  In this paper we are investigating the long time behaviour of the solution of
a mutation competition model of Lotka-Volterra's type. Our main motivation
comes from the analysis of the Lotka-Volterra's competition system with
mutation which simulates the demo-genetic dynamics of diverse virus in their
host : $$ \frac{dv_{i}(t)}{dt}=v_i\[r_i-\frac{1}{K}\Psi_i(v)\]+\sum_{j=1}^{N}
\mu_{ij}(v_j-v_i). $$ In a first part we analyse the case where the competition
terms $\Psi_i$ are independent of the virus type $i$. In this situation and
under some rather general assumptions on the functions $\Psi_i$, the
coefficients $r_i$ and the mutation matrix $\mu_{ij}$ we prove the existence of
a unique positive globally stable stationary solution i.e. the solution
attracts the trajectory initiated from any nonnegative initial datum. Moreover
the unique steady state $\bar v$ is strictly positive in the sense that $\bar
v_i>0$ for all $i$. These results are in sharp contrast with the behaviour of
Lotka-Volterra without mutation term where it is known that multiple non
negative stationary solutions exist and an exclusion principle occurs (i.e For
all $i\neq i_0, \bar v_{i}=0$ and $\bar v_{i_0}>0$). Then we explore a typical
example that has been proposed to explain some experimental data. For such
particular models we characterise the speed of convergence to the equilibrium.
In a second part, under some additional assumption, we prove the existence of a
positive steady state for the full system and we analyse the long term
dynamics. The proofs mainly rely on the construction of a relative entropy
which plays the role of a Lyapunov functional.
