Math 121 - Calculus for Biology I Spring Semester, 2009 Linear Discrete Dynamical Models
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1. Lung Diseases and Modeling Breathing
2. Equilibrium and Cobwebbing
3. Malthusian Growth Model with Emigration
4. Stability of a Linear Discrete Dynamical Model
5. Worked Examples
6. References

• Discrete Malthusian growth models are linear models
  • A constant growth rate times the dependent discrete variable, P_n
  • A time varying growth rate times the dependent discrete variable, P_n
• This section examines a model for breathing and population models that include either immigration or emigration
• These models are linear, but include a constant term
• Begin the study of qualitative analysis of these models

Lung Diseases and Modeling Breathing

• Pulmonary ventilation or breathing brings oxygen to the cells of the body and removes metabolic waste product, carbon dioxide.
• Inspiration or inflow of fresh air results from the contracting the muscles of the diaphragm
• Relaxation of these muscles or contraction of the abdominals causes expiration of air with the waste product CO₂
• Normal respiration in the lungs exchanges about 500 ml of air 12 times a minute
• Tidal volume is the normal volume of air inspired or expired
• The inspiratory reserve volume is about 3000 ml that can be inspired above the tidal volume
• The expiratory reserve volume is about 1100 ml, which can be forcefully expired
• The vital capacity includes all of the above yielding about 4600 ml
• Well-trained athletes may have values 30-40% higher
• Females generally have 20-25% less for the quantities listed above
• The lungs contain surfactants, which prevent them from totally collapsing
  • Expelling all air requires too much energy to reinflate them from the collapsed state
  • Premature babies born before their genes for producing surfactants have turned on use more energy to breath than they derive from the process of breathing
• The residual volume is the amount of air that cannot be expelled and averages about 1200 ml
• The functional residual capacity is the amount of air that remains behind during normal breathing, which amounts to 2300 ml 

• The body depends on an adequate supply of oxygen to the tissues, which depends on respiration through the lungs.
• Several respiratory diseases jeopardize the vital function of the lungs.
  • The respiratory muscles damaged by spinal paralysis or poliomyelitis decreases the vital capacity to as low as 500 ml
  • The pulmonary compliance reduces vital capacity in diseases like tuberculosis, emphysema, chronic asthma, lung cancer, chronic bronchitis, cystic fibrosis, or fibrotic pleurisy.
  • Several of the diseases above and heart disease can cause pulmonary edema, decreasing vital capacity from fluid build up
• The alveoli are where the oxygen actually enters the blood
• When the alveoli are damaged or filled with fluid (one result of smoking), the exchange of oxygen is inhibited

• The vital capacity and residual volume help physiologists determine the health of the pulmonary system
• The vital capacity is easily measured by taking a deep breath and expiring into a spirometer
• Some of the diseases, like emphysema, are determined by the tidal volume and the functional residual capacity, the average or minute respiratory volume
• When the ratio of the tidal volume to the functional residual capacity becomes too low, then there is insufficient exchange of air to maintain adequate supplies of oxygen to the body

Discrete Model for Breathing

• Tidal volume and the functional residual capacity is found by breathing a mixture including an inert gas
  • The subject breathes the mixture until the lungs are essentially filled with this mixture
  • A physiologist measures the amount of the inert gas in a series of breaths after the subject is removed from the gas mixture to normal air
• The mathematical model for this experiment is a discrete dynamical system

• Professor Bruce Wingerd ran dilution experiments with the inert gas argon (Ar)
• Argon is a noble gas, so is totally non-reactive
• It is the third most common gas, comprising 0.93% of Earth's atmosphere (dry air has Nitrogen, N₂, 78%, Oxygen, O₂, 21%, with CO₂ a distant fourth, 0.03%)
• The subjects breathe an air mixture with 10% Ar
• Then the subjects resume breathing normal air at a normal rate
• Below are tables that show the percent of Ar in each of the next six breaths along with the tidal volume is given by the average volume in each breath.

• Use these data to determine the functional reserve volume for our subjects
• Determine the vital capacity using a spirometer
• A physiologist uses this information to find the health of a subject's lungs.

• This breathing experiment is a dynamic exchange of gases, which occurs at discrete intervals of time
• Suggests a discrete dynamical model tracking the concentration of Ar in the lungs
• Define the concentration of Ar at the end of the n^th inspiration cycle as c_n
• The concentration at the end of the (n + 1)^st inspiration cycle depends on remaining air in the lungs from the previous cycle and inhaling fresh air from the atmosphere
  • Assume the gases become well-mixed during this process, which ignores some of the complications caused by the actual physiological structures in the lungs, such as the "anatomical dead space" in the pharynx, trachea, and larger bronchi or weak mixing from slow gas flow in the alveoli
  • In fact, of the 500 ml of fresh air brought in by inspiration, only about 350 ml reaches the alveoli, which means that there is less than a seventh exchange of gases with a normal breath

• The physiological parameters for this model
  • V_i for the tidal volume (air normally inhaled and exhaled)
  • V_r for the functional residual volume
  • g for the concentration of Ar in the atmosphere
• Let q = V_i/(V_i + V_r) be the fraction of atmospheric air exchanged in each breath
• Thus, V_r/(V_i + V_r) = (1 - q) is the residual air fraction
• Upon exhaling, there remains behind the functional residual volume with the amount of Ar given by V_r c_n
• The inhaled air during this cycle contains the amount of Ar given by V_i g
• Quantities or amounts of Ar are given by the volume times concentration
• The amount of Ar in the next breath is given by

• The concentration of Ar is found by dividing by the total volume, V_i + V_r
• The model becomes

• Use the value of q to produce the linear discrete dynamical model for breathing
• Linear Discrete Dynamical Model for Breathing an Inert Gas

Below is a graph of the data above with the best fitting model for breathing

Finding the Functional Reserve Capacity

• The object of the experiments was to find the functional reserve capacity
• The diseased states are often characterized by a decreased ratio between the tidal volume and the functional reserve capacity
• Emphysema is characterized by a loss of elasticity in the lungs and a decrease in the alveolar surface/volume ratio

• The discrete dynamical model for breathing an inert gas is solved for the parameter q

 

• From the data for the normal subject, q = (0.1 - 0.084)/(0.1 - 0.0093) = 0.18
• The volume of the functional reserve capacity, V_r , satisfies

• From the data above, V_r = 0.82(550)/0.18 = 2500 ml
• The ratio of the tidal volume to the functional reserve capacity is 0.22

• A similar analysis of the subject with emphysema gives q = (0.1 - 0.088)/(0.1 - 0.0093) = 0.13
• The functional reserve capacity for the subject with emphysema is V_r = 0.87(250)/0.13 = 1670 ml
• The ratio of the tidal volume to the functional reserve capacity is 0.15, significantly smaller than the one for the normal subject

Equilibrium and Cobwebbing

• The discrete dynamical model for breathing shows the concentration of Ar decreasing
• If the simulation for the normal individual is carried out for about 3 minutes or 36 breaths, it can be seen that the concentration of Ar drops to 0.0094, which is within 1% of the atmospheric concentration
• Since Ar is an inert gas, we expect that after breathing an enriched source of Ar, then the concentration should return to the same value as normally found in the atmosphere
• This value is the equilibrium value of Ar for the model

• Consider a discrete dynamical system given by the equation

where f(x_n) is any function describing the dynamics of the model

• An equilibrium, x_e, for this discrete dynamical system is achieved if x_n+1 = x_n = x_e
• The dynamic variable settles into a constant value for all n.

• The mathematical model for breathing should reach an equilibrium value that corresponds the the value of Ar in the atmosphere
• Let c_n+1 = c_e and c_n = c_e
• The discrete dynamical model for breathing becomes

• This is easily solved and gives c_e = g, as expected

Cobwebbing

• Graphic depiction of discrete dynamical models
• The general model above states that x_n+1 = f(x_n)
  • The (n + 1)^st state of the model, x_n+1, is a function depending on the n^th state of the model, x_n
  • Graph with the variable x_n+1 on the vertical axis and x_n on the horizontal axis
  • Draw the graph of x_n+1 = f(x_n) and the line x_n+1 = x_n
• A process called cobwebbing allows us to view the dynamics of this discrete dynamical model

• Start at some point x₀ on the horizontal axis, then go vertically to f(x₀) to find x₁
• Next go horizontally until we hit the line x_n+1 = x_n
• From here go vertically to f(x₁) to find x₂
• The process is repeated to give a geometric view of the discrete dynamical model
• At any point where the function f(x_n) crosses the line x_n+1 = x_n, there is an equilibrium for the model

• The cobwebbing technique is illustrated with the discrete dynamical model for breathing
• Below is a graph showing the simulation for the normal subject listed in the table above

• Notice that for the breathing model, the concentration, c_n, tends towards the equilibrium concentration, c_e = g
• When the solution approaches the equilibrium for large n, then the equilibrium is said to be stable

Malthusian Growth Models with Immigration or Emigration

• In the previous section, we examined the discrete Malthusian growth model for the U. S. population
• A simple Malthusian growth model has a limited value for studying the U. S. population though the nonautonomous Malthusian growth model substantially improved our predictions
• These models only account for the net growth of the population in what is considered a closed system, since it acts as if the population is totally dependent on the population being studied
• Throughout U. S. history, our population has been significantly affected by the rate of immigration
• Through much of the 20^th century, the government has regulated legal immigration to 250,000 people per year.

• The discrete Malthusian growth model is easily modified to account for either immigration or emigration
• Suppose that a population, P_n, grows according to the discrete Malthusian growth model
• Assume that a constant number of the population leaves or emigrates in each time interval
• The mathematical model for this behavior is given by the equation:

where r is the rate of growth and m is the constant number emigrating

• Assuming that the constants are known and if the initial population , P₀, is given, then it is easy to determine all subsequent populations by iteration
• This model is similar to the breathing model above in that the discrete dynamical model is linear, that is the right hand side of the equation is only a linear function of P_n

• Notice what happens if we attempt to iterate this model starting with P₀
• We obtain

• It is not hard to work the algebra and see that

which simplifies to

• This solution is clearly more complicated and harder to obtain than the previous section for the discrete Malthusian growth model
• However, few other discrete dynamical models have a solution that can be written as a formula depending only on the parameters, n, and P₀, a closed form solution
• In the next section, we will study a model with a simple quadratic term on the right side of the equation, yet this discrete dynamical model will have no closed form solution
• It can only be simulated one step at a time to determine the exact value of P_n.

• When the solution becomes complicated or impossible to find exactly, then we still would like to obtain some information about the qualitative behavior of the model
• The cobwebbing technique illustrated above gives us some ideas on studying the behavior
• Below is the cobwebbing diagram for the discrete Malthusian growth model with emigration, where r = 0.2 and m = 500

• The green line represents the model, P_n+1 = 1.2 P_n - 500, while the blue line is what is known as the identity map, P_n+1 = P_n
• These lines intersect at the equilibrium point, which solves the equation,

• One significant difference between this dynamical model and one for the breathing is that iterations of the solution are going away from the equilibrium
• Thus, if the population begins above 2500, then it grows increasingly larger, much like we saw in the previous section for the Malthusian growth model
• However, if the population starts below 2500, then more animals leave than can be replaced, so the population is driven to extinction
• The equilibrium for the discrete Malthusian growth model with emigration is said to be unstable.

Stability of a Linear Discrete Dynamical Model

• Stability of an equilibrium for a Discrete Dynamical Model is important for understanding how the model behaves
• Consider the Linear Discrete Dynamical Model

y_n+1 = a y_n + b

• The equilibrium satisfies

  

• For modeling situations, the equilibrium (if it exists) must be positive (or zero), thus
  • Either a < 1 and b > 0
  • Or a > 1 and b < 0
• Linear discrete dynamical models have a single unique equilibrium if the slope of the linear function, a, is not 1
  • If a = 1, then either there are no equilibria or all points are equilibria (b = 0)
• An equilibrium of a linear discrete dynamical model is stable if either of the following conditions hold:
  1. Successive iterations of the model approach the equilibrium.
  2. The slope |a| is less than 1.

• An equilibrium of a linear discrete dynamical model is unstable if either of the following conditions hold:
  1. Successive iterations of the model move away from the equilibrium.
  2. The slope |a| is greater than 1.

Worked Examples

References:

[1] A. C. Guyton and Hall (1997), Textbook of Medical Physiology, W. B. Saunders Co., Philadelphia