Prey-Predators systems 
Malthus Law
The differential equation \(z^{'}=Kz\), where \(K\) is a nonzero constant and \(z\) is a function of \(t\geq0\), models the evolution of a species that has not constraints on the food source and that is not the target of predators. The sign of \(K\) determines whether the species extinguishes or can take arbitrarily large dimensions. The solutions of the equation can be obtained from some considerations about the function \[\begin{array}{cccc} f: & t & \rightarrow & \frac{z(t)}{e^{Kt}}\end{array}\]
Since \(z^{'}=Kz\), the derivative of \(f\) is given by \[f'(t)=\frac{z'(t)e^{Kt}-z(t)Ke^{Kt}}{(e^{Kt})^{2}}=\frac{Kz(t)e^{Kt}-z(t)Ke^{Kt}}{(e^{Kt})^{2}}=0\]
Then \(f\) is constant and therefore with an initial condition \(z(0)=z_{0}>0\), we obtain \(z(t)=z_{0}e^{Kt}\). Thus, the sign of \(K\) determines the monotony of \(z\), informing us about the future of the species:
