Effective dimension of the Lotka–Volterra parameter space I am working on a project about the Lotka–Volterra equations for simple predation:
$$A \xrightarrow{k} 2A,\\
 A + B \xrightarrow{l} 2B,\\
 B \xrightarrow{m} 0.$$
Here I am asked about the “effective dimension of the parameter space”, but I am not sure about what could possibly be meant with the parameter space. Would it be the dimensions of each parameter $k$, $l$, $m$? 
The corresponding differential equations would be 
\begin{aligned}
\dot A &=kA-lAB,\\
\dot B &=lAB - Bm.
\end{aligned}
but this doesn’t make it any more clear to me.
 A: You can rescale $A,B$ and the time, $t=ws$, $A(t)=ua(s)$, $B(t)=vb(s)$. Then
$$
a'(s)=\frac wu A'(ws)=\frac wu (kA−lAB)=\frac wu(k\,ua−l\,ua\,vb)=wk\,a -vwl\,ab\\
b'(s)=\frac wv B'(ws)=\frac wv (lAB−mB)=\frac wv (l\,ua\,vb-m\,vb)=uwl\,ab-wm\,b
$$
Now you can try to set as many coefficients to $1$ as possible, starting with $wk=1$, net $vl=k$ so that $vwl=1$, the same with $ul=k$ so that $uwl=1$ which leaves us with $wm=\frac{m}k$ so that the normalized system is
$$
a'=a-ab\\
b'=ab-\frac mk b
$$
so that the normal forms have one free parameter.
A: I suspect that whoever gave you this tasks was aiming at the following:
In many contexts, it can happen that two (or more) parameters have an identical or translatable effect.
For a simple example, consider the following differential equation:
$$\dot{y} = α·β·\exp(δ)·y + ε,$$
where $α$, $β$, $δ$, and $ε$ are control parameters.
While the parameters $a$ and $b$ may correspond to different physical aspects, their effect on the dynamics is equivalent.
For example it does not matter whether you double $a$ or $b$. Thus for an abstract analysis of the equation, treating them separately just complicates things without any gain.
The same applies to the parameter $c$, which is not equivalent to the others, but whose effect can be easily translated to a change of $a$ or $b$. Thus, for an analysis of the dynamics, you could as well write this equation as:
$$ \dot{y} = λ·y+ε,$$
where $λ=α·β·\exp(δ)$.
But that’s not all. Suppose, you substitute $y$ with $ε·z$. Then you would have the following differential equation for $z$:
$$
  ε·\dot{z} = \dot{y} = λ·y + ε = λ·ε·z+ε \\
⇔   \dot{z} =                     λ·  z+1.
$$
So, the parameter $ε$ just corresponds to a rescaling of the entire dynamics, but doesn’t add any new dynamical phenomena.
Hence it is obsolete as well.
Finally, the parameter $λ$ is actually necessary.
If you change it, the dynamical behaviour may change completely, e.g., from an escalating to a converging one.
With other words, you can describe the range of dynamical phenomena exhibited by the initial equation with just one parameter.
The effective dimension of the parameter space is thus 1.
Now, all you need to do is to translate this to the Lotka–Volterra equation.
