Suppose that Fibonacci's rabbits live for only eight years,...

Name: Problem 26, Chapter 9: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2020-07-29T22:33:05-08:00
Duration: 1 min 3 s
Channel: Ankit Gupta
Description: Suppose that Fibonacci's rabbits live for only eight years, [44]. (a) Write out an iterative equation to describe the rabbit population. (b) Write down the first few terms. (c) Convert your equation into a first order iterative system, using the method of Exercise 9.1.21. (d) At what rate does the rabbit population grow?

Suppose that Fibonacci's rabbits live for only eight years, [44]. (a) Write out an iterative equation to describe the rabbit population. (b) Write down the first few terms. (c) Convert your equation into a first order iterative system, using the method of Exercise 9.1.21. (d) At what rate does the rabbit population grow?

Suppose that Fibonacci's rabbits live for only eight years, [44]. (a) Write out an iterative equation to describe the rabbit population. (b) Write down the first few terms.
(c) Convert your equation into a first order iterative system, using the method of Exercise 9.1.21. (d) At what rate does the rabbit population grow?

Key Concepts

Recurrence Relations

Recurrence relations are equations that define a sequence recursively; each term of the sequence is expressed as a function of its preceding terms. In population dynamics, they capture the idea that the current population depends on the past populations, often incorporating features such as reproduction and death. These relations are essential for modeling discrete-time processes where the state at each time step is determined by the previous states.

Age-Structured Population Models

Age-structured models divide a population into different 'age classes' to account for differences in birth and death rates among individuals at various life stages. When individuals have a finite lifespan, as in this problem, the population model must account for the fact that older individuals die after a certain age. This consideration leads to higher-order recurrences where the population at a given time depends only on contributions from individuals of reproductive age or those who have not yet expired.

Conversion to First-Order Iterative Systems

A higher-order recurrence relation can be transformed into a system of first-order equations by defining a state vector that includes consecutive past population values. This state-space or iterative system representation allows the use of linear algebra techniques to study the dynamics of the model. Such a conversion is particularly useful for analyzing the stability of equilibria and determining long?term behavior, especially through matrix representation.

Growth Rate and Dominant Eigenvalue

The growth rate of a population modeled by a discrete-time system is determined by the dominant eigenvalue of the matrix that represents the first-order iterative system. According to the Perron–Frobenius theorem, for non-negative matrices arising in population models, this dominant eigenvalue is real and reflects the asymptotic rate at which the population grows (or decays). It is a key parameter that characterizes the long-term behavior of the system.

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