The Lotka-Voltera equations are a very simple model of predator-prey dynamics. They lead to oscillatory behaviour in both populations.

This is a simple model of a predator-prey system. The key assumptions is that the prey, \(x\), have unlimited food and will grow exponentially at a rate \(A\) unless killed by predators, \(y\), which happens at a rate \(B\) when they encounter them. The predators die off at a rate \(C\) and can increase their numbers at a rate \(D\) by eating prey . The rate of change of both population numbers is proportional to the size of the population. These assumptions can be encoded in the following differential equations, \begin{align}\label{eq:lotka-voltera} \dot{x} &= Ax-Bxy &\\ \dot{y} &= Dxy-Cy, \end{align} which are a pair of coupled first-order nonlinear differential equations.

Although the nonlinearity make a solution difficult, we can numerically integrate the equations to plot their time evolution (prey in green and predators in red), notice the predator populating lags behind the prey population:

Instead of trying to solve the equations we can easily construct the dynamical map:

Notice the strange feature around which the dynamics seems to flow? It would have been hard to predict that feature just by looking at the equations. It's a

Say we were to start the simulation close to the fixed point. We can see from the graph that the system would oscillate around the fixed point in something resembling an ellipse — i.e. the population numbers will oscillate like a sinusoid function. In such a region the population numbers would behave like a simple oscillator with a fixed phase between the two populations (see harmonic-oscillator).

Let's examine the motion near this fixed point by changing variables to
\begin{align*}
x(t) &= x_0 +\delta_x(t) \\
y(t) &= y_0 +\delta_y(t)
\end{align*}
What we are doing is a process called *linearisation*. Substituting the above into \eqref{eq:lotka-voltera} and expanding yields:
\begin{align*}
\dot{\delta_x} &= -\frac{BC}{D}\delta_y - B\delta_x\delta_y \\
\dot{\delta_y} &= \frac{AD}{B}\delta_x + D\delta_x\delta_y.
\end{align*}
Since \(\delta_x\) and \(\delta_y\) are both small, \(\delta_x\delta_y\) will be even smaller and we can neglect it, Giving
\begin{align*}
\dot{\delta_x} &= -\frac{BC}{D}\delta_y \\
\dot{\delta_y} &= \frac{AD}{B}\delta_x .
\end{align*}
These equations may not appear familiar it this form so lets take the first equation and differentiate it:
\begin{equation*}
\ddot{\delta_x} + \frac{BC}{D}\dot{\delta_y} = 0.
\end{equation*}
Now substitute in \(\dot{\delta_y}\),
\begin{equation*}
\ddot{\delta_x} + AC\delta_x = 0.
\end{equation*}
This is nothing more than the equation of motion for simple harmonic motion with an angular frequency of \(\omega_0=\sqrt{AC}\).
Interestingly the oscillation frequency does not depend on \(B\), the rate predators kill their prey, or \(D\), the rate the predators reproduce. For the parameters in the plot above the period should be \(T=2\pi/\omega_0=6.3\), have a close look at the plot.

Here is a model you can play with

Click figure to download the CDF demo.

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Duration: 5 min

A thin loop is hung on a horizontal nail. If the period of small angle oscillations is 2.0s what is the radius of the loop?

2
Duration: 10 min

A bullet of mass \(m\) is fired at a wooden block of mass \(M\) that rests on a frictionless surface and is attached to a wall by an ideal spring of spring constant \(k\). The block is initially at rest. Assume the bullet effectively instantaneously embeds itself in the block and sets the combined system into motion. Note that this is an inelastic collision so kinetic energy is *not* conserved. If the maximum amplitude of the spring is observed to be \(x_0\) after the collision, what was the initial velocity of the bullet, \(v_0\)?

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Duration: 5 min

Show that the following different ways of writing the solutions to the harmonic oscillator are all equivalent to each other:

- \(x(t) = A \cos(\omega_0 t + \phi_1)\),
- \(x(t) = B \sin(\omega_0 t + \phi_2)\),
- \(x(t) = C \cos(\omega_0 t) + D \sin(\omega_0 t)\),
- \(x(t) = E e^{i \omega_0 t} + E^* e^{-i \omega_0 t}\), for \(E\) complex where \(^*\equiv\) complex conjugation.

2
Duration: 5 min

If we were to suspend a mass on a spring vertically and have gravity act on the mass as well, how would the resulting oscillations change?