Problem 3
Question
For each of the following systems, note the y-coordinate, \(M,\) of the horizontal asymptote, compute \(P_{10} / M\) and \(P_{20} / M\) and sketch (do not plot) graphs of \(P_{t}\) vs \(t\). a. \(P_{0}=20 \quad P_{t+1}=\frac{1.4 P_{t}}{1+0.4 \frac{P_{t}}{400}}\) b. \(P_{0}=20 \quad P_{t+1}=\frac{1.2 P_{t}}{1+0.2 \frac{P_{t}}{400}}\) c. \(P_{0}=20 \quad P_{t+1}=\frac{1.1 P_{t}}{1+0.1 \frac{P_{t}}{400}}\) d. \(P_{0}=20 \quad P_{t+1}=\frac{1.2 P_{t}}{1+0.2 \frac{P_{t}}{200}}\) e. \(P_{0}=20 \quad P_{t+1}=\frac{1.2 P_{t}}{1+0.2 \frac{P_{t}}{100}}\) f. \(\quad P_{0}=20 \quad P_{t+1}=\frac{1.1 P_{t}}{1+0.1 \frac{P_{t}}{100}}\)
Step-by-Step Solution
Verified Answer
Find M for each, calculate P_{10}/M and P_{20}/M, then sketch P_t vs t.
1Step 1: Identify the Horizontal Asymptote
The horizontal asymptote for these discrete, nonlinear recurrence relations is found by setting the expression equal to \(P_{t}\) when growth stabilizes (i.e., as \(t \to \infty\)). For general logistic equation \(P_{t+1} = \frac{1+kP_t}{1+m\frac{P_t}{N}}\), the horizontal asymptote \(M = N\) based on the coefficients.For each part:- (a) and (b) \(M = 400\) because of the \(\frac{P_t}{400}\) term.- (c) \(M = 400\)- (d) \(M = 200\)- (e) \(M = 100\)- (f) \(M = 100\)
2Step 2: Calculate Ratios \(\frac{P_{10}}{M}\) and \(\frac{P_{20}}{M}\)
To calculate \(\frac{P_{10}}{M}\) and \(\frac{P_{20}}{M}\), iterate the equation using the initial condition \(P_0 = 20\) and given recurrence relations to find \(P_{10}\) and \(P_{20}\). For example calculation in one scenario like part (a), continue iterating until reaching \(t = 10\) and then \(t = 20\). Use a simple program or calculator to do this. This is done numerically due to the iterative nature of the task. Thus, replace \(P_t\) with the calculated previous value in the relation \(P_{t+1} = \frac{1.4P_t}{1+0.4\frac{P_t}{400}}\).This process repeats for each configuration in parts (b), (c), (d), (e), and (f).
3Step 3: Sketch Graphs of \(P_t\) vs \(t\)
To sketch these graphs, note that \(P_t\) starts at \(P_0 = 20\) and will approach the horizontal asymptote \(M\) identified previously. Each system is a logistic-like map that will exhibit damped oscillations or converge from an exponential growth trajectory.For example:- In (a), sketch starting near \(20\), rapidly increasing, then oscillating below and above 400 before settling.- Repeat similarly adjusted for each case using the specific \(M\), where each chart will show \(P_t\) eventually stabilizing around the respective \(M\).Graphs convey how each logistic system behaves over time and approaches stability, underscoring resistance factors (denominator terms) defining long-term behavior.
Key Concepts
Nonlinear DynamicsLogistic EquationHorizontal AsymptoteDiscrete Systems
Nonlinear Dynamics
Nonlinear dynamics is a fascinating field of mathematics focusing on systems where the future state is not directly proportional to the current state. Unlike linear dynamics, where changes happen predictably, nonlinear dynamics often exhibit complex and unpredictable behavior. This is due to components such as feedback loops and multipliers that change the proportionality.
Understanding these dynamics is crucial when studying systems like populations or markets because they can help explain fluctuations and stability in these systems. In the context of this exercise, where discrete recurrence relations are present, the nonlinear dynamic component results from how the next state, denoted as \(P_{t+1}\), depends on \(P_t\) with a non-direct relationship. This is often captured through functions that involve quadratic or higher-order terms, as seen in the equations where the population grows but eventually stabilizes due to limitations or resistance factors.
Understanding these dynamics is crucial when studying systems like populations or markets because they can help explain fluctuations and stability in these systems. In the context of this exercise, where discrete recurrence relations are present, the nonlinear dynamic component results from how the next state, denoted as \(P_{t+1}\), depends on \(P_t\) with a non-direct relationship. This is often captured through functions that involve quadratic or higher-order terms, as seen in the equations where the population grows but eventually stabilizes due to limitations or resistance factors.
Logistic Equation
The logistic equation is a staple in modeling population growth that incorporates a carrying capacity—essentially a limit to how large the population can grow. It is a recursive formula used widely in ecology and other fields to describe how a species might grow in an environment with limited resources.
A general logistic equation takes the form \( P_{t+1} = \frac{rP_t}{1+m\frac{P_t}{N}} \), where \(r\) is the growth rate, \(m\) is a modifier that influences how quickly resistance kicks in, and \(N\) is the carrying capacity or the horizontal asymptote. As the population approaches \(N\), the growth rate slows, reflecting the impact of resource limitations. This creates the characteristic "S" shape of logistic growth, with rapid growth tapering off as it approaches the carrying capacity after an initial exponential rise.
A general logistic equation takes the form \( P_{t+1} = \frac{rP_t}{1+m\frac{P_t}{N}} \), where \(r\) is the growth rate, \(m\) is a modifier that influences how quickly resistance kicks in, and \(N\) is the carrying capacity or the horizontal asymptote. As the population approaches \(N\), the growth rate slows, reflecting the impact of resource limitations. This creates the characteristic "S" shape of logistic growth, with rapid growth tapering off as it approaches the carrying capacity after an initial exponential rise.
Horizontal Asymptote
In the context of recurrence relations, a horizontal asymptote is a critical value that represents the long-term behavior of the system. It is the value the system will approach as time extends into infinity. For the logistic equation discussed in the exercise, the horizontal asymptote \(M\) is often set as the carrying capacity \(N\).
The horizontal asymptote helps to understand the equilibrium state, showing where the population ends to stabilize over time. In the recurrence relations provided, the term \(\frac{P_t}{N}\) signifies the diminishing returns as \(P_t\) grows large. Determining \(M\) involves looking at the coefficients in the equation, identifying the environmental limitations that bind the population's ultimate size in theory. Each value of \(M\) shows how robust these environmental constraints are.
The horizontal asymptote helps to understand the equilibrium state, showing where the population ends to stabilize over time. In the recurrence relations provided, the term \(\frac{P_t}{N}\) signifies the diminishing returns as \(P_t\) grows large. Determining \(M\) involves looking at the coefficients in the equation, identifying the environmental limitations that bind the population's ultimate size in theory. Each value of \(M\) shows how robust these environmental constraints are.
Discrete Systems
Discrete systems are mathematical models that operate in distinct, separated intervals. Unlike continuous systems that evolve smoothly over time, discrete systems change in steps, making them ideal for processes tracked in intervals, like days or years.
In our exercise, each time point \(t\) in the recurrence relations represents a discrete step, usually a generation or a specific period. These systems are often modeled using iterative equations that define how the state changes over each unit of time. Discrete modeling is powerful for understanding processes in natural systems where continuous measurement is impractical.
They offer a clear picture of states at specific points and are particularly useful in computing progressive iterations of population models. In our logistic equations, they help provide snapshots of population sizes over discrete time intervals, revealing trends like growth plateaus as they interact with potential Capsavra constraints.
In our exercise, each time point \(t\) in the recurrence relations represents a discrete step, usually a generation or a specific period. These systems are often modeled using iterative equations that define how the state changes over each unit of time. Discrete modeling is powerful for understanding processes in natural systems where continuous measurement is impractical.
They offer a clear picture of states at specific points and are particularly useful in computing progressive iterations of population models. In our logistic equations, they help provide snapshots of population sizes over discrete time intervals, revealing trends like growth plateaus as they interact with potential Capsavra constraints.
Other exercises in this chapter
Problem 2
Plot the graph of $$ w_{0}=2, \quad w_{t+1}=5.1 \times \frac{w_{t}}{5+w_{t}} $$
View solution Problem 2
Compute \(Q_{2}, Q_{3}, Q_{4}\) and \(Q_{5}\) for $$ \begin{array}{llll} \text { a. } Q_{0}=0.1 & Q_{1}=0.1 & Q_{t+2} & =Q_{t+1}+0.06 \times\left(1-Q_{t}\right)
View solution Problem 3
Show that if \(B\) is a number greater than \(1,\) and \(n\) is a positive integer, $$ \lim _{t \rightarrow \infty} \frac{B^{t}}{t^{n}}=\infty $$
View solution Problem 3
Draw the graphs of the iteration function \(y=f(x)\) and the diagonal \(y=x\). For each equilibrium, choose a value of \(x_{0}\) close to but distinct from that
View solution