Since you are asking for the biological interpretations about these parameters, it is important to realize that the model you are presenting is a non-dimensionalized version of this model:
$$ \frac{dx_1}{dt} =b_1x_1\Big(1- \Big(\frac{x_1 + \beta_{12}x_2}{K_1} \Big)\Big) $$ $$ \frac{dx_2}{dt} =b_2x_2\Big(1- \Big(\frac{x_2 + \beta_{21}x_1}{K_2} \Big)\Big), $$
which can also be written as:
$$ \frac{dx_1}{dt} =b_1x_1\Big(1- \frac{x_1}{K_1} - \beta_{12}\frac{x_2}{K_1}\Big) $$ $$ \frac{dx_2}{dt} =b_2x_2\Big(1- \frac{x_2}{K_2} - \beta_{21}\frac{x_1}{K_2}\Big). $$
This clearly shows how this competition model is merely the logistic <PERSON> model, with an extra term attached. This extra term is simply describing to what extent each individual in "species" 2 is using up the resources of "species" 1 ($K_1$), where $\beta_{12}$ describes this relationship (~proportion of species 2 that is counted as species 1). The parameters in the model formulations above are also more easily interpreted in ecological/biological terms than the $a_{ij}$s in your model.
To go from this latter formulation to yours you substitute: $$a_{11} = b_1/K_1$$ $$a_{12} = \frac{b_1\beta_{12}}{K_1}$$ $$a_{22} = b_2/K_2$$ $$a_{21} = \frac{b_2\beta_{21}}{K_2}$$
As you can see, this means that the $a_{ij}$s are composites of other parameters, and perturbing $a_{ij}$s could correspond to either $b_i$, $K_i$ or $\beta_{ij}$ being perturbed. 374
It also means that perturbing e.g. $b_1$ in your model while holding $a_{11}$ fixed means that you are actually also perturbing $K_1$ to offset the change in $b_1$ (since $a_{11} = b_1/K_1$), which will introduce a positive temporal correlation between $K_1$ and $b_1$. This is a strong assumption to make. This also means that $a_{ij}$s are not strictly "interaction parameters" as you call them, but they include other aspects as well.
Since realistic cases and scenarios for perturbations may often be expressed in terms of the parametrization I give above (e.g. random fluctuations in population growth rate or carrying capacity), these relationships are essential to keep in mind. 915