Equation (2.2.1) tells us how a solution curve changes in the \(x\) direction, while equation (2.2.2) tells us how a solution curve changes in the \(y\) direction.
The direction field is relatively easy to understand. After plotting only few vectors, we can very quickly see that the vectors are tangent to circles centered at the origin (Figure 2.2.2). Since the solutions to the undamped harmonic oscillator \(x'' + x = 0\) are of the form
\begin{equation*}
x(t) = A \cos t + B \sin t
\end{equation*}
for arbitrary constants \(A\) and \(B\text{,}\) this should not be too surprising.
also gives us a direction field where the vectors point towards the origin; however, we shall soon see that there are important differences between this direction field and the direction field of the previous system (Figure 2.2.5).
Let us recall the modified predator-prey system that we developed in the last section. That is, we will assume that the prey in our model has logistic growth,
where \(N\) is the carrying capacity. In order to investigate the geometric properties of our system, we will rewrite our system in vector form. For each value of \(t\text{,}\) we will use \({\mathbf x}(t)\) to denote the vector-valued function
This vector-valued function, \(\mathbf x(t)\) corresponds to our solution curve \((R(t), F(t))\) in the \(RF\)-plane. Now we can write our predator-prey model as a single vector equation,
We can associate a vector in the \(RF\)-plane for each value of \(R\) and \(F\text{.}\) For example, if we let \((R, F) = (10, 10)\text{,}\) we have \((R', F') = (-100, 50)\text{.}\) At this particular point, the population of rabbits is falling rapidly while the number of foxes is climbing very quickly, We can represent this vector in the phase plane by drawing an arrow in the proper direction. Thus, we obtain a direction field for our system of differential equations (Figure 2.2.8).
Suppose that \(x\) and \(y\) are the population of two distinct species that compete for the same resources. For example, two species of fish may compete for the same food in a lake, leopards and lions may compete for the same prey, or cattle and sheep may compete for the same grazing land. We can model two competing species using the following system of first-order differential equations,
The first term in each equation is the logistic growth of each species. The second term is how the species is affected by interacting with the competing species.
Suppose that \(x = x(t)\) and \(y = y(t)\) are two populations, competing for the same resources, are governed by the following system of differential equations,
then the values of \(x\) and \(y\) satisfying \(dx/dt = f(x, y) = 0\) are called the \(x\)-nullclines of the system and the values of \(x\) and \(y\) satisfying the equation \(dy/dt = g(x, y) = 0\) are called the \(y\)-nullclines of the system.
In our example, we can plot the \(x\) and \(y\)-nullclines to help us understand our system (Figure 2.2.10). Of course, we have an equilibrium solution whenever an \(x\)-nullcline intersects with a \(y\)-nullcline. Thus, the equilibrium points for our particular system are \((0,0)\text{,}\)\((1, 0)\text{,}\)\((0, 1)\text{,}\) and \((1/3, 1/3)\text{.}\) Furthermore, we can choose a representative point in each region to find how the direction field is oriented. If we are given the initial condition \((1.4, 1.3)\) for example, we can see that
Since both of these numbers are negative, we can see that our initial trajectory is headed down and to the left—slightly more to the left than down. However, we have no guarantee that the trajectory will continue in this direction.
Before we proceed further with our analysis, let us determine what happens on the nullclines themselves. That is, we will examine the case when \(dx/dt = 0\) or \(dy/dt = 0\text{.}\) If \(dx/dt = 0\text{,}\) then any solution crossing this \(x\)-nullcline must be moving vertically. There can be no right or left movement at this point in the phase portrait. We can indicate this fact along with the direction of vertical movement by drawing a vertical slash with an arrow on the \(x\)-nullcline (Figure 2.2.10). For example, the point \((0.5, 0.25)\) lies on the \(x\)-nullcline and
Therefore, the trajectory that crosses the \(R\)-nullcline at \((0.5, 0.25)\) is moving down. As we move along the \(x\)-nullcline, the direction of this arrow can only change when we cross an \(y\)-nullcline.
Similarly, there can be no vertical motion on an \(y\)-nullcline. We will indicate the direction of horizontal motion by drawing a horizontal line with an arrow pointing to the right if \(dx/dt \gt 0\) on the \(y\)-nullcline and an arrow pointing to the left if \(dx/dt \lt 0\) on the \(y\)-nullcline.
We can also determine the basic direction of the solution curve by checking what happens at a point in each of the regions bounded by the nullclines. For example, at the point \((0.1, 0.1)\) we find that
What happens to the initial condition \((1.4, 1.3)\text{?}\) We see three possible scenarios if we follow the nullclines for large values of \(t\text{.}\)
Only species \(x\) survives and species \(y\) becomes extinct.
The nullclines will divide the first quadrant of the plane, \(x \geq 0\) and \(y \geq 0\text{,}\) into different regions. In each of these regions find a slope vector at some point \((x_0, y_0)\) in the region. For example, at the point \((1,2)\text{,}\) we can attach slope vector \((-1, -3)\text{.}\)
There is no reason why our nullclines should be limited to straight lines. The system
\begin{align*}
x' & = y - x^2\\
y' & = x - 2
\end{align*}
has an \(x\)-nullcline \(y = x^2\) and a \(y\)-nullcline \(x = 2\text{.}\) These two nullclines intersect at \((2, 4)\text{,}\) and the system has a single equilibrium solution. The nullclines divide the plane into four basic regions (Figure 2.2.13). By choosing a point in each of these regions and determining the direction of the slope field at that point, we can decide the direction of the slope field in the entire region.
Subsection2.2.4Plotting Direction Fields with Sage
It is easy to plot direction fields using Sage. We can plot the direction field for the system
\begin{align*}
x' & = y - 2x,\\
y' & = x - xy - x^3
\end{align*}
using the following Sage interact. You can evaluate the cell below to get a menu driven applet for drawing direction fields for systems of two differential equations.
Step 1. Draw the curves where \(f(x, y) = 0\text{.}\) These curves are called the \(x\)-nullclines. When a solution curve \({\mathbf x}(t)\) lies on one of these curves, \(dx/dt = 0\text{,}\) draw vertical slash marks on the \(x\)-nullclines to remind yourself that a trajectory crossing the nullcline can only do so if it is moving in a vertical direction at the instant of crossing.
Step 2. Draw the curves where \(g(x, y) = 0\text{.}\) These curves are called the \(y\)-nullclines. When \({\mathbf x}(t)\) lies on one of these curves, \(dg/dt = 0\text{.}\) Draw horizontal slash marks on the \(y\)-nullclines to remind yourself that a trajectory crossing the nullcline can only do so if it is moving in a horizontal direction at the instant of crossing.
Step 3. Label the points where the \(x\) and \(y\)-nullclines intersect. These intersections are the equilibrium points. If \({\mathbf x}(t)\) is ever at one of these points, then both \(dx/dt\) and \(dy/dt\) vanish. This means that the trajectory stays at the equilibrium point for all time. If our system is going to tend towards a steady state, then \({\mathbf x}(t)\) will approach on of the equilibrium points as \(t \to \infty\text{.}\)
Step 4. Label the regions of the \(xy\)-plane where \(dx/dt \lt 0\) and where \(dx/dt \gt 0\text{.}\) These regions are always separated by \(x\)-nullclines. Likewise, label the regions where \(dy/dt\) is positive and negative.
Step 5.Go back and put arrows on the vertical hash marks of the \(x\) nullclines. These arrows indicate whether the motion across the nullcline is up or down. The arrows are up on the parts of the \(x\)-nullclines that are in the \(dy/dt \gt 0\) region, and down on those parts of the \(x\)-nullclines in the \(dy/dt \lt 0\) regions. Likewise, draw arrows on the horizontal slash marks of the \(y\)-nullclines. These arrows are pointing right on the parts of the \(y\)-nullclines in the \(dx/dt \gt 0\) regions and left point on the parts in the \(dx/dt \lt 0\) regions.
Part (d). Starting at \((x(0), y(0)) = (2.4, 0.3)\text{,}\) the trajectory lies in the basin of attraction of the stable equilibrium \((2,2)\text{.}\) As \(t \to \infty\text{,}\) both populations converge to
Equilibria. Setting \(x' = 3x + 4y = 0\) and \(y' = -x + y = 0\text{,}\) the second equation gives \(y = x\text{.}\) Substituting into the first gives \(7x = 0\text{,}\) so \(x = 0\) and \(y = 0\text{.}\) The only equilibrium is \((0, 0)\text{.}\)
Behavior of typical solutions. Every solution except the equilibrium \((0,0)\) spirals outward away from the origin as \(t \to \infty\text{,}\) with both \(|x(t)|\) and \(|y(t)|\) growing without bound. The rotation is counterclockwise.
Equilibria. Setting \(x' = -3x + 4y = 0\) and \(y' = -x + y = 0\text{,}\) the second equation gives \(y = x\text{.}\) Substituting into the first gives \(x = 0\) and \(y = 0\text{.}\) The only equilibrium is \((0, 0)\text{.}\)
Behavior of typical solutions. Since \(\lambda = -1 \lt 0\text{,}\) every solution tends to the origin as \(t \to \infty\text{.}\) Trajectories curve and approach the origin. No solution oscillates; all decay monotonically to zero.
Equilibria. Setting \(x' = -3x + 4y = 0\) and \(y' = x + 6y = 0\text{,}\) the second equation gives \(x = -6y\text{.}\) Substituting into the first gives \(18y + 4y = 22y = 0\text{,}\) so \(y = 0\) and \(x = 0\text{.}\) The only equilibrium is \((0, 0)\text{.}\)
Behavior of typical solutions. The origin is an unstable saddle. The stable manifold, tangent to \(\mathbf{v}_2 \approx (4, -0.42)^T\text{,}\) consists of two trajectories that approach the origin as \(t \to \infty\text{;}\) all other solutions are repelled. Generic trajectories approach the origin. Both \(|x(t)|\) and \(|y(t)|\) grow without bound for almost all initial conditions.
Behavior of typical solutions. The system has infinitely many equilibria arranged periodically along the line \(x = y\text{,}\) alternating between saddles (at even \(n\)) and unstable spirals (at odd \(n\)). Trajectories near an unstable spiral wind outward and are eventually captured by the stable manifold of an adjacent saddle, then expelled along its unstable manifold toward the next unstable spiral. Solutions generically diverge to infinity, though their paths are guided by the repeating saddle-spiral structure. No bounded attracting regions exist for this system.
Behavior of typical solutions. The origin is a stable improper node with a basin of attraction bounded by the stable manifolds of the two saddle points. Trajectories starting near \((0,0)\) converge to the origin tangent to the repeated eigenvector direction. The saddle at \((2-\sqrt{5},\,(\sqrt{5}-3)/4)\) lies close to the origin and partitions the phase plane; its stable manifold forms part of the boundary of the basin of attraction of the origin. Trajectories outside this basin diverge to infinity, guided by the unstable manifolds of the two saddles. Solutions starting far from the origin in the direction of positive \(x\) generally diverge rapidly due to the \(x^2\) term.
Consider an epidemic that moves through an isolated population. We will make the following assumptions about the epidemic.
Individuals are infected at a rate proportional to the product of the number of infected and susceptible individuals. We assume that the constant of proportionality is \(\alpha\text{.}\)
The constant \(\alpha\) is a measure of the relative infectivity of the disease. Some diseases such as Ebola, a viral hemorrhagic fever, are extremely infectious with a mortaility rate of up to 90%. On the other hand, AIDS which is caused by the HIV virus, has a much lower transmission rate. The goal of this exercise is to examine the differences between the two.
If \(\alpha = 0.05\text{,}\) draw the phase portrait. Be sure to label all nullclines and equilibrium solutions. Suppose that \(S(0) = 1000\) and \(I(0) = 1\text{.}\) What happens to the solution curve as \(t \to \infty\text{?}\)
If \(\alpha = 0.000001\text{,}\) draw the phase portrait. Be sure to label all nullclines and equilibrium solutions. Suppose that \(S(0) = 30000\) and \(I(0) = 1\text{.}\) What happens to the solution curve as \(t \to \infty\text{?}\)
The \(S\)-nullclines are \(S = 0\) and \(I = 0.0003/\alpha\text{;}\) the \(I\)-nullclines are \(I = 0\) and \(S = 0.1/\alpha\text{.}\) The biologically relevant equilibria are the disease-free states \((S_0, 0)\) for any \(S_0 \geq 0\) and the interior endemic equilibrium
Whether the disease spreads or dies out depends on whether the initial susceptible population \(S(0)\) exceeds the threshold \(S^* = 0.1/\alpha\text{:}\) if \(S(0) \gt S^*\) then \(dI/dt \gt 0\) initially and an epidemic occurs; if \(S(0) \lt S^*\) the infection dies out.
The threshold is \(S^* = 0.1/0.05 = 2\) and the endemic equilibrium is at \((2,\; 0.006)\text{.}\) Since \(S(0) = 1000 \gg 2\text{,}\) the epidemic explodes immediately: \(I\) rises rapidly while \(S\) is driven sharply downward. The solution curve sweeps to the left and downward, eventually approaching the \(S\)-axis (disease dies out after nearly the entire susceptible population has been exposed). As \(t \to \infty\text{,}\)\(I(t) \to 0\) and \(S(t)\) approaches a small positive value near zero.
The threshold is \(S^* = 0.1/0.000001 = 100{,}000\) and the endemic equilibrium is at \((100000,\; 300)\text{.}\) Since \(S(0) = 30{,}000 \lt 100{,}000\text{,}\) we have \(\alpha S(0) - 0.1 = 0.03 - 0.1 = -0.07 \lt 0\text{,}\) so \(dI/dt \lt 0\) immediately. The infected population decreases monotonically to zero while the susceptible population grows due to births. As \(t \to \infty\text{,}\)\(I(t) \to 0\) and \(S(t)\) grows (the disease dies out and the population recovers).
Part (c): Conclusions. The key quantity governing epidemic behavior is the threshold \(S^* = 0.1/\alpha\text{:}\)
When \(\alpha\) is large (highly infectious disease like Ebola, \(\alpha = 0.05\)), the threshold \(S^* = 2\) is extremely low. Almost any population will exceed this threshold, so an epidemic is virtually inevitable and spreads explosively, infecting nearly the entire susceptible population before dying out.
When \(\alpha\) is small (low-transmission disease like HIV, \(\alpha = 0.000001\)), the threshold \(S^* = 100{,}000\) is very high. Populations below this threshold see the disease die out naturally; only very large susceptible populations sustain an epidemic. Birth rates can maintain susceptible populations, but the slow transmission means the disease spreads gradually rather than explosively.
In both cases, the long-term fate of the epidemic is determined by the relationship between the initial susceptible population and the threshold \(S^* = 0.1/\alpha\text{.}\) Public health interventions that reduce \(\alpha\) (e.g., protective equipment, quarantine) or reduce \(S\) (e.g., vaccination) can push a population below the threshold and cause the disease to die out.
