Modeling Nonlinear Discrete Dynamical Systems
In this section, we build nonlinear DDS to describe the change in behavior of the quantities we study. We also will study systems of DDS to describe the changes in various quantities that act together in some way or influence each other in deterministic ways. We define a nonlinear DDS by
If the DDS a(n + 1) = f(a(n),... ,n) involves powers of a(k) [such as a2(k), or a functional relationship [such as a(k) ja(k — 1) or sin(«(A~))], we say the discrete dynamical system is nonlinear.
We will restrict our investigations to numerical and graphical solutions of nonlinear models. Analytical solutions of nonlinear models are studied in more advanced mathematics courses.
We often model population growth by assuming that the change in population is directly proportional to the current size of the given population.
This produces a simple, first-order DDS similar to those seen earlier. It might appear reasonable at first examination, but the long-term behavior of growth without bound is disturbing. Why would growth without bound of a yeast culture in a jar (or confined space) be alarming?[1]
There are certain factors that affect population growth. These factors include resources: food, oxygen, space, etc. The available resources can support some maximum population. As this number is approached, the change, or growth rate, should decrease and the population should never exceed its resource supported amount. If the population does exceed the maximum supported amount, the growth should become negative.
Example 2.11. Growth of a Cancer Culture.
Problem.
To determine whether our treatments are successful, we need to predict the growth of cancer cells in a controlled environment as a function of the resources available and the current cell 'population.'
Assumptions and Variables.
We assume that the population size is best described by the weight of the biomass of the culture. Define y{n) as the population size of the cell culture after period n. There exists a maximum carrying capacity M that is sustainable by the resources available. The culture is growing under the conditions established.
Model.
where к is the constant of proportionality (a function of growth rate).
Using the data collected in our experiment, we first plot y(n) versus n and find a stable ev of approximately 670. Next, we plot y(n + 1) — y(n) versus t/(n)(670 — y(n)) to find the slope, which gives k, is approximately 0.00090. With к = 0.00090 and carrying capacity in biomass is 670. Our specific model[2] is then
Again, this is nonlinear because of the y2 (n) term (after expanding the right side of the model). There is no closed-form analytical solution for this model. The numerical solution iterated in Maple from an initial condition of biomass = 30.0 follows.
We use option remember in the procedure to ‘remember’ (store) values of biomass as they are computed in order to dramatically reduce the number of calculations needed. Try it without the option!
The model shows stability in that the population (biomass) of the cell culture quickly approaches 670 as n gets large. Thus, the population is eventually stable at approximately 670 units. We would then proceed with treatments, collect new data, and evaluate the new trend of the biomass to determine whether the treatment is successful or not.
Example 2.12. Spread of a Contagious Disease.
There are 1000 students in a college dormitory. Several students have returned from a Spring Break trip where they were exposed to mumps, a highly contagious disease. The Health Center wants to build a model to determine how fast the disease will spread in order to determine effective vaccination and quarantine procedures.
Problem.
Predict the number of students affected with mumps as a function of time. Assumptions and Variables.
We assume all students are susceptible to the disease. The possible contacts of infected and susceptible students are proportional to their product (as an interaction term). Let m(n) be the number of students affected with mumps after n days.
Model.
Two students have come down with mumps. The rate of spreading per day is characterized by к = 0.0005. A vaccine can be delivered and students vaccinated within 1-2 weeks. Hence
This model matches the cancer model above. We modify its Maple code accordingly.
We rounded the value of Mumps as it represents an integer: the number of infected students.
Graph the points to see the trend of the disease.
By 21 days, essentially all the students are infected. Let’s look at the weekly counts.
Interpretation.
The results show that essentially all the students will be infected by 3 weeks. Since only about 3% will be infected within one week, every effort must be made to get the vaccine to the school and to vaccinate the students within one week.
We can model the effect of the vaccine on the spread of mumps by reducing к in the model and using Mumps(7) as the initial value of the new curve.
Exercises
Iterate and graph the following DDSs. Explain their long-term behavior. For each DDS, find a realistic scenario that it might explain/model.
- 1. Consider the model a(n+l) = r-a(n)(l —o(n)). Let a(0) = 0.2. Determine the numerical and graphical solutions for the following values of r. Find the pattern in the solution.
- (a) r = 2
- (b) r = 3
- (c) г = 3.6
- (d) г = 3.7
- 2. Find the equilibrium value of the given DDS by iteration. Determine if the equilibrium value is stable or unstable.
- (a) a(n + 1) = 1.7a(n) — 0.14a(n)2
- (b) a(n + 1) = 0.8a(n) + 0.1a(n)2
- (c) a(n + 1) = 0.2a(n) — 0.2a(n)3
- (d) a(n + 1) = 0.1a(n)2 + 0.9a(n) — 0.2
3. A rumor spreads through a company of 1000 employees all working in the same building. We assume that the spread of a rumor is similar to the spread of a contagious disease in that the number of people hearing the rumor each day is proportional to the product of the number hearing the rumor and the number who have not yet heard the rumor. This model is given by
where к is the parameter that depends on how fast the rumor spreads. Assume к = 0.001. Further assume that 4 people initially know the rumor. How soon will everyone know the rumor?
Projects
Project 2.1. Consider the highly contagious and deadly Ebola virus, which in 2018 appeared to be spreading again. Determine how deadly this virus actually is. (Visit https://www.cdc.gov/vhf/ebola/index.html.) Consider an animal research laboratory in Restin, VA (pop. ~ 58,000), a suburb of Washington, DC, (pop. ~ 602,000). A monkey carrying the Ebola virus has escaped its cage and infected one employee (unknown at the time) during its escape from the research laboratory. This employee reports to University hospital later with Ebola symptoms. The Centers for Disease Control and Prevention (CDC) in Atlanta gets a call and begins to model the spread of the disease. Build a model for the CDC with the following growth rates to determine the number infected after 2 weeks:
- (a) к = 0.00025
- (b) k = 0.000025
- (c) к = 0.00005
- (d) k = 0.000009
List some possible ways of controlling the spread of the virus.
Project 2.2. A rumor concerning termination spreads among 10,000 employees of a major company. Assume that the spreading of a rumor is similar to the spread of contagious disease in that the number hearing the rumor each day is proportional to the product of those who have heard the rumor and those who have not yet heard the rumor. Build a model for the company with the following rumor growth rates to determine the number having heard the rumor after 1 week:
- (a) k = 0.25
- (b) k = 0.025
- (c) lb = 0.0025
- (d) k = 0.00025
List some possible ways of controlling the spread of the rumor.
