814 research outputs found
Asymptotic behavior of age-structured and delayed Lotka-Volterra models
In this work we investigate some asymptotic properties of an age-structured
Lotka-Volterra model, where a specific choice of the functional parameters
allows us to formulate it as a delayed problem, for which we prove the
existence of a unique coexistence equilibrium and characterize the existence of
a periodic solution. We also exhibit a Lyapunov functional that enables us to
reduce the attractive set to either the nontrivial equilibrium or to a periodic
solution. We then prove the asymptotic stability of the nontrivial equilibrium
where, depending on the existence of the periodic trajectory, we make explicit
the basin of attraction of the equilibrium. Finally, we prove that these
results can be extended to the initial PDE problem.Comment: 29 page
Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation
In this paper, we consider a non-autonomous stochastic Lotka-Volterra competitive system dxi(t) = xi(t)[(bi(t)Ā” nPj=1aij (t)xj (t))dt+Ā¾i(t)dBi(t)], where Bi(t) (i = 1; 2; Ā¢ Ā¢ Ā¢ ; n) are independent standard Brownian motions. Some dynamical properties are discussed and the suĀ±cient conditions for the existence of global positive solutions, stochastic permanence, extinction as well as global attractivity are obtained. In addition, the limit of the average in time of the sample paths of solutions is estimated
Asymptotic properties of stochastic population dynamics
In this paper we stochastically perturb the classical Lotka{Volterra model x_ (t) = diag(x1(t); ; xn(t))[b + Ax(t)] into the stochastic dierential equation dx(t) = diag(x1(t); ; xn(t))[(b + Ax(t))dt + dw(t)]: The main aim is to study the asymptotic properties of the solution. It is known (see e.g. [3, 20]) if the noise is too large then the population may become extinct with probability one. Our main aim here is to nd out what happens if the noise is relatively small. In this paper we will establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss the limit of the average in time of the sample paths
Competitive Lotka-Volterra Population Dynamics with Jumps
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 th
moment with 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 -dimensional model.Comment: 25 page
Stochastic population dynamics under regime switching II
This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69-84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population
- ā¦