## 2 A Preliminary Basis the New Solution Developed  
### Scaling of the Model Equation  
At the beginning, we are obliged to say some words about the consequences of steps towards the solution we will develop, precisely discussing the model's dimensional coherence and its physical adequacy. First, the dimensional coherence of the model of interest is examined in the following section, which reveals distinctive scales that enable the creation of dimensionless groups that control the diffusion process. This step is essential because we can only create the approximate solution by applying the chosen method and, most importantly, physically recognizing the results. The last means that the solutions developed will reveal naturally appearing scales and dimensional groups that should be recognized and interpreted.  
Let us analyze the dimensional coherency of the model Equation (1). In general, \(u(x,t)\) is called (dimensionless) by a certain scale (characteristic, reference value) \(u_{ref}\), so that \(0\leq u(x,t)\leq 1\). However, working with \(x\) and \(t\) strongly indicates that the equation is semi-scaled, and the task now is to define the time and length scale allowing us to complete its non-dimensionalization. The simple inspection of both sides of Equation (1) reveals that both the coefficients \(b\) and \(c\) have dimensions of the inverse of time, i.e., \(s^{-1}\). Thus, either \(1/b\) or \(1/c\) may serve as a characteristic time scale \(t_{0}\). To avoid ambiguity, the presentation of the model equation as a Fisher-type one (Equation (3) removes any doubts) because the ratio \(d=c/b\) is dimensionless. Hence, from this moment, we accept that the tim escale \(t_{0}\) is defined as \(t_{0}=1/b\). In this context, the dimensionless time, i.e., the Fourier number \(Fo=t/t_{0}\), is defined as \(Fo=bt\). There is an exception, when \(b=0\), i.e., the model considers only a production (sink) term, and then the time scale is defined by \(t_{0}=1/c\).  
Taking into account that diffusivity \(D_{0}\) has a dimension of \(m^{2}s^{-1}\), a simple recalling of Equation (1) reveals the length scale \(L\) as \(\sqrt{D_{0}/b}\). Thus, the model can be presented in two versions as a complete dimensionless equation:  
\[\frac{\partial u}{\partial Fo}=\frac{\partial u}{\partial x}\bigg{(}u^{m} \frac{\partial u}{\partial x}\bigg{)}+u^{p}-u^{q} \tag{5}\]or as  
\[\frac{\partial u}{\partial F_{0}}=\frac{\partial u}{\partial x}\bigg{(}u^{m}\frac{ \partial u}{\partial x}\bigg{)}+u^{p}(1-du^{r}),\quad d=\frac{c}{d},\quad r=(q-p )>0,\quad m>1 \tag{6}\]  
In this context, the version (2) could be considered as completely scaled, at a glance, but the solutions developed in [1; 2] reveal that the complete non-dimensional was not conducted.  
An important outcome of the scaling analysis, which will be useful in the interpretation of the solution speed, is that the characteristic velocity scale is defined by \(V_{0}=\sqrt{D_{0}b}\).  
It is worth noting that when the diffusivity \(D_{0}\) is accepted as unity, as in Equation (2), it does not lose its physical meaning. Neglecting the diffusivity as a coefficient in the modeling equation makes it dimensionally incoherent, and to a greater extent, even though mathematically it is not so important, it yields hardly physically interpretable results. The solutions developed next and all consequent subcases will be developed based on the semi-scaled Equation (1), and we will see the benefits of this approach.