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Chapter 2: Problem 46

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.1, x_{0}=0\)

Short Answer

Expert verified
For \(x_0 = 0\), all \(x_t\) values remain zero. The graph is a flat line at zero.

Step by step solution

01

Understand the Logistic Equation

The discrete logistic equation is a mathematical model used to describe population dynamics. Given by \(x_{t+1} = R_{0} x_{t}(1-x_{t})\), it models the changing population \(x_{t}\) over time \(t\). The variable \(R_{0}\) is a growth rate parameter, and \(x_{0}\) is the initial population at \(t=0\). In this problem, \(R_{0} = 3.1\) and \(x_{0} = 0\).
02

Calculate First Iteration

Calculate \(x_{1}\) using \(x_{t+1} = R_{0} x_{t}(1-x_{t})\). With \(x_{0} = 0\), we have:\[x_{1} = 3.1 \times 0 \times (1-0) = 0.\]The population remains zero at the next time step.
03

Recognize the Pattern

Notice that when \(x_{t}=0\), it leads to \(x_{t+1} = 0\) because any number multiplied by zero results in zero. Therefore, \(x_{2}, x_{3}, \ldots, x_{20}\) will all remain zero.
04

Graph the Solution

Since \(x_{t}\) evaluates to zero for all \(t\), plot \(x_{t}\) on the vertical axis and time \(t\) on the horizontal axis. The graph will be a horizontal line at zero, illustrating that the population does not change from its initial state.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Dynamics
Population dynamics refer to the branch of life sciences that studies short and long term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes. In this context, the discrete logistic equation is fundamental as it models how populations grow and reach their carrying capacity, or the maximum population size that an environment can sustainably support. This particular equation, expressed as \( x_{t+1} = R_{0} x_{t}(1-x_{t}) \), assumes that the population changes at discrete time intervals and considers both the growth potential and the limitations imposed by environmental factors.

For instance, if the initial population \( x_0 \) starts at zero, and considering that any value multiplied by zero remains zero, each subsequent calculation of the population size over time will remain at zero. This can help you understand scenarios where the initial population is not sufficient enough to lead to growth, which might reflect real-world cases where resources or conditions are inadequate for sustaining life.
Growth Rate Parameter
The growth rate parameter, denoted as \( R_0 \) in the discrete logistic equation, plays a crucial role in determining how quickly a population can grow. This parameter represents the intrinsic growth rate of the population, indicative of how fast the population can increase without any limits initially imposed by environmental resistance. In the given problem, \( R_0 = 3.1 \), suggesting a high growth potential under optimal conditions.

However, this growth potential does not always equate to a rising population, as seen when \( x_0 = 0 \). Even with a theoretical model that allows rapid growth, actual growth is dependent on initial quantities and conditions, like availability of resources and space.
  • A high \( R_0 \) can suggest faster growth in early stages before reaching saturation.
  • An \( R_0 \) less than 1 can indicate a declining population.
Ultimately, \( R_0 \) helps predict possible outcomes of population changes under different settings.
Mathematical Model
A mathematical model is a systematic approach used in science to represent real-world systems using mathematical language and concepts. The discrete logistic equation is one such model that effectively represents the theory of population dynamics. These models allow scientists and researchers to predict future changes and trends within populations using mathematical expressions.

The equation \( x_{t+1} = R_{0} x_{t}(1-x_{t}) \) specifically models populations under constraints, taking into account growth that is initially exponential but ultimately limited by environmental capacities. This makes it useful not only in theoretical studies but also in practical applications like wildlife management, resource allocation, and ecological forecasting.
  • Models help simplify complex biological systems.
  • They provide insights into population stability and potential changes over time.
Though simplified, they offer valuable predictions that can be used for planning and decision-making in sustainability and conservation projects.

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Most popular questions from this chapter

Mountain Gorilla Conservation You are trying to build a mathematical model for the size of the population of mountain gorillas in a national park in Uganda. The data in this question are taken from Robbins et al. (2009). (a) You start by writing a word equation relating the population of gorillas \(t\) years after the study begins, \(N_{t}\), to the population \(N_{t+1}\) in the next year: \(N_{t+1}=N_{t}+\begin{array}{l}\text { number of gorillas } \\ \text { born in one year }\end{array}-\begin{array}{l}\text { number of gorillas } \\\ \text { that die in one yeai }\end{array}\) We will derive together formulas for the number of births and the number of deaths. (i) Around half of gorillas are female, \(75 \%\) of females are of reproductive age, and in a given year \(22 \%\) of the females of reproductive age will give birth. Explain why the number of births is equal to: \(\quad 0.5 \cdot 0.75 \cdot 0.22 \cdot N_{t}=0.0825 \cdot N_{t}\) (ii) In a given year \(4.5 \%\) of gorillas will die. Write down a formula for the number of deaths. (iii) Write down a recurrence equation for the number of gorillas in the national park. Assuming that there are 300 gorillas initially (that is \(N_{0}=300\) ), derive an explicit formula for the number of gorillas after \(t\) years. (iv) Calculate the population size after \(1,2,5\), and 10 years. (v) According to the model, how long will it take for population size to double to 600 gorillas? (b) In reality the population size is almost totally stagnant (i.e., \(N_{t}\) changes very little from year to year). Robbins et al. (2009) consider three different explanations for this effect: (i) Increased mortality: Gorillas are dying sooner than was thought. What percentage of gorillas would have to die each year for the population size to not change from year to year? (ii) Decreased female fecundity: Gorillas are having fewer offspring than was thought. Calculate the female birth rate (percentage of reproductive age females that give birth) that would lead to the population size not changing from year to year. Assume that all other values used in part (a) are correct. (iii) Emigration: Gorillas are leaving the national park. What number of gorillas would have to leave the national park each year for the population to not change from year to year?

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=4, a=1 / 40, N_{0}=2\)

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=e^{-3 n}, \epsilon=0.001 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.05 $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{3}{a_{n}+2}, a_{0}=0 $$
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