Chaînes alimentaires de type Lotka-Volterra bruitées et quasi-stationnarité avec frontières mobiles aléatoires

In the first part of this thesis, we study Lotka-Volterra food chains. This model considers n species whose interactions are governed by Lotka-Volterra equations. More precisely, species i is the prey of species i+1 and the predator of species i−1. In addition, there can be intra-specific interactions but there can be no other interactions than those mentioned above. First, we consider this model as a stochastic differential equation, i.e. we assume that the drift is a Lotka-Volterra type food chain. In this model, we also assume that the noise is degenerate and, more precisely, that it only affects species 1 or species n. We show that, under these assumptions, species persistence is equivalent to the drift having an equilibrium point with all its coordinates positive. Under the condition that all species are present, we understand persistence to mean that the semigroup converges in total variation to a single invariant probability measure whose support is contained in the positive orthant. If we neglect the intra-specific competition of at least one species, then we show that the speed of convergence is polynomial. However, if all species have intra-specific interactions, then the speed of convergence is exponential. In a second step, we consider this model as a piecewise deterministic Markov process. I.e. we consider the model generated by the random switching between N food chains of the Lotka-Volterra type. We also assume that there are two food chains that differ only in the resources allocated to the first species. Under these conditions, we show that the persistence of species is equivalent to the positivity of the coordinates of the equilibrium point of the average chain. Furthermore, we show that the speed of convergence is exponential. In the case of extinction, we also determine which species become extinct and which species survive. We also show that the extinction rate is exponential while the semi-group converges in law to an invariant probability measure putting weight only on the surviving species. In addition, we also discuss the critical case and the sensitivity of the model to parameters. The second part of this thesis deals with quasi-stationarity with random moving boundaries. More precisely, we assume that the boundary can only take a finite number of possible values and that these changes are governed by a jump process. Under new Champagnat-Villemonais conditions, we show the existence of a Q-process and a quasiergodic measure. Moreover, we also show the ergodicity of the flow induced by the marginal law of the process conditioned not to be absorbed.