Prey-Predators systems 
Properties of the solution curves
The image of some elements of the vector field \(V\) suggests that the solution curves with initial conditions close to the equilibrium point \((\frac{C}{D},\frac{A}{B})\) stay in a neighbourhood of this point and evolve around it. Thus, taking into account the properties of tangent curves to the vector fields \(V\), we expect to observe one of the following scenarios:
The fact that the curves \((x,y)\) are closed tells us that the functions \(x\) and \(y\) are periodic with time, which means that the populations of sharks and pouts recover losses and manage the variations that result from their relationship in the food chain.
(Click on the figure to observe the relationship between the two species over time)
The plot of the pairs \((x,y)\) shows us the relationship between the pout and shark values at the same time, but it hides an important variable: time. To visualize it, let us add to this plot another dimension. The next three-dimensional figure represents the curve \((t,x(t),y(t))\), from which the initial plot is its the projection in the plane \(xOy\).
(Click on the figure to observe the relationship between the two species over time.
