Discrete Lotka-Volterra System Research Articles

Time-delay version of the integrable discrete Lotka-Volterra system in terms of the LR transformations

To introduce time delay into integrable discrete systems, we present a time-delay version of the discrete Lotka-Volterra (dLV) system, which is a time-discretization of the famous predator-prey Lotka-Volterra system. Focusing on the LR transformations, which has been designed for solving symmetric eigenvalue problems, plays a key role in deriving the essential properties of the original dLV system. We also clarify asymptotic convergence in the resulting time-delay system and present an application for computing matrix eigenvalues and singular values.

Accurate similarity transformation derived from the discrete Lotka–Volterra system for bidiagonal singular values

In this paper, we derive a new similarity transformation from the discrete Lotka---Volterra system by taking an infinite limit of the discretization parameter. The proposed similarity transformatio...

Asymptotic analysis for an extended discrete Lotka–Volterra system related to matrix eigenvalues

The integrable discrete hungry Lotka–Volterra (dhLV) system is easily transformed to the qd-type dhLV system, which resembles the recursion formula of the qd algorithm for computing matrix eigenvalues. Some of the qd-type dhLV variables play a role for assisting the time evolution of the others. This property does not appear in the original dhLV system. In this article, we first show the existence of a centre manifold for the qd-type dhLV system. With the help of the centre manifold theory, we next investigate the local convergence of the qd-type dhLV system, and then clarify the monotonicity related to the qd-type dhLV variables at the final phase of the convergence.

Spiral patterns near Turing instability in a discrete reaction diffusion system

In this paper, linear stability analysis is applied to an exponential discrete Lotka–Volterra system, which describes the competition between two identical species. Conditions for the Turing instability are obtained and the emergence of spiral patterns is demonstrated by means of numerical simulations in the vicinity of the bifurcation point. Moreover, the impact of crucial system parameters on the stability and coherence of spiral patterns is illustrated on several examples.

On m-step Fibonacci sequence in discrete Lotka-Volterra system

The integrable Lotka-Volterra (LV) system stands for a prey-predator model in mathematical biology. The discrete LV (dLV) system is derived from a time-variable discretization of the LV system. The solution to the dLV system is known to be represented by using the Hankel determinants. In this paper, we show that, if the entries of the Hankel determinants become m-step Fibonacci sequences at the initial discrete-time, then those are also so at any discrete time. In other words, the m-step Fibonacci sequences always arrange in the entries of the Hankel determinants under the time-variable evolution of the dLV system with suitable initial setting. Here the 2-step and the 3-step Fibonacci sequences are the famous Fibonacci and Tribonacci sequences, respectively. We also prove that one of the dLV variables converges to the ratio of two successive and sufficiently large m-step Fibonacci numbers, for example, the golden ratio in the case where m=2, as the discrete-time goes to infinity. Some examples are numerically given.

Center manifold approach to discrete integrable systems related to eigenvalues and singular values

The existence of center manifolds is closedly related to local behavior of dynamical systems. In this paper we consider center manifolds both of the discrete Toda equation and the discrete Lotka-Volterra system. Their solutions converge to eigenvalues anti singular values of certain structured matrices. A free parameter plays a key role to show the existence of a center manifold of the discrete Lotka-Volterra system. A monotone convergence of the solution of the discrete Lotka-volterra system is proved with the help of the existence of a center manifold. In contrast, a center manifold of the discrete Toda equation does not always exist.

Probabilities of extinction, weak extinction,permanence, and mutual exclusion in discrete, competitive, Lotka-Volterra systems that involve invading species

The probabilities of various biological asymptotic dynamics are computed for a stable system that is invaded by another competing species. The asymptotic behaviors studied include extinction, weak extinction, permanence, and mutual exclusion. The model used is a discrete Lotka-Volterra system that models species that compete for the same resources. Among the results found are that the chance of permanence occurring in the invaded system is significantly higher than the probability of permanence in a purely random system, and that multiple extinctions that include the invading species and one of the original species are impossible.

An application of the discrete Lotka–Volterra system with variable step-size to singular value computation

The discrete Lotka–Volterra system with variable step-size is applied to a numerical algorithm for computing singular values. A new version of integrable singular value decomposition algorithm is designed, where the step-size δ is then a stepwise parameter δ(n). Some examples demonstrate that a better choice of the step-size gives a benefit in both convergence speed and numerical accuracy.

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.

A Study of Solutions to Discrete Time Lotka-Volterra Equation: I. Integrable Case

The time evolution of a class of completely integrable discrete Lotka-Volterra system is shown not unique but have two different ways chosen randomly at every step of generation. This uncertainty is consistent with the existence of constants of motion and disappears in both continuous time and ultra discrete limits.

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.