Interpretation of the growth rate of consumption in a simple one-good model In Perman et al. "The efficient and optimal use of natural resources" it is presented a simple single-good model of welfare dynamic optimisation with non-renewable natural resource where:

Obj:
$$
max \int_{t=0}^{t=inf}{U(C) e^{-\rho t} dt} 
$$
s.t.
$$
\dot S_t = -R_t
$$
$$
\dot K_t = Q(K_t,R_t)-C_t
$$

That is, the economy is comprised of a single output Q that can be either consumed or added to the capital stock K (that doesn't depreciate).

Solving the model and rearranging the first order conditions, the book found that among the optimal path the growth rate of consumption is:

$$
\frac{\dot C} {C} = \frac {Q_k - \rho}{\eta}
$$

where $Q_k$ is the marginal product of the capital, $\rho$ is the social discount rate and $\eta$ being the elasticity of the marginal utility with respect to consumption is, under bland assumptions, guaranteed to be positive.

Now, my problem is the interpretation of the sign implications of this equation.

If $Q_K$ is greater than $\rho$, the consumption rate increases (and the opposite). What does it mean?

The books explains it considering that $\rho$ (the social discount rate) reflects impatience for future consumption, and $Q_K$ (the marginal product of capital) is the pay-off to delayed consumption. 

Under this interpretation the relation implies that along an optimal path
when ‘pay-off’ is greater than ‘impatience’ "consumption is increasing", or that "the economy will be accumulating K and hence growing".

But in my experience if $Q_K$ is higher than the discount rate, I have an higher incentive to stock K rather than consume, so how the consumption could increase ?
Also, it is said that "when ‘pay-off’ is less than ‘impatience’, the economy will be running down K.", but how can the K reduce in this simple model where there is no capital depreciation?
I got stock with this problem for days.

The fact is that in a statical analysis, increasing $Q_K$ reduces consumption. But in a dynamic analysis, increasing $Q_K$ means more capital is saved and hence next year an higher production function that allows for more consumption than today, hence, a positive growth rate of consumption.