## 3 Logistic Equation  
[PERSON] [10] considered the possibility of representing the [PERSON] law, written in the form (2), as the Euler-Lagrange equation. The Lagrangian formulation of [PERSON] law is interesting in many ways. In particular, it provides a basis for searching for possible generalizations of the law [11]. But the search for a suitable Lagrangian is not the only way to derive the evolution equation. One can proceed, for example, from a fairly general integro-differential equation:  
\[\frac{dn}{dt}=\int_{0}^{\infty}K\big{(}t-t^{\prime}\big{)}\cdot F\big{[}n\big{(} t^{\prime}\big{)}\big{]}\cdot dt^{\prime}. \tag{5}\]If we put \(F(n)=-\sigma n^{2}\), then, when choosing the trivial kernel \(K(t-t^{\prime})=\delta(t-t^{\prime})\) from (5) follows the Omori law in the form (2). Derivation of (2) from (5) is useful in the sense that a natural generalization of the Omori law is suggested to us. The need for generalization is dictated by the following consideration. It follows from (2) that \(\lim n(t)=0\) for \(t\to\infty\). Meanwhile, experience shows that the flow of aftershocks ends with a transition to a certain background seismicity of the source. It is desirable for us to take this circumstance into account by using minimal changes in the form of the classical Omori law. It turns out that for this it is enough to take into account the linear term in the formula \(F(n)\): \(F(n)=\gamma n-\sigma n^{2}\). Here \(\gamma\) is the second phenomenological parameter of our theory. As a result, we get the master equation in the following form:  
\[\frac{dn}{dt}=n(\gamma-\sigma n). \tag{6}\]  
This is the logistic equation of [PERSON] [12], well known in biology, chemistry, and sociology. It turns out to be useful in the physics of earthquakes [9; 11]. We divide the family of solutions to logistic Equation (6) into two classes. The first class includes growing, and the second, falling functions of time. The separation principle is easiest to show in the phase portrait shown in Figure 1, where we have used the dimensionless quantities  
\[X=\frac{n}{n_{max}},\ \ P=\frac{n_{\infty}}{\gamma n_{max}}.\frac{dX}{dT}\,\ \ T=\gamma t \tag{7}\]  
instead of the original quantities. Here \(n_{\infty}=\gamma/\sigma\). [PERSON] [10] proposed the introduction of the phase plane of the dynamic system simulating the evolution of aftershocks according to the Omori law (2). The corresponding phase portrait is shown in Figure 1 with the red line. The point (0,0) corresponds to the equilibrium state. The representative point moves from bottom to top along the phase  
Figure 1: Phase portrait on the phase plane of Equation (6). The red, green, and blue phase trajectories are plotted at \(X_{\infty}=0,\ 0.5\) and \(1\), respectively (see text). trajectory with deceleration. The portrait consists of one phase trajectory that starts at point \((1,-1)\) and ends at point (0,0). However, we are interested in the family of phase trajectories for Equation (6), constructed for different values of the parameter \(X_{\infty}=n_{\infty}/n_{max}\). The red, green, and blue trajectories in Figure 1 are plotted with \(X_{\infty}\) values of 0, 0.5, and 1, respectively. Equilibrium point (0,0) is stable at \(X_{\infty}=0\) and unstable at \(X_{\infty}>0\). Equilibrium point \((X_{\infty},0)\) is stable at any values of parameter \(X_{\infty}>0\). It can be shown that the velocity of motion of the imaging point along the phase trajectory asymptotically tends to zero with approaching \((X_{\infty},0)\). The segment of the trajectory located above the horizontal axis corresponds to the Verhulst logistic curve, widely known in biology, chemistry, sociology, and other sciences. The segment located below the horizontal axis corresponds to the evolution of aftershocks (Figure 2). The choice between the logistic and aftershock branches is made when setting the Cauchy problem for Equation (6). Evolution proceeds along the aftershock branch if the initial condition satisfies the inequality \(n(0)=n_{0}>n_{\infty}\), where \(n_{\infty}=\gamma/\sigma\).