Problem 31
Question
Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium point of $$ \begin{array}{l} x_{1}(t+1)=a x_{2}(t) \\ x_{2}(t+1)=2 x_{1}(t)-\cos \left(x_{2}(t)\right)+1 \end{array} $$ Assume that \(a>0\). For which values of \(a\) is \(\left[\begin{array}{c}0 \\\ 0\end{array}\right]\) locally stable?
Step-by-Step Solution
Verified Answer
The equilibrium point is \(\left[\begin{array}{c}0 \\
0\end{array}\right]\) when \(a > 0\), and it is locally stable if \(-1 < a < 1\).
1Step 1: Understand the question
The question asks: Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium point of $$ \begin{array}{l} x_{1}(t+1)=a x_{2}(t) \\ x_{2}(t+1)=2 x_{1}(t)-\cos \left(x_{2}(t)\right)+1 \end{array} $$ Assume that \(a>0\). For which values of \(a\) is \(\left[\begin{array}{c}0 \\\ 0\end{array}\right]\)
2Step 2: Apply Biology knowledge
Using fundamental Biology concepts to analyze the problem.
3Step 3: Formulate the answer
The equilibrium point is \(\left[\begin{array}{c}0 \\ 0\end{array}\right]\) when \(a > 0\), and it is locally stable if \(-1 < a < 1\).
4Step 4: Conclude
The answer is: The equilibrium point is \(\left[\begin{array}{c}0 \\ 0\end{array}\right]\) when \(a > 0\), and it is locally stable if \(-1 < a < 1\).
Key Concepts
Stability AnalysisSystem of EquationsMathematical Biology
Stability Analysis
Stability analysis is a crucial step in understanding the behavior of equilibrium points in a system of equations. When analyzing stability, we want to know if a small disturbance around the equilibrium point will vanish over time or if it will grow and push the system away from equilibrium.
For a point to be stable, small deviations from this point must not result in the system moving away from the original point indefinitely. In this exercise, the stability of the point \( \begin{pmatrix} 0 \ 0 \end{pmatrix} \)in the system described is tested.
To determine stability, one approach is to perform a linearization of the system around the equilibrium. This involves finding the Jacobian matrix at the equilibrium point and analyzing its eigenvalues. If all the eigenvalues have absolute values less than 1, the equilibrium is considered locally stable. In thisway, stability analysis helps us understand the local behavior around equilibrium points and predict the system's response to small perturbations.
For a point to be stable, small deviations from this point must not result in the system moving away from the original point indefinitely. In this exercise, the stability of the point \( \begin{pmatrix} 0 \ 0 \end{pmatrix} \)in the system described is tested.
To determine stability, one approach is to perform a linearization of the system around the equilibrium. This involves finding the Jacobian matrix at the equilibrium point and analyzing its eigenvalues. If all the eigenvalues have absolute values less than 1, the equilibrium is considered locally stable. In thisway, stability analysis helps us understand the local behavior around equilibrium points and predict the system's response to small perturbations.
System of Equations
A system of equations is a set of equations with multiple variables that are solved together. In mathematical biology and other fields, such systems often arise to model dynamic changes in populations or other biological phenomena over time.
In this exercise, the given system is expressed as:
\[\begin{align*}x_{1}(t+1) &= ax_{2}(t) \x_{2}(t+1) &= 2x_{1}(t) - \cos(x_{2}(t)) + 1\end{align*}\] This represents a discreet-time recursion, where the next state of the variables \( x_1 \) and \( x_2 \) depends on their current states. Understanding such a system often involves finding fixed points or equilibrium points where the system does not change as time progresses.
One of the essential tasks is determining the conditions under which these points are stable. In practice, systems of equations like these are often used to model interactions in ecological systems or changes in concentrations of competing species.
In this exercise, the given system is expressed as:
\[\begin{align*}x_{1}(t+1) &= ax_{2}(t) \x_{2}(t+1) &= 2x_{1}(t) - \cos(x_{2}(t)) + 1\end{align*}\] This represents a discreet-time recursion, where the next state of the variables \( x_1 \) and \( x_2 \) depends on their current states. Understanding such a system often involves finding fixed points or equilibrium points where the system does not change as time progresses.
One of the essential tasks is determining the conditions under which these points are stable. In practice, systems of equations like these are often used to model interactions in ecological systems or changes in concentrations of competing species.
Mathematical Biology
Mathematical biology applies mathematical techniques to understand biological processes and systems, often involving complex interactions represented by systems of equations. It helps us make sense of intricate biological phenomena by providing precise and quantitative insights.
In this context, the system of equations focuses on modeling dynamics that could reflect biological processes, like competition between species or cellular interactions.
Analyzing such systems helps in understanding, for instance, how changing a parameter (such as \(a>0\) in this exercise) can affect population stability or predict long-term outcomes of biological processes. Moreover, integrating mathematical models into biology enables researchers to simulate scenarios that are otherwise challenging to manipulate experimentally.
For students of mathematical biology, learning to find equilibrium points and conducting stability analyses are critical steps in interpreting systems that have applications in anything from population dynamics to disease modeling.
In this context, the system of equations focuses on modeling dynamics that could reflect biological processes, like competition between species or cellular interactions.
Analyzing such systems helps in understanding, for instance, how changing a parameter (such as \(a>0\) in this exercise) can affect population stability or predict long-term outcomes of biological processes. Moreover, integrating mathematical models into biology enables researchers to simulate scenarios that are otherwise challenging to manipulate experimentally.
For students of mathematical biology, learning to find equilibrium points and conducting stability analyses are critical steps in interpreting systems that have applications in anything from population dynamics to disease modeling.
Other exercises in this chapter
Problem 30
a) Write $$ h(x, y)=\cos (y-x) $$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?
View solution Problem 31
Compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q\). $$ f(x, y)=2 x^{2} y-3 x, P=(2,1), Q=(3,2) $$
View solution Problem 31
Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{l} e^{x-y} \\ e^{x+y} \end{array}\right] $$
View solution Problem 31
In Problems \(31-38\), find \(\partial f / \partial x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. $$ f(x, y, z)=x^{2} z
View solution