Mathematical Biology (non-dimensionalise)

[ra_forever8]

Consider the predator-prey system defined by
\begin{equation} \frac{dP}{dt}= aP(1-\frac{N}{K})-bPN= f(N,P)----------(1)
\end{equation}
\begin{equation} \frac{dN}{dt}=cNP-dP= g(N,P)--------------(2)
\end{equation}
with initial conditions $P=P_{0}$ and $N=N_{0}$,
Where $N=N(t)$ is the prey density and $P=P(t)$ is the predator density. Here $a,b,c,d$ and $K$ are all positive constants.
Non-dimensionalise equations (1) and (2) to obtain
\begin{equation} \frac{du}{d\tau}=u(1-u-v)= f(u,v)
\end{equation}
\begin{equation} \frac{dv}{d\tau}=\alpha v(u-\beta)= g(u,v)
\end{equation}
with $u=u_{0}$ and $v=v_{0}$,
where $\alpha = \frac{cK}{a}$ and $\beta= \frac{d}{cK}$=> I try to do by saying Non-dimensionalise parameter to be
$P= au$, $N=Xv$ and $t=\frac{\tau}{a}$, where I can choose $X= \frac{K}{b}$, but it did not works. can anyone please help to get Non-dimensionalise equations.

 

[jvicens]

Thank you for your question. Non-dimensionalization is a common technique used in mathematical modeling to simplify and analyze complex systems, and it can be very useful in studying predator-prey systems.

To non-dimensionalize the given system, we can follow a few steps:

1. Define new variables: Let us define new variables $u=\frac{P}{P_0}$ and $v=\frac{N}{N_0}$. These variables represent the relative densities of predators and prey, respectively, compared to their initial values $P_0$ and $N_0$.

2. Rewrite the equations: Using the new variables, the original equations (1) and (2) can be rewritten as:
\begin{equation}
\frac{dP}{dt} = aP_0(1-\frac{N_0}{K}v)-bP_0PN = aP_0(1-v)-bP_0Pv = aP_0(1-u-v)
\end{equation}
\begin{equation}
\frac{dN}{dt} = cN_0P-dP = cN_0P_0uv-dP_0u = cN_0P_0(uv-\frac{d}{c}u)
\end{equation}

3. Non-dimensionalize the parameters: To non-dimensionalize the parameters, we can use the following relations:
\begin{equation}
u = \frac{P}{P_0} = \frac{aP}{aP_0} = \frac{a}{aP_0}P
\end{equation}
\begin{equation}
v = \frac{N}{N_0} = \frac{cN}{cN_0} = \frac{c}{cN_0}N
\end{equation}

Substituting these relations in the original equations, we get:
\begin{equation}
\frac{du}{dt} = \frac{a}{aP_0}P(1-\frac{c}{cN_0}N) - \frac{b}{aP_0}PN = u(1-\alpha v) - \beta uv
\end{equation}
\begin{equation}
\frac{dv}{dt} = \frac{c}{cN_