I am dealing with the following matrix.
$A=\begin{pmatrix} 0&a & a & a & c & c &c & c\\ a& 0& a &a & c& c& c& c\\ a& a &0 &a & c& c & c & c\\ a& a& a &0 & c &c &c &c \\ c& c & c & c & 0& b& b &b \\ c& c & c & c& b&0 & b & b\\ c& c & c & c & b & b &0 & b\\ c& c & c & c & b& b& b &0 \end{pmatrix}$
We have $a > b >1/2$ and $c<1/2$, and $ a, b, c \ge 0$ and $ a, b, c \le 1$. The typical replicator dynamics equation is: \begin{equation} \frac{d x}{d t} = [ Ax - (x, Ax)] x = (Ax) x - (x, Ax) x, \label{A1} \end{equation}
where $\frac{d x}{d t}$ denotes derivative with respect to the time variable $t$, $(x, Ax)$ denotes the usual inner product, i.e. the dot product, of the vectors $x$ and $Ax$, and $(Ax) x$ is the vector whose $i$-th component is the product of the $i$-th components of $(Ax)$ and $x$ (i.e. the ``pointwise product" of two vectors). The matrix $A$ is called the payoff matrix or fitness matrix.
Is it possible show that the dynamics converge to $x_1^*=(1/4, 1/4, 1/4, 1/4, 0, 0, 0,0)$ or $x_2^*=(0, 0, 0, 0, 1/4, 1/4, 1/4,1/4)$. I am not sure what are the stable points of the equation. We suppose we are working in simplex, which means sum of the components of x are equal to 1.
A start:
characteristic polynomial is $$ \left( \lambda^2 -3(a+b) \lambda + 9ab-16c^2 \right) (\lambda + a)^3 (\lambda + b)^3 $$
Two of the eigenvalues are $$ \frac{3(a+b)\pm \sqrt {9(a-b)^2 + 64 c^2}}{2} $$ With your inequalities, the largest eigenvalue is larger than $3/2.$