Mathematical Biology and Modelling

[ra_forever8]

Consider the two species competition model given by
$$ \frac{da}{dt }= \frac {λ_1 a} {a+K_1} - r_{ab}\cdot ab - da, \ \ \ \ \ \ \ \ \ \ (1)$$
$$\frac{db}{dt }= λ_2 b (1-\frac{b}{K_2}) - r_{ba}\cdot ab , \ \ \ \ \ \ \ t>0,\ \ \ \ \ \ \ \ (2)$$

for two interacting species denoted a=a(t) and b=b(t), with initial conditions a=a0 and b=b0 at t=0. Here λ1, λ2, K1,K2, $r_{ab}$,$r_{ba}$ and d are all positive parameters.,
NOW it is decided that the model is to non-dimensionalised such that
$a= Xu$, $b=K_2 v$, and $t= \frac{\tau}{Y}$,
where X and Y are parameter scalings to be determined. Here u,v and $\tau$ are all non-dimensional variables.
The application of this non-dimensionalisation leads to the equation for a(t) becoming \begin{equation} \frac {du}{d\tau}=\frac{\overline{\lambda}_{1} u}{u+1} -\overline{r}_{ab}uv -\overline{d}u
\end{equation}
where $\overline{\lambda}_{1}= \frac {\lambda_{1}}{\lambda_{2} K_{1}}$
Determine X and Y, all of the remaining non-dimensional parameters and their relationship to the dimensional parameters and the non-dimensional equation describing b(t)

=>i have started by finding b(t), but sure whether is right or wrong,\begin{equation} \frac {dv}{d\tau}={\overline{\lambda}_{2}}(1-v) -\overline{r}_{ba}uv
\end{equation}
where I think $ \overline{\lambda_{2}}=\lambda_{2}K_{2}$
I need to determine X and Y and also all the remaining non-dimensional parameters like $\overline{r}_{ba},\overline{\lambda}_{2}, \overline{\lambda}_{1},\overline{r}_{ab},\overline{d}$ and their relationship to the dimensional parameters, by using the two equations of a(t) and b(t).
please help me, really confuse . thanks alot

 

[jvicens]

Hi there,

First of all, let's start by looking at the non-dimensionalization process. The goal of non-dimensionalization is to simplify the equations and eliminate any unnecessary parameters, making it easier to analyze and interpret the model.

In this case, we have three variables (a, b, and t) and seven parameters (λ1, λ2, K1, K2, $r_{ab}$, $r_{ba}$, and d). To non-dimensionalize the model, we need to find three scaling parameters: X, Y, and Z.

The first step is to choose the scaling parameters for each variable. We have already been given the scaling for a and b, so we just need to determine the scaling for t. Let's choose t=Yτ, where Y is a scaling parameter that we need to determine and τ is a non-dimensional variable.

Next, we substitute these scalings into the original equations and simplify to get the non-dimensionalized equations:

\begin{equation} \frac{X}{Y}\frac{du}{d\tau}=\frac{\lambda_{1}Xu}{Xu+K_{1}Y}-r_{ab}XK_{2}uv-dXu
\end{equation}
\begin{equation} \frac{K_{2}}{Y}\frac{dv}{d\tau}=\lambda_{2}K_{2}v(1-\frac{K_{2}v}{K_{2}})-r_{ba}XK_{2}uv
\end{equation}

Now, we can see that the scaling parameters for u and v are X/Y and K2/Y, respectively. To simplify the equations further, we can choose X=YK2, which will eliminate the Y parameter from the equations. This means that X and Y are related by X=YK2, or Y=X/K2.

Substituting this into the equations, we get:

\begin{equation} \frac{du}{d\tau}=\frac{\lambda_{1}u}{u+K_{1}}-r_{ab}uv-du
\end{equation}
\begin{equation} \frac{dv}{d\tau}=\lambda_{2}(1-v)-r_{ba}uv
\end{equation}

Now, let's look at the non-dimensional parameters. We have the