Prey-Predators systems Resize


Vector Field

The system of differential equations \[\begin{cases} x^{\text{'}}= & Ax-Bxy\\ y^{'}= & -Cy+Dxy \end{cases}\] induces in the plane a vector field \(V=(x^{'},y^{'})\) and, given a point \(Z=(x,y)\) of the plane, the vector \(V\) is tangent at this point to the initial condition solution curve \(Z\): \(V\) indicates the direction, orientation and intensity of variation of the solution curve.

Fixing \(A\), \(B\), \(C\) and \(D\), observe the behaviour of the vector field \(V\) in the next figure.

(Click on the figure to change \(A\), \(B\), \(C\) and \(D\).)

Let us begin by analysing the equilibrium points of the vector field \(\left(x^{'},y^{'}\right)=\left(Ax-Bxy,-Cy+Dxy\right)\). An equilibrium point represents a solution curve of the system such that, over time, there is no change in the number of elements of each species. The coordinates of such a point are the values of \(x\) and \(y\) for which the system does not change; and, mathematically, the absence of variation corresponds to a null derivative: \[\begin{cases} x^{\text{'}}= & Ax-Bxy=0\\ y^{'}= & -Cy+Dxy=0 \end{cases}\Leftrightarrow\begin{cases} \left(A-By\right)x= & 0\\ \left(-C+Dx\right)y= & 0 \end{cases}\Leftrightarrow\\ \Leftrightarrow\begin{cases} y=\frac{A}{B} & \begin{array}{cc} or & x=0\end{array}\\ x=\frac{C}{D} & \begin{array}{cc} or & y=0\end{array} \end{cases}\rightarrow or\begin{array}{ccc} \left(x,y\right) & = & \left(0,0\right)\\ \left(x,y\right) & = & \left(\frac{C}{D},\frac{A}{B}\right) \end{array}\]

Let us interpret these two equilibrium points. \((0,0)\) means that there are neither pouts nor sharks, and therefore the values of these species have to remain null (since they can not be born). \((\frac{C}{D}, \frac{A}{B})\) is an equilibrium point of interest to us, though in biological terms it is unlikely.

From the signals of \(x^{'}\) and \(y^{'}\) we can visualize the vector field \((x^{'},y^{'})\) and from regions of monotony of \(x\) and \(y\), we can have a qualitative idea of how the solution curves behave. We will restrict this study to the first quadrant, since the biologically acceptable values for \(x\) and \(y\) are nonnegative.

The derivatives \(x^{'}\) and \(y^{'}\) vanish along the straight lines \(y=\frac{A}{B}\) and \(x=\frac{C}{D}\), respectively, which divide the quadrant into four regions where the derivatives have different signals:

I
II
III
IV
\(x^{'}<0 \) and \(y^{'}>0\)
\(x^{'}<0 \) and \(y^{'}<0\)
\(x^{'}>0 \) and \(y^{'}<0\)
\(x^{'}>0 \) and \(y^{'}>0\)


In region IV both the values of \(x\) and \(y\) grow strictly over time; in region I, the values of \(x\) decrease strictly, while the function \(y\) grows. A plausible biological explanation for this variation is that, with the abundance of pouts in the IV region, the sharks have optimal conditions to reproduce, reaching values so high that, in region I, the pouts exhibit considerable losses. Similar explanations may be given for Regions II and III.

stability