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Chapter 3: Problem 11

Investigating parameter change. Using Maple or MATLAB examine the effect of increasing the parameter \(r\) on the solution to the equation $$ X_{n+1}=X_{n} e^{a\left(1-X_{n} / K\right)}, \quad \text { where } \quad a=\ell \mathrm{n}(r+1) $$ Establish (roughly) for what values of \(r\) the system undergoes its first two bifurcations. (Code can be adapted from that in Section 3.6.)

Short Answer

Expert verified
The first bifurcation typically occurs around \(r \approx 1.0\) and the second around \(r \approx 2.0\) depending on initial conditions.

Step by step solution

01

Understanding the Equation

The given equation is a discrete-time dynamical system represented by \(X_{n+1} = X_n e^{a(1-X_n/K)}\). The parameter \(a = \ln(r+1)\) determines the growth rate. The equation models scenarios such as population growth, where \(r\) is a key parameter that affects the behavior of the system.
02

Setting Up the Equation in MATLAB/Maple

To study the effect of changing \(r\), we will write code in MATLAB or Maple that defines the given recursive formula. We will increment \(r\) and observe the behavior of \(X_n\) for each value, tracking how the solution evolves over time.
03

Implementing Bifurcation Analysis

We track the system's behavior by plotting \(X_n\) against \(r\). For a stable solution, \(X_n\) converges to a constant value. Bifurcations occur when a small change in \(r\) causes the system's behavior to change drastically, such as periodic doubling or chaos.
04

Identifying First Bifurcation Point

By observing the plot, the first bifurcation typically occurs when a stable fixed point becomes unstable and gives rise to a periodic orbit. This "doubling" happens when \(r\) reaches a critical value, often \(r \approx 1.0\) or slightly above, depending on other parameters and initial conditions.
05

Identifying Second Bifurcation Point

The second bifurcation can be identified as another critical value of \(r\) where the period of the orbits doubles again or they become chaotic. This may often happen as \(r\) is further increased, generally around a value like \(r \approx 2.0\) but precise values can vary based on the specific setup.
06

Conclusion and Results

By gradually incrementing \(r\), analyze the plots to estimate the bifurcation points. If \(X_n\) begins to oscillate with a different period or shows chaotic behavior, it indicates a bifurcation. Using this method, we estimate the first two bifurcations.

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Key Concepts

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

Dynamical Systems
A dynamical system is a mathematical formulation that describes a process that evolves over time according to a set of defined rules. They are pivotal in understanding complex systems that change over certain conditions like time. In our exercise, the discrete-time equation \( X_{n+1} = X_n e^{a(1-X_n/K)} \) is a classic example of a dynamical system. Here, the evolution of \(X_n\) depends not only on the current state, represented by \(X_n\), but also on parameters like \(a\) and \(K\). Such systems can model various natural and engineered processes, with applications ranging from population dynamics to financial systems.

One of the fascinating aspects of dynamical systems is their ability to undergo bifurcations, where a small change in a parameter (like \(r\) in our equation) can lead to a dramatic change in behavior. This makes the study of dynamical systems essential in predicting and understanding potential outcomes based on different parameters.
Parameter Variation
Parameter variation is a crucial aspect of analyzing dynamical systems. In the context of our exercise, the parameter \(r\) is varied to observe how the system's behavior changes. The parameter \(r\) in the equation influences the growth rate since it impacts the parameter \(a\), where \(a = \ln(r+1)\).

By carefully adjusting \(r\), we can track specific changes in the system, identifying when and how the system undergoes bifurcations. This approach helps in understanding the sensitivity of the system to initial conditions and parameter changes. Tracking the evolution of \(r\) allows us to predict critical points where the system transitions from stable to periodic or even chaotic behavior, providing insights into the inherent complexities of dynamical systems.
MATLAB
MATLAB is a powerful computing environment often used for numerical simulations and data visualization, making it perfect for exploring dynamical systems. By using MATLAB, students can set up the recursive formula given in the exercise to study the effects of parameter changes on the discrete-time model.

With MATLAB, we can implement bifurcation analysis numerically by incrementing the parameter \(r\) and plotting \(X_n\) against it. MATLAB鈥檚 visualization tools allow us to readily observe how the system's behavior evolves as \(r\) increases. Code implementation becomes essential as it provides a concrete method to examine theoretical predictions. It transforms abstract equations into visual graphs, illustrating the bifurcation points and the eventual shift to chaos as parameters continue to vary.
  • Simplifies complex calculations
  • Provides clear graphical representations
  • Enables efficient simulations of parameter variations
Discrete-time Models
Discrete-time models are used to analyze systems that evolve in distinct steps, rather than continuously. In our exercise, the equation \( X_{n+1} = X_n e^{a(1-X_n/K)} \) is a discrete-time model because the system state at each step \(n\) is calculated only once, based on the previous state \(X_n\).

These models are prevalent in scenarios like population dynamics, where growth happens in jumps, for instance after each breeding season. In addition to biological systems, discrete-time models are also relevant in economics, meteorology, and digital signal processing, where changes happen at specific intervals. Understanding discrete-time models helps us visualize and predict the potential future states of a system by expressing changes step-by-step, offering a snapshot of the system's dynamics at distinct moments.

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Most popular questions from this chapter

Models for the spread of technology are very similar to the logistic model for population growth. Let \(N(t)\) be the number of ranchers who have adopted an improved pasture technology in Uruguay. Then \(N(t)\) satisfies the differential equation $$ \frac{d N}{d t}=a N\left(1-\frac{N}{N^{*}}\right) $$ where \(N^{*}\) is the total population of ranchers. It is assumed that the rate of adoption is proportional to both the number who have adopted the technology and the fraction of the population of ranchers who have not adopted the technology. (a) Which terms correspond to the fraction of the population who have not yet adopted the improved pasture technology? (b) According to Banks \((1994), N^{*}=17015, a=0.490\) and \(N_{0}=141 .\) Determine how long it takes for the improved pasture technology to spread to \(80 \%\) of the population. Note: This same model can be used to describe the spread of a rumour within an organisation or population.

Consider the population of a country. Assume constant per-capita birth and death rates, and that the population follows an exponential growth (or decay) process. Assume there to be significant immigration and emigration of people into and out of the country. (a) Assuming the overall immigration and emigration rates are constant, formulate a single differential equation to describe the population size over time. (b) Suppose instead that all immigration and emigration occurs with a neighbouring country, such that the net movement from one country to the other is proportional to the population difference between the two countries and such that people move to the country with the larger population. Formulate a coupled system of equations as a model for this situation. In both (a) and (b) start with appropriate word equations and ensure all variables are defined. Give clear explanations of how the differential equations are obtained from the word equations.

Many animal populations have decreasing per-capita birth rates when the population density increases, as well as increasing per-capita death rates. Suppose the density dependent per-capita birth rate \(B(X)\) and density dependent death rate \(A(X)\) is given by $$ B(X)=\beta-(\beta-\alpha) \delta \frac{X}{K}, \quad A(X)=\alpha+(\beta-\alpha)(1-\delta) \frac{X}{K} $$ where \(K\) is the population carrying capacity, \(\beta\) is the intrinsic per- capita birth rate, \(\alpha\) is the intrinsic per-capita death rate and \(\delta\), where \(0 \leq \delta \leq 1\), is a parameter describing the extent that density dependence is expressed in births or deaths. Show that this still gives rise to the standard logistic differential equation $$ \frac{d X}{d t}=r X\left(1-\frac{X}{K}\right) $$

Linear differential-delay equation. Consider the linear differential-delay equation $$ \frac{d X}{d t}=X(t-1), \quad X(0)=1 $$ Look for an exponential solution, of the form \(X(t)=C e^{m t}\) where \(m\) is a constant you must determine, and \(C\) is an arbitrary constant.

In view of the potentially disastrous effects of overfishing causing a population to become extinct, some governments impose quotas which vary depending on estimates of the population at the current time. One harvesting model that takes this into account is $$ \frac{d X}{d t}=r X\left(1-\frac{X}{K}\right)-h_{0} X $$ (a) Show that the only non-zero equilibrium population is $$ X_{e}=K\left(1-\frac{h}{r}\right) $$ (b) At what critical harvesting rate can extinction occur? Although extinction can occur with this model, as the harvesting parameter \(h\) increases towards the critical value the equilibrium population tends to zero. This contrasts with the constant harvesting model in Section \(3.3\) and Question 5 , where a sudden population crash (from a large population to extinction) can occur as the harvesting rate increases beyond a critical value.
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