Convergence to the equilibrium in a Lokta-Volterra ode competition system with mutations

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 : $$ \frac{dv_{i}(t)}{dt}=v_i\left[r_i-\frac{1}{K}\Psi_i(v)\right]+\sum_{j=1}^{N} \mu_{ij}(v_j-v_i). $$ 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.