Constructing Systems of Nonlinear First-Order Differential Equations to Model Population Dynamics

Build a system with desired behavior.

The Formula

The nonlinear system of differential equations,

\[\begin{align*} \frac{dx}{dt}=&\frac{x}{x_0}\big[F_{x1}(x-x_0)+F_{y1}(y-y_0)\big]\\ \frac{dy}{dt}=&\frac{y}{y_0}\big[G_{x1}(x-x_0)+G_{y1}(y-y_0)\big]\\ \end{align*}\]

will have a locally linear critical point at \((0,0)\) and \((x_0,y_0)\), for which the latter has a corresponding linear system

\[\begin{equation} \frac{d}{dt}\begin{bmatrix}x-x_0\\y-y_0\end{bmatrix} =\begin{bmatrix}F_{x1}&F_{y1}\\G_{x1}&G_{y1}\end{bmatrix}\begin{bmatrix}x-x_0\\y-y_0\end{bmatrix} \end{equation}\]

If you have specific eigenvectors and eigenvalues in mind rather than a matrix, you can always turn to diagonalization:

\[\begin{equation} \begin{bmatrix}F_{x1}&F_{y1}\\G_{x1}&G_{y1}\end{bmatrix} =\bigg[\textbf{v}_1\quad \textbf{v}_2\bigg] \begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix} \bigg[\textbf{v}_1\quad \textbf{v}_2\bigg]^{-1} \end{equation}\]

Example

Let’s take an example.

I want a system which behaves like the system

\[\begin{equation} \frac{d}{dt}\begin{bmatrix}x\\y\end{bmatrix} =\begin{bmatrix}0&1\\-1&0\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} \end{equation}\]

at the point \((1,1)\). This linear system has the behavior of concentric circles around the origin, so I expect there to be some cycling around my center point.

Using the formula that would give me

\[\begin{align*} \frac{dx}{dt}=&x(y-1)\\ \frac{dy}{dt}=&-y(x-1)\\ \\ \frac{dx}{dt}=&-x+xy\\ \frac{dy}{dt}=&y-xy\\ \end{align*}\]

This happens to be a lucky case where we can fairly easily get an implicit solution

Since

\[\begin{equation} \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \end{equation}\] \[\begin{equation} \frac{dy}{dx}=\frac{y(1-x)}{x(y-1)} \end{equation}\]

And this is a separable equation.

\[\begin{equation} x+y-\ln(xy)=C \end{equation}\]

If we suppose an arbitrary condition such as \(y(x_1)=y_1\), then

\[\begin{equation} (x-x_1)+(y-y_1)-\ln\left(\frac{xy}{x_1y_1}\right)=0 \end{equation}\]

Plotting various solutions with initial points in the first quadrant indeed, as we predicted, shows some cycling around the critical point \((1,1)\). Most initial points result in some pretty mishapen ellipse, but initial points very close to the critical point indeed approach something very close to a circle.