We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutati...

We study a cyclic Lotka-Volterra model of N interacting species populating a d-dimensional lattice. In the realm of a Kirkwood approximation, a critical number of species Nc(d) above which the system fixates is determined analytically. We find Nc = 5, 14, 23 in dimensions d = 1, 2, 3, in remarkably good agreement with simulation results in two dimensions. A cyclic variant of the Lotka-Volterra ...

Since the recent research has shown the importance of biological control in many biological systems appearing in nature, this research paper investigates research in the dynamic and chaotic analysis of the generalized Lotka-Volterra three-species biological system, which was studied by Samardzija and Greller (1988). The generalized Lotka-Voterra biological system consists of two predator and on...

It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to nonautonomous systems of arbitrary finite dimension. That is, for the n species nonautono...

The population control of the Lotka-Volterra model is one of the most important and widely investigated issues in mathematical ecology. In this study, assuming that birth rate is controllable and using the Z-type dynamic method, we develop Z-type control laws to drive the prey population and/or predator population to a desired state to keep species away from extinction and to improve ecosystem ...

A mathematically simple example of a high-dimensional (many-species) Lotka-Volterra model that exhibits spatiotemporal chaos in one spatial dimension is described. The model consists of a closed ring of identical agents, each competing for fixed finite resources with two of its four nearest neighbors. The model is prototypical of more complicated models in its quasiperiodic route to chaos (incl...