New! Section 2.3 – Recurrence Equations (Recursions)
Instructions
- The following videos were recorded specifically for MATH 147.
- The first videos below explain the concepts in this section.
- This page also includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.
Concepts
- Explaining the definition of a recurrence equation, ratio of successive population, and the reproductive rate
- Explaining density-dependent population growth, difference equations, the Beverton-Holt Model, and Discrete Logistic Equations
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. A specific species of bird has a reproductive rate of \(\dfrac{7}{8}.\)
- What is the growth constant for this species of bird?
- Write the recursion equation for the population of this bird in implicit form, given 50 birds were placed in a sanctuary in 2020.
- Write the recursion equation for the population of this bird in the sanctuary in explicit form.
- How many birds are predicted to be in the sanctuary in 2026?
- \(R=\dfrac{15}{8}\)
- \(N_{t+1}=\dfrac{15}{8}N_{t},\) with \(N_0=50\) and 2020 is \(t=0\)
- \(N_{t}=50\cdot \left(\dfrac{15}{8}\right)^t\)
- \(N_6\approx 2,172\) birds
2. A university bird sanctuary is conducting a study of the ruby-throated hummingbirds which migrate through College Station each year. The sanctuary currently has 36 ruby-throated hummingbirds.
- Write a word equation relating the population of ruby-throated hummingbirds \(t\) years after the study begins, \(N_{t}\), to the population \(N_{t+1}\) in the next year.
- The sanctuary has noted that half of the hummingbirds are females and 83% of the females are of reproductive age. Each female of reproductive age will lay 4 eggs a year. Write an expression for the number of births per year.
- Ruby-throated hummingbirds females live approximately 6 years, while the males live only around 4 years. The bird sanctuary is also home to several hummingbird predators, which accounts for a loss of 5% of the hummingbirds each year. Write an expression for the number of deaths per year.
- Assuming no additional hummingbirds are introduced into the sanctuary or removed from the sanctuary, write a recurrence equation for the number of ruby-throated hummingbirds in the bird sanctuary.
- Using the recurrence equation we just wrote, calculate the population size of the red-throated hummingbirds in the sanctuary after 5 years.
- \(N_{t+1}=N_t+\) (births or immigration into sanctuary)\(-\) (deaths or emigration out of sanctuary)
- \(B=\dfrac{83}{50}N_t\)
- \(D=\dfrac{31}{120}N_t\)
- \(\displaystyle{N_{t+1}} = \displaystyle{\frac{1441}{600}N_{t}}\); \(N_{0} = 36\)
- \(2876\) or \(2877\) birds
3. Assume that the population growth for a population is described by the discrete logistic model:
\[\displaystyle{ N_{t+1} = \left\{ \begin{array}{lll} 2.5N_t - \frac{1}{30}\left(N_t \right)^{2} & \, & \text{if } N_{t} < 75 \\[0.25em] 0 & & \text{otherwise} \end{array} \right.}\]
Determine all fixed points for the model.
\(n=0\) or \(n=45\)
4. A drug company is developing a new antidepressant. Their mathematical model for the concentration, \(c_t,\) of the antidepressant in the blood \(t\) hours after a pill is taken has first order elimination kinetics. 7% of the antidepressant is eliminated from the blood each hour.
- Write down the recursion relation for \(c_{t+1}\) in terms of \(c_{t},\) if each pill is 61 mg per liter.
- If the drug company recommends patients take one pill every 12 hours, how much of the drug is in a patient's bloodstream 1 hour prior to them taking their second dosage?
- If the drug company recommends patients take one pill every 12 hours, how much of the drug is in a patient's bloodstream one hour after taking their second dosage?
- \(C_{t+1}=0.93C_t,\) \(C_0=61\)
- \(C_{11}\approx 27.5\) mg/liter
- \(C_{13}\approx 80.477\) mg/liter