Discrete Lotka Research Articles

Stable periodic solution of a discrete periodic Lotka–Volterra competition system

This paper discusses a discrete Lotka–Volterra competition system. We first obtain the persistence of the system. Assuming that the coefficients in the system are periodic, we obtain the existence of a periodic solution. Moreover, under some additional conditions, this periodic solution is globally stable. Our results not only reduce to those for the scalar equation when there is no coupling but also improve and complement some in the literature.

On the convergence of a solution of the discrete Lotka–Volterra system

For arbitrary positive discrete step size δ adeterminantal solution of the discrete-time Lotka–Volterra (dLV) system isshown to asymptotically converge to the square of some singular value of agiven bidiagonal matrix, where the initial value of the dLV system isuniquely determined by the entries of the matrix. Some basic propertiesof the solution are proved which are important for designing a newnumerical algorithm for computing all of the singular values. They are apositivity of solution, a dependence of the correct initial value on δ, asorting property and an acceleration of convergence speed by enlarging δ.

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.

Discrete Competitive and Cooperative Models of Lotka–Volterra Type

The dynamics of discrete Lotka–Volterra system of two species is investigated. It is shown that the proposed discrete models for competitive and cooperative systems possess “dynamical consistency” with their continuous counterparts.