpha_{i}>0\) and all \(\sigma_{i}>0\).) (b) The characteristic equation of a Leslie matrix is $$ p_{n}(\lambda)=\operatorname{det}(\lambda \mathbf{I}-\mathbf{A})=0 $$ Show that this leads to the equation $$ \lambda^{n}-\alpha_{1} \lambda^{n-1}-\alpha_{2} \sigma_{1} \lambda^{n-2}-\alpha_{3} \sigma_{1} \sigma_{2} \lambda^{n-3}-\cdots-\alpha_{n} \sigma_{1} \sigma_{2} \cdots \sigma_{n-1}=0 $$ (c) A Leslie matrix has a unique positive eigenvalue \(\lambda^{*}\). To prove this assertion, define the function $$ f(\lambda)=1-\frac{p_{n}(\lambda)}{\lambda^{n}} $$ Show that \(f(\lambda)\) is a monotone decreasing function with \(f(\lambda) \rightarrow+\infty\) for \(\lambda \rightarrow 0, f(\lambda) \rightarrow 0\) for \(\lambda \rightarrow \infty\). Conclude that there is a unique value of \(\lambda\) \(\lambda^{*}\) such that \(f\left(\lambda^{*}\right)=1\), and use this observation in proving the assertion. (Note: This can also be easily proved by Descartes' Rule of Signs.) (d) Suppose that \(\mathbf{v}^{*}\) is the eigenvector corresponding to \(\lambda^{*}\). Further suppose that \(\left|\lambda^{*}\right|\) is strictly greater than \(|\lambda|\) for any other eigenvalue \(\lambda\) of the Leslie matrix. Reason that successive generations will eventually produce an age distribution in which the ages are proportional to elements of \(\mathrm{v}^{*}\). This is called a stable age distribution. (Note: Some confusion in the literature stems from the fact that Leslie matrices are formulated for systems in which a census of the population is taken after births occur. Compare with the plant-seed example, which was done in this way in problem 19 but in another way in Section 1.2. A good reference on this point is M. R. Cullen (1985), Linear Models in Biology, Halstead Press, New York.

Convert the following systems of difference equations to single higher-order equations and find their solutions. Sketch the behavior of each solution, indicating whether it increases or decreases and whether oscillations occur. (a) \(\begin{aligned} x_{n+1} &=3 x_{n}+2 y_{n} \\ y_{n+1} &=x_{n}+4 y_{n} \end{aligned}\) (b) \(\begin{aligned} x_{n+1} &=\frac{x_{n}}{4}+y_{n} \\ y_{n+1} &=\frac{3 x_{n}}{16}-\frac{y_{n}}{4} \end{aligned}\) (c) \(\begin{aligned} x_{n+1} &=\sigma_{1} x_{n}+\sigma_{2} y_{n} \\ & y_{n+1} &=\beta x_{n} \end{aligned}\) (d) \(x_{n+1}=x_{n}+y_{n}\) \(y_{n+1}=2 x_{n}\) (e) \(\begin{aligned} x_{n+1} &=-x_{n}+3 y_{n} \\ y_{n+1} &=\frac{y_{n}}{3} \end{aligned}\) (f) \(\begin{aligned} x_{n+1} &=\frac{x_{n}}{4}+3 y_{n} \\ y_{n+1} &=-\frac{x_{n}}{8}+y_{n} \end{aligned}\)

Consider the difference equation \(x_{n+2}-3 x_{n+1}+2 x_{n}=0\). (a) Show that the general solution to this equation is $$ x_{n}=A_{1}+2^{n} A_{2} $$ Now suppose that \(x_{0}=10\) and \(x_{1}=20\). Then \(A_{1}\) and \(A_{2}\) must satisfy the system of equations $$ \begin{aligned} &A_{1}+2^{0} A_{2}=x_{0}=10 \\ &A_{1}+2^{1} A_{2}=x_{1}=20 \end{aligned} $$ (b) Solve for \(A_{1}\) and \(A_{2}\) and find the solution to the above initial value problem.
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