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In this paper we are investigating the long time behaviour of the solution ofa mutation competition model of Lotka-Volterra's type. Our main motivationcomes from the analysis of the Lotka-Volterra's competition system withmutation which simulates the demo-genetic dynamics of diverse virus in theirhost : $$ \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 competitionterms $\Psi_i$ are independent of the virus type $i$. In this situation andunder some rather general assumptions on the functions $\Psi_i$, thecoefficients $r_i$ and the mutation matrix $\mu_{ij}$ we prove the existence ofa unique positive globally stable stationary solution i.e. the solutionattracts the trajectory initiated from any nonnegative initial datum. Moreoverthe unique steady state $\bar v$ is strictly positive in the sense that $\barv_i>0$ for all $i$. These results are in sharp contrast with the behaviour ofLotka-Volterra without mutation term where it is known that multiple nonnegative stationary solutions exist and an exclusion principle occurs (i.e Forall $i\neq i_0, \bar v_{i}=0$ and $\bar v_{i_0}>0$). Then we explore a typicalexample that has been proposed to explain some experimental data. For suchparticular models we characterise the speed of convergence to the equilibrium.In a second part, under some additional assumption, we prove the existence of apositive steady state for the full system and we analyse the long termdynamics. The proofs mainly rely on the construction of a relative entropywhich plays the role of a Lyapunov functional.

Convergence to the Equilibrium in a Lotka-Volterra Ode Competition System with Mutations