Dynamics of some classes of Lotka-Volterra stochastic operators on low dimensional simplex

Abstract

Lotka-Volterra (LV) operator has been the subject of study in dynamical systems, notably on the asymptotic behaviour of its trajectory. In this thesis, we introduce a general class of LV operator defined on a simplex. This class of LV operator encompasses most of the previously studied LV operators. Our aim is to study the dynamics of operators derived from it for lower dimensional simplex. Firstly, we provide conditions, under which the operator is a bijection when restricted to 2-dimensional simplex (2-simplex). We showed that under such conditions the operator is a homeomorphism, i.e., the determinant of its Jacobian is non-zero. Previously, it was shown that any LV operator which satisfies f-monotonicity condition is bijective. Here, we disprove its converse by giving an example of a bijective LV operator which is not f-monotone. In the second part, we consider and study the limiting behaviour of a class of LV operator defined on the 2-simplex. We find that its interior fixed point is unique. Then, by constructing a Lyapunov function we show that the limiting set lies in the boundary. We also provide a description of the path taken by its trajectory as the number of iterations tends to infinity. We estimate the time the trajectory spent within a neighbourhood of a vertex, and we find that the trajectory does not remain in any of the neighbourhoods. Thereupon, we proceed to show that the operator has the property of being non-ergodic. Then, a special case of the operator above is considered in the next part, in which we consider a convex combination of this non-ergodic higher order LV operator and a regular quadratic LV operator. We find that fixed points exist at the edge of the 2-simplex when its respective parameter is above a critical value. Furthermore, they are saddle-nodes when parameter exceeds the critical value by certain amount or repelling if otherwise. Then, by imposing a condition to its parameter we construct a Lyapunov function for the operator. We show that, under such condition; the LV operator derived from such combination has the property of being regular, and its trajectory converges to one of the vertices for any initial point other than the edge fixed points and interior fixed point. A full description of the limit set of such a trajectory is also obtained. In the final part, we consider a class of LV operator defined on the 3-dimensional simplex (3-simplex). Besides having a unique interior fixed point, we also show the existence of uncountable fixed points on two of its edges. We construct several Lyapunov functions, by which we find that its trajectory converges to the edges. Then, we prove the existence of a subset of 3-simplex with positive measure such that the trajectory of the operator does not converge for any initial point taken from the set, and the eventual path taken by such trajectory is described. Using similar technique as in the case of 2-simplex, we show that the Cesaro mean of its trajectory diverges, i.e., the operator is non-ergodic. At the end, the dynamics on its 2-dimensional faces (2-faces) are studied. The operator is found to be regular when restricted on its 2-faces, and the limiting set is estimated.