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. $$ \begin{array}{l}{d x / d t=x(1-0.5 y)} \\ {d y / d t=y(-0.25+0.5 x)}\end{array} $$

Short answer

#Short answer# The populations of the two species will oscillate around the equilibrium point (0.5, 2) as time goes to infinity. This indicates that the species will coexist in an oscillatory manner, with fluctuations in their populations over time.

Step-by-step solution

Draw a direction field and describe how solutions seem to behave

First, we notice that the given system of differential equations are: $$ \frac{dx}{dt} = x(1 - 0.5y), \\ \frac{dy}{dt} = y(-0.25 + 0.5x). $$ To draw a direction field, we plug in values for \(x\) and \(y\) to determine the behavior of the trajectories in different regions of the \(xy\)-plane. For example, if \(x>0, y>0\) then both \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) will be positive, implying that trajectories move upwards and to the right. We can do this for other regions of the \(xy\)-plane and prepare a direction field accordingly.

Find the critical points

A critical point \((x_0, y_0)\) is when both \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) are zero. Equating the given equations to zero and solving for \(x\) and \(y\), we find that the critical points are: \((x_0, y_0) = (0, 0)\) and \((x_1, y_1) = (0.5, 2)\).

Determine eigenvalues, eigenvectors, classify critical points and their stability

At each critical point, we can linearize the system of differential equations by computing the matrix \(J\) of partial derivatives: $$ J = \begin{pmatrix} \frac{\partial (1 - 0.5 y)}{\partial x} & \frac{\partial (1 - 0.5 y)}{\partial y} \\ \frac{\partial (-0.25 + 0.5 x)}{\partial x} & \frac{\partial (-0.25 + 0.5 x)}{\partial y} \end{pmatrix} = \begin{pmatrix} 1-0.5y & -0.5x \\ 0.5y & -0.25+0.5x \end{pmatrix}. $$ Now, we evaluate the Jacobian matrix \(J\) at each critical point and find the eigenvalues and eigenvectors. At \((x_0, y_0) = (0, 0)\): $$ J = \begin{pmatrix} 1 & 0 \\ 0 & -0.25 \end{pmatrix}. $$ Eigenvalues are \(\lambda_1 = 1\) and \(\lambda_2 = -0.25\). Since the eigenvalues have different signs, the critical point \((0, 0)\) is a saddle point, which is unstable. At \((x_1, y_1) = (0.5, 2)\): $$ J = \begin{pmatrix} 0 & -0.25 \\ 1 & 0 \end{pmatrix}. $$ The characteristic equation for this matrix is: $$ \lambda^2 + \frac{1}{4} = 0. $$ This gives the complex eigenvalues \(\lambda_3 = i/2\) and \(\lambda_4 = -i/2\). A pair of purely imaginary eigenvalues indicates that the critical point \((0.5, 2)\) is a center, and the stability cannot be determined from the eigenvalues.

Sketch trajectories in the neighborhood of critical points

Near the saddle point \((0, 0)\), trajectories move away from the point, indicating the population will not stabilize at (0, 0). Near the center \((0.5, 2)\), the trajectories will appear as closed curves, implying the oscillatory behavior of the populations of the two species around this point.

Draw a phase portrait for the system

Using the information about the trajectories and the critical points, we can now draw a phase portrait for the system. The phase portrait will show the direction of the trajectories in each region of the \(xy\)-plane and the behavior near the critical points mentioned in Step 4.

Determine the limiting behavior of \(x\) and \(y\) as \(t \rightarrow \infty\) and interpret the results

As \(t \rightarrow \infty\), the system tends to the critical point \((0.5, 2)\), which is a center. This indicates that the two populations will oscillate around the equilibrium point \((0.5, 2)\). Therefore, both species will coexist in an oscillatory manner, with fluctuations in their populations over time.

Key concepts

Direction Field

The direction field is a visual tool that provides insight into the behavior of solutions to a system of differential equations without actually solving the equations. It consists of tiny vectors (or arrows) at various points in the variables' plane, indicating the direction in which the solution moves at that point. The direction and magnitude of the vectors are determined by the differential equations. For example, in our population dynamics problem, by evaluating and plotting the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) at various points, we can visualize how the population densities are expected to change over time. This approach reveals whether the populations are increasing, decreasing, or remaining constant at each point in the plane. The direction field is especially helpful for identifying general trends and patterns in the system's behavior.

Critical Points

Critical points, also known as equilibrium points, are vital in analyzing differential equations as they represent states where the system does not change, hence no movement occurs. In population dynamics, these points indicate stable conditions where the species' populations remain constant. To find the critical points, we set the time derivatives \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = 0 \) and solve the system of equations. The solution yields points in the plane where the vectors in the direction field are zero-length, indicating no movement. In our exercise, the critical points are identified as \( (x_0, y_0) = (0, 0) \) and \( (x_1, y_1) = (0.5, 2) \) which represent two different potential long-term behaviors of the populations.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are powerful mathematical concepts used in the analysis of linear systems which arise from linearizing nonlinear systems at critical points. They help in classifying the type and determining the stability of these critical points. An eigenvalue gives you information about the potential growth, decay, or oscillation in magnitude of solutions nearby the critical points, while eigenvectors indicate the direction in which these behaviors occur. For stable systems, the real parts of all eigenvalues are negative, causing solutions to decay towards the critical point over time, suggesting a return to equilibrium after a disturbance. Our exercise demonstrates the process of finding eigenvalues and eigenvectors for the Jacobian matrix at each critical point and thus classifying them.

Phase Portrait

A phase portrait is an advanced graph that encompasses a set of trajectories representing possible states of a dynamical system. Each trajectory shows a possible path that system could follow over time in the state-space. It is a comprehensive picture which combines the information from the direction field and the behavior near critical points. By examining a phase portrait, one can deduce the stability of an ecosystem and predict how it may respond to different starting conditions. In our population model, the phase portrait reveals that trajectories near the saddle point move away, indicating instability, whereas near the center, trajectories form closed loops, suggesting a stable, oscillating behavior.

Asymptotic Stability

Asymptotic stability is a fundamental concept concerning the long-term behavior of dynamical systems. A critical point is considered asymptotically stable if solutions starting near the point tend to it as time goes to infinity. This indicates a strong form of stability where not only does the system return to equilibrium after perturbations, but it also remains there. Conversely, an unstable critical point will lead to solutions that diverge away. In our case study, we look at the eigenvalues of the Jacobian matrix to determine stability. If all eigenvalues have negative real parts, the critical point is asymptotically stable, whereas if any eigenvalue has a positive real part, it is considered unstable. Purely imaginary eigenvalues, however, require further investigation as they alone cannot determine stability.