I dont know whether this math can be more clarifying or not. If you want of the reason behind It is strongly influenced by the most abundant species. It is all about finding the best p's that does the minimization for $D$. You can skip to conclusion of you feel like skipping all these.
You want to find the critical point (where the function attains it minimum and maximum) of $D = p_{1}^2+p_{2}^2+...+p_{n}^2$ given $p_{1}+p_{2}+...+p_{n}=1$. 172
where $p_{i}$ of species.
Lagrange Multiplier method is one of the techniques to find the values given to $p_{1}, p_{2}, ...p_{n}$ to make $p_{1}^2+p_{2}^2+...+p_{n}^2$ go to maximum or minimum given some constraint on this.
so assume
$f = p_{1}^2+p_{2}^2+...+p_{n}^2$ & $g = p_{1}+p_{2}+...p_{n}$
According to <PERSON> multiplier
$\partial_{p_{1},..p_{n}} f = \lambda \partial_{p_{1},..p_{n}} g$
We have to find the value of $\lambda$.
So after partial fractions step, we will end up with four equations below
$np_{1} = \lambda ; np_{2} = \lambda; ... np_{n} = \lambda$ and we already know that $p_{1}+p_{2}+...+p_{n} = 1$
From above the value of $p_{1} = p_{2} = ... = p_{n} = \lambda/n$ we get $\lambda = 2/n$ and from this the minimum values of the function
$f = D = p_{1}^2+p_{2}^2+...+p_{n}^2$ is when all $p_{i} = 1/n $ for i is denoting each species in ${1,...,n}$
Conclusion: So $D$ value attains its minimum when all terms $p_{i}$ are equal. As any value goes away from $1/n$ ('n' is the total number of individuals of species you got)the value increases drastically as it is being squared in $D$.
Sample example will be like $0.5^2 + 0.5^2 = 0.25$ but a change from this say $0.6,0.4$ increases $0.5$ term by $0.11$ but decreases $0.4$ term by only $0.09$.
Hope this helps. 172