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

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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: dvᵢ(t)dt=vᵢ rᵢ-1/KΨᵢ(v) +∑ⱼ₌₁ⁿ μij(vⱼ-vᵢ). In a first part we analyse the case where the competition terms Ψᵢ are independent of the virus type i. In this situation and under some rather general assumptions on the functions Ψᵢ, the coefficients rᵢ and the mutation matrix μij 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 v is strictly positive in the sense that vᵢ>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≠ i₀, vᵢ=0 and vᵢ₀>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.