Abstract:
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 : $$ \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 competition terms $\Psi_i$ are independent of the virus type $i$. In this situation and under some rather general assumptions on the functions $\Psi_i$, the coefficients $r_i$ and the mutation matrix $\mu_{ij}$ 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 $\bar v$ is strictly positive in the sense that $\bar v_i>0$ 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 $i\neq i_0, \bar v_{i}=0$ and $\bar v_{i_0}>0$). 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.

Abstract:
This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) We discuss the uniform boundedness of $p$th moment with $p>0$ and reveal the sample Lyapunov exponents; (c) Using a variation-of-constants formula for a class of SDEs with jumps, we provide explicit solution for 1-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our $n$-dimensional model.

Abstract:
We study the properties of Lotka-Volterra competitive models in which the intensity of the interaction among species depends on their position along an abstract niche space through a competition kernel. We show analytically and numerically that the properties of these models change dramatically when the Fourier transform of this kernel is not positive definite, due to a pattern forming instability. We estimate properties of the species distributions, such as the steady number of species and their spacings, for different types of kernels.

Abstract:
This paper studies multispecies nonautonomous Lotka-Volterra competitive systems with delays and fixed-time impulsive effects. The sufficient conditions of integrable form on the permanence of species are established.

Abstract:
For autonomous Lotka-Volterra systems of differential equations modelling the dynamics of n competing species, new criteria are established for the existence of a single point global attractor. Under the conditions of these criteria, some of the species will survive and stabilise at a steady state whereas the others, if any, will die out.

Abstract:
Global dynamical behaviors of the competitive Lotka-Volterra system even in 3-dimension are not fully understood. The Lyapunov function can provide us such knowledge once it is constructed. In this paper, we construct explicitly the Lyapunov function in three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-dimensional case; (2) a 3-dimensional model; (3) the model of May-Leonard. The dynamics of these examples include bistable case and cyclical behavior. The first two examples are the generalized gradient system defined in the Appendixes, while the model of May-Leonard is not. Our method is helpful to understand the limit cycle problems in general 3-dimensional case.

Abstract:
The paper discusses a nonautonomous discrete time Lotka-Volterra competitive system with pure delays and feedback controls. New sufficient conditions for which a part of the -species is driven to extinction are established by using the method of multiple discrete Lyapunov functionals.

Abstract:
In this paper, by using Mawhin's continuation theorem of coincidence degree theory, we establish the existence of at least four positive periodic solutions for a discrete time Lotka-Volterra competitive system with harvesting terms. An example is given to illustrate the effectiveness of our results.

Abstract:
A nonautonomous -species discrete Lotka-Volterra competitive system with delays and feedback controls is considered in this work. Sufficient conditions on the coefficients are given to guarantee that all the species are permanent. It is shown that these conditions are weaker than those of Liao et al. 2008.

Abstract:
In this work we focus on a natural class of population protocols whose dynamics are modelled by the discrete version of Lotka-Volterra equations. In such protocols, when an agent $a$ of type (species) $i$ interacts with an agent $b$ of type (species) $j$ with $a$ as the initiator, then $b$'s type becomes $i$ with probability $P\_{ij}$. In such an interaction, we think of $a$ as the predator, $b$ as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an $n$-agent population converges to some absorbing state in time polynomial in $n$, w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We start by presenting a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but "control" convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a distributed system. Some of our techniques may be of independent value.