Question

Can be interpreted as describing the interaction of two species with population densities \(x\) and \(y .\) In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Draw a phase portrait for the system. (f) Determine the limiting behavior of \(x\) and \(y\) as \(t \rightarrow \infty\) and interpret the results in terms of the populations of the two species. $$ \begin{array}{l}{d x / d t=x\left(-1+2.5 x-0.3 y-x^{2}\right)} \\ {d y / d t=y(-1.5+x)}\end{array} $$

Short answer

Question: Determine the limiting behavior of the given system of differential equations describing the interaction of two species with population densities \(x\) and \(y\) as \(t \rightarrow \infty\), and interpret the results in terms of the populations of the species. Step 1: Draw a direction field and describe how solutions seem to behave. Step 2: Find the critical points. Step 3: Linearize, find eigenvalues and eigenvectors, and classify stability. Step 4: Sketch the trajectories near critical points. Step 5: Draw a phase portrait for the system. Step 6: Analyze the phase portrait and study the trajectories to determine the limiting behavior of \(x\) and \(y\) as \(t\) approaches infinity. Interpret the results in terms of the populations of the two species.

Step-by-step solution

Draw a direction field and describe how solutions seem to behave.

In order to plot the direction field, we need to evaluate the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) at various points on the plane. The direction field will give us an idea of how solutions might behave in the plane. Computer software like Matlab, Mathematica, or an online direction field generator can help visualize the direction field by plotting arrows. After drawing a direction field, look for patterns and general solution behavior. For example, you may notice trajectories converging, diverging, or cycling around certain points.

Find the critical points.

Critical points occur when both \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\). To find these, set both expressions to 0 and solve for \(x\) and \(y\). This will give us a set of points \((x, y)\) where the system is stationary, meaning the population densities don't change at these points. $$ \begin{cases} 0=x(-1+2.5x-0.3y-x^2)\\ 0=y(-1.5+x) \end{cases} $$ Solve these equations simultaneously to find the critical points.

Linearize, find eigenvalues and eigenvectors, and classify stability.

First, compute the Jacobian matrix \(J(x,y)\) of the given system, which represents the partial derivatives of the system with respect to \(x\) and \(y\): $$ J(x,y)= \begin{bmatrix} \frac{\partial}{\partial x}(x(-1+2.5x-0.3y-x^2)) & \frac{\partial}{\partial y}(x(-1+2.5x-0.3y-x^2))\\ \frac{\partial}{\partial x}(y(-1.5+x)) & \frac{\partial}{\partial y}(y(-1.5+x)) \end{bmatrix} $$ Evaluate \(J(x,y)\) at each critical point found in Step 2. Next, find eigenvalues and eigenvectors for each linearized system. Analyze eigenvalues to classify stability; if both eigenvalues have negative real parts, the critical point is asymptotically stable. If at least one eigenvalue has a positive real part, the critical point is unstable.

Sketch the trajectories near critical points.

Using the information gathered from Steps 1 and 3, sketch the trajectories near each critical point. Take into account the type of critical point (stable, unstable, etc.) and the direction field.

Draw a phase portrait for the system.

Combine the information from Steps 1-4, including the direction field and the behavior near critical points. Draw a phase portrait, which represents the possible trajectories of the system.

Determine limiting behavior as \(t \rightarrow \infty\) and interpret the results.

Analyze the phase portrait and study the trajectories to determine the limiting behavior of \(x\) and \(y\) as \(t\) approaches infinity. Interpret the results in terms of the populations of the two species; for example, if a trajectory converges to a stable critical point, you may conclude that as time goes on, the population densities of both species will reach equilibrium at the corresponding values. If a trajectory diverges, then the populations may become unbounded, and one or both species may eventually die out. In conclusion, following these six steps will help analyze the given system of differential equations, and describe the interaction of the two species modeled by the given population densities \(x\) and \(y\).

Key concepts

Critical Points

Critical points in the context of a system of differential equations are vital to understand the system's behavior. They serve as the 'crossroads' in a population model, indicating where the population densities cease changing, thus predicting potential equilibrium states. To find these, one must set the rate of change to zero, which usually leads to a set of algebraic equations.

For instance, in a system where populations are denoted by variables like \(x\) and \(y\), the equilibrium points are found by solving \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\). The solutions for \(x\) and \(y\) yield the coordinates of these critical points. Understanding where these points lie and how they behave is central to predicting long-term outcomes in population dynamics.

Jacobian Matrix

The Jacobian matrix plays a quintessential role in studying the local behavior of a system of differential equations near critical points. It is a matrix of first-order partial derivatives and is used to linearize the system around these points.

In essence, the Jacobian offers a snapshot of the system's dynamics by quantifying the rate at which variables affect each other. For a system like \( \frac{dx}{dt} = f(x, y)\) and \(\frac{dy}{dt} = g(x, y)\), the Jacobian matrix is given by: \[ J(x,y)= \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix} \] It provides vital clues about stability and behavior at the critical points.

Phase Portrait

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each curve in a phase portrait represents a possible pathway that the system’s state can follow over time.

In population dynamics for two species, visualizing a phase portrait allows for an easy grasp of complex interactions. For example, orbits closing in on a critical point suggest a stable population equilibrium, while spirals or outward trajectories might represent oscillation or instability. By adding arrows to indicate directionality and stability, a phase portrait becomes a powerful tool for visualizing and understanding the flow and long-term behavior of the given populations.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors offer crucial insights into the nature of critical points. In a nutshell, eigenvalues tell us whether the populations will return to equilibrium following a perturbation, remain constant, or diverge away. The associated eigenvectors, on the other hand, inform us about the direction of this movement.

To delve deeper, the sign and magnitude of eigenvalues from the Jacobian matrix at a critical point determine the point's stability. When all eigenvalues have negative real parts, populations are likely to settle down to a steady state. If any eigenvalue is positive, the equilibrium is unstable, indicating a tendency for the populations to grow uncontrolled or crash. This binary nature of stability makes eigenvalues an indispensable tool in population dynamics studies.

Asymptotic Stability

Asymptotic stability is a term that conveys the destiny of populations as time proceeds to infinity. In layman's terms, it tells us whether the species' populations will endure or face extinction.

For a critical point to be asymptotically stable, any small deviation from this point must decay over time, causing the populations to inevitably return to this equilibrium state. Mathematically, this implies that all eigenvalues of the Jacobian matrix at the critical point have strict negative real parts. Populations associated with an asymptotically stable equilibrium are bound to settle into a consistent pattern, characterized by unchanging density levels. Conversely, populations associated with unstable points are likely to deviate significantly over time, making long-term predictions uncertain.