Modeling - Math 056 - Homework 2

1. In class, we determined that the model for annual plants could be condensed into a single equation:

p_n+1 = ασγp_n + βσ²(1 - α)γp_n-1.

(a) Show that the model can be written as a single equation in s¹_n. (Write s¹_n+1 as a function of s¹_n and s¹_n-1).

(b) Explain the term ασγp_n.

2. Suppose that we define the beginning of a generation at the time when seeds are produced.

(a) Define the variables and write new equations linking them.

(b) Derive the system as two equations for S⁰_n+1 and S¯¹_n+1. Write the solution in matrix form.

i. Solve the system found in (b).

ii. Show that p_n+2 - ασγp_n+1 - βσ²γ(1 - α)p_n = 0. Is it important when we define the start of a generation?

iii. Find the solution of (ii).

iv. Find the condition for survival of the adult plant population. Biologically interpret this condition.

(c) Consider the problem in which seeds S⁰_n can now survive and germinate in generations n, n+ 1, n+ 2, and n+ 3, with germination fractions α, β, δ, and ∈. Derive the corresponding system of equations in matrix form,

Interpret the terms in the matrix.

3. Solve the first order equation x_n+1 = ax_n + b (a, b constants).

4. Solve and describe the solutions to the following difference equations. (You can check solutions with matlab simulations.)

(a) x_n+1 = 3x_n + 2yn, y_n+1 = x_n + 4y_n

(b) x_n+1 = -x_n + 3y_n, y_n+1 = y_n/3

(c) x_n+2 - x_n+1 + x_n = 0

5. Consider the difference equation x_n+2 - 3x_n+1 + 2x_n = 0.

(a) Show that the general solution to this equation is

x_n = A₁ + 2ⁿA₂.

(b) Now suppose that x₀ = 10 and x₁ = 20. Then A₁ and A₂ must satisfy the system of equations

A₁ + 2⁰A₂ = x₀ = 10,

A₁ + 2¹A₂ = x₁ = 20.

(c) Solve for A₁ and A₂ and find the solution to the above initial value problem.

6. In the circulatory system, the red blood cells (RBCs) are constantly being destroyed and replaced. Since these cells carry oxygen throughout the body, their number must be maintained at some fixed level. Assume that the spleen filters out and destroys a certain fraction of the cells daily and that the bone marrow produces a number proportional to the number lost of the previous day. What would be the cell count on the nth day?

(a) Derive a model for the number of RBCs in circulation and the number of RBCs produced by the marrow. Don't forget to define your variables and parameters.

(b) For homeostasis in the red cell count, the total number of RBCs should remain roughly constant. Explain why one way of achieving this is by letting λ₁ = 1.

(c) Using the results of part (c), find λ₂. What then is the behavior of the solution?

7. In class we discussed solving homogeneous equations (ax_n+2 + bx_n+1 + cx_n = 0). This problem will show you how to solve nonhomogeneous equations.

(a) Consider the difference equation

aX_n+2 + bX_n+1 + cX_n = d.

If d ≠ 0, the equation is called nonhomogeneous. Show that X_n = K, where K is a particular constant, will solve this second order nonhomogenous equation (provided λ = 1 is not an eigenvalue) and find the value of K. This is called a particular solution.

(b) Suppose the solution to the corresponding homogeneous equation aX_n+2 + bX_n+1 + cX_n = 0 is

X_n = c₁λⁿ₁ + c₂λⁿ₂,

where c₁ and c₂ are arbitrary constants. This is called the complementary or homogenous solution. Show that

X_n = K + c₁λⁿ₁ + c₂λⁿ₂

will be a solution to the nonhomogeneous problem (provided λ₁ ≠ λ₂ and λ_i ≠ 1). This solution is called the general solution.

8. In the blood there is a steady production of CO₂ that results from the basal metabolic rate. CO₂ is lost by way of the lungs at a ventilation rate governed by CO₂-sensitive chemoreceptors located in the brainstem. In reality, both the rate of breathing and the depth of breathing (volume of a breath) are controlled physiologically. We will simplify the problem by assuming that breathing takes place at constant intervals t, t + τ , t + 2τ , ... and that the volume, V_n is controlled by the CO₂ concentration in the blood in the previous time interval, C_n-1. Keeping track of the two variables C_n and V_n might lead to the following equations:

C_n+1 = C_n - ?(V_n, C_n) + m,

V_n+1 = ρ(C_n)

where ? is the amount of CO₂ lost as a function of volume and concentration, ρ is the sensitivity to blood CO₂ as a function of concentration, and m is the constant production rate of CO₂ in the blood.

(a) As a first model, assume that the amount of CO₂ lost, ?(V_n, C_n) is simply proportional to the ventilation volume V_n with constant factor β (and does not depend on C_n). Further assume that the ventilation at time n + 1 is directly proportional to C_n (with factor α), i.e., that ρ(C_n) = αC_n. Write down the system of equations and show that it corresponds to a single equation

C_n+1 - C_n + αβC_n-1 = m.

(b) For m ≠ 0 the equation in part (a) is a nonhomogeneous problem. Use the steps outlined in problem 7 to solve it.

i. Show that C_n = m/αβ is a particular solution.

ii. Find the general solution.

(c) Now consider the nature of this solution in two stages.

i. First assume that 4αβ < 1. Interpret this inequality in terms of the biological process. Give evidence for the assertion that under this condition, a steady blood CO₂ level C equal to m/αβ will eventually be established, regardless of the initial conditions. What will the steady ventilation rate then be?

ii. Now suppose 4aαβ > 1. Show that the CO₂ level will undergo oscillations. If αβ is large enough, show that the oscillations may increase in magnitude. Comment on the biological relevance of this solution.

(d) Suggest how the model might be made more realistic.