Dimensional Lotka Research Articles

Integrable Symmetric (2 + 1)-dimensional Lotka–Volterra Equation with Self-consistent Sources

Integrable Symmetric (2 + 1)-dimensional Lotka–Volterra Equation with Self-consistent Sources

Rank Two Integral Aspects of Three Dimensional Lotka–Volterra Equations with Nonstandard Analysis

Rank Two Integral Aspects of Three Dimensional Lotka–Volterra Equations with Nonstandard Analysis

Global observer for Lotka–Volterra systems

This paper develops a new approach to the long-standing problem of global asymptotic positive observer design for n -dimensional Lotka–Volterra (LV) systems. In this approach, instead of the state error dynamics, the notion of state ratio dynamics is used by exploiting the positivity property of the LV systems. A novel class of observers is proposed and the global asymptotic convergence of such observers is studied via a well-known logarithmic Lyapunov function. The obtained convergence conditions of the proposed observer are formulated as a linear matrix inequality (LMI), which can be used to determine the unknown parameters of the observer. This design procedure is modified for state estimation of time-varying LV systems, too. The applicability of the proposed method is investigated through two numerical examples.

A general theory of coexistence and extinction for stochastic ecological communities.

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

Integrability of a class of N–dimensional Lotka–Volterra and Kolmogorov systems

We study the integrability of an N–dimensional differential Kolmogorov systems of the formx˙j=xj(aj+∑k=1Najkxk)+xjΨ(x1,…,xN),j=1,…,N, where aj, and ajk are constants for j,k=1,…,N and Ψ(x1,…,xN) is a homogeneous polynomial of degree n>2, with either one additional invariant hyperplane, or with one exponential factor. We also study the integrability of the N–dimensional classical Lotka-Volterra systems (when Ψ(x1,…,xN)=0). In particular we consider the integrability of the asymmetric May–Leonard systems.

Empirical estimation of urban life cycle using the three dimensional Lotka-Volterra model

This paper addresses the issues of urban growth sustainability by emphasizing on urban life cycle dynamics and its determinants with respect to the issues of environmental quality. A number of qualitative models have been discussed in order to understand urban dynamics. This paper empirically estimates the parameters of the mathematical model by the interaction of urban population, urban rent and urban income in the framework of the three dimensional Lotka-Volterra model. Numerical simulations of international cities are divided into five categories. The results confirm the cyclic behavior of the three variables with little difference in significance level. The findings of this paper and the application of the three dimensional Lotka Volterra Model can be used for policy formulation for future urban planning.

Integrability and Linearizability Problems of Three Dimensional Lotka–Volterra Equations of Rank-2

This paper deals with both integrability and linearizability problems of three dimensional Lotka–Volterra systems at the origin with rank-2. We give necessary conditions of a system with quadratic nonlinearities with $$(1:3:-\,1)$$ -resonance. We proved the sufficiency of these conditions by showing the existence of two functionally independent first integrals via the Darboux method with inverse Jacobi multiplier together with some other techniques such as two equations define a linearizable node and the third equation is linearizable using the power series argument as well as the monodromy method.

A concrete example with three limit cycles in Zeeman’s class 29 for three dimensional Lotka–Volterra competitive systems

The number of limit cycles for three dimensional Lotka–Volterra competitive systems is an open problem. Recently, we have presented a concrete example with three limit cycles in Zeeman’s class 27 [6].In this paper, we present a concrete example with three limit cycles which belongs to Zeeman’s class 29. We explicitly give the critical parameter values such that the interior equilibrium is an exact unstable weak focus of order two. Also we verify that the system is permanent. This implies that there can exist three limit cycles around the interior equilibrium under suitable perturbations. We actually generate multiple limit cycles, and confirm them by numerical simulation. In addition, we present some other examples with three limit cycles in Zeeman’s class 27.

A concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems

The number of limit cycles for three dimensional Lotka–Volterra systems is an open problem. Recently, Yu et al. (2016) constructed some examples with the possibility of the existence of four limit cycles. Unfortunately, multiple limit cycles are not visible by numerical simulations, because all of them are very close to the interior equilibrium and extremely small. We present a concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems which we can confirm them by numerical simulations. First we prepare the modified formula to compute coefficients of the normal form for the generalized Hopf bifurcation. Applying this formula to three dimensional Lotka–Volterra competitive systems with the aid of the computer algebra system, we derive the critical parameter values explicitly such that the interior equilibrium is exactly an unstable weak focus. Also we show that the heteroclinic cycle on the boundary of R+3 is repelling. This implies that there exists a stable limit cycle by the Poincare–Bendixson theorem. Then, adding some suitable perturbations to parameters, we generate additional two limit cycles near the interior equilibrium by the generalized Hopf bifurcation. Finally we confirm that there exist three limit cycles by numerical simulations.

A Geometric Investigation of the Invariant Algebraic Curves in Two Dimensional Lotka–Volterra Systems

The aim of this work is to investigate the geometric behaviour of the invariant algebraic curves in two dimensional Lotka–Volterra systems, first classified by Moulin-Ollagnier.

New Integrable Dynamical Systems Derived from the Lie Algebraic Structure of the Homogeneous Lotka–Volterra Ladder Equation

The \(n\)-dimensional homogeneous Lotka–Volterra (HLV) ladder equation possesses Lie symmetries that generate a Lie algebra isomorphic to \(\mathfrak{sl}_{n}(\mathbb{K})\) [J. Phys. Soc. Jpn. 71 (2002) 2396 and 72 (2003) 973], where \(\mathbb{K}=\mathbb{C}\) or \(\mathbb{R}\). In this work, using the Lie algebraic structure of the HLV ladder equation, we derive new \(n\)-dimensional dynamical systems and prove their integrability from the viewpoint of Lie symmetries.

Local integrability and linearizability of three-dimensional Lotka–Volterra systems

We investigate the local integrability and linearizability of three dimensional Lotka–Volterra equations at the origin. Necessary and sufficient conditions for both integrability and linearizability are obtained for (1,-1,1),(2,-1,1) and (1,-2,1)-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable.

Average conditions for competitive exclusion in a nonautonomous two dimensional Lotka–Volterra system

A nonautonomous two dimensional competitive Lotka–Volterra system is considered in this paper. Average conditions on the coefficients are given to guarantee the principle of competitive exclusion in the system. It is shown that our results are extensions of those of the counterpart of periodic system.

A normal form of completely integrable systems

The purpose of this article is to show that a C1 differential system on Rn which admits a set of n−1 independent C2 conservation laws defined on an open subset Ω⊆Rn, is essentially C1 equivalent on an open and dense subset of Ω, with the linear differential system u1′=u1,u2′=u2,…,un′=un. The main results are illustrated in the case of two concrete dynamical systems, namely the three dimensional Lotka–Volterra system, and respectively the Euler equations from the free rigid body dynamics.

Limit cycles and singular point quantities for a 3D Lotka–Volterra system

Four limit cycles are constructed for a three dimensional Lotka–Volterra system. This gives a good example to the cyclicity of 3D Lotka–Volterra systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. What is worth mentioning is that the expressions of focal values are simpler, and the formula is readily done with using computer symbol operation system such as Mathematica due to its linearity.

Invariant Torus in 3D Lotka–Volterra Systems Appearing After Perturbation of Hopf Center

We study a three dimensional Lotka–Volterra systems. In the paper Bobienski and Żolądek (J Ergod Theory Dyn Syst 25:759–791, 2005) four cases of center (i.e. an invariant surface supporting a center) were found. In this paper, we study a codimension 2 component LV Hopf and its versal deformation. We prove that at most one invariant torus may appear. This invariant torus corresponds to the limit cycle bifurcating in the amplitude system.

On certain new integrable second order nonlinear differential equations and their connection with two dimensional Lotka–Volterra system

In this paper, we consider a second order nonlinear ordinary differential equation of the form ẍ+k1(ẋ2/x)+(k2+k3x)ẋ+k4x3+k5x2+k6x=0, where ki’s, i=1,2,…,6, are arbitrary parameters. By using the modified Prelle–Singer procedure, we identify five new integrable cases in this equation besides two known integrable cases, namely (i) k2=0, k3=0 and (ii) k1=0, k2=0, k5=0. Among these five, four equations admit time-dependent first integrals and the remaining one admits time-independent first integral. From the time-independent first integral, nonstandard Hamiltonian structure is deduced, thereby proving the Liouville sense of integrability. In the case of time-dependent integrals, we either explicitly integrate the system or transform to a time-independent case and deduce the underlying Hamiltonian structure. We also demonstrate that the above second order ordinary differential equation is intimately related to the two dimensional Lotka–Volterra system. From the integrable parametric choices of the above nonlinear equation all the known integrable cases of the LV system can be deduced.

Four limit cycles for a three-dimensional competitive Lotka–Volterra system with a heteroclinic cycle

For three-dimensional competitive Lotka–Volterra systems, Zeeman [M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dynamics and Stability of Systems 8 (1993) 189–217] identified 33 stable equivalence classes. Among these, only classes 26–31 may have limit cycles. It is known that all these classes may possess two limit cycles and in classes 27 and 29 three limit cycles have been constructed. It has been conjectured that the maximum number of limit cycles is three. In this paper, we disprove the conjecture by constructing four limit cycles in the three-dimensional competitive Lotka–Volterra system with a heteroclinic cycle (class 27 in Zeeman’s classification). Furthermore, in the case of a heteroclinic cycle on the boundary of the carrying simplex of three-dimensional competitive Lotka–Volterra systems, we show that: (i) the conditions (a) there is a pair of purely imaginary eigenvalues at a positive equilibrium, (b) the first focal values vanishes, and ( c ′ ) the second focal values vanishes do not imply that the heteroclinic cycle is neutrally stable, and hence they do not imply that the interior equilibrium is a center; (ii) the conditions (a) there is a pair of purely imaginary eigenvalues at a positive equilibrium, (b) the first focal value vanishes, and (c) the heteroclinic cycle is neutrally stable do not imply the second focal value vanishes, and hence they do not imply that the interior equilibrium is a center. This refutes a conjecture by Hofbauer and So [J. Hofbauer, J.W.-H. So, Multiple limit cycles for three dimensional Lotka–Volterra Equations, Appl. Math. Lett. 7 (1994) 65–70].

Partial permanence and extinction in an N-species nonautonomous Lotka–Volterra competitive system

In this paper, we first consider a general N-species nonautonomous Lotka–Volterra system. We show that certain average conditions involving the respective averages of the growth rates and the interaction coefficients imply the total persistence of all species. Using this result, we then give computable conditions on the coefficients which imply the ultimate extinction of certain species in a N -dimensional Lotka–Volterra system.

Asymptotic stability of an N-dimensional impulsive competitive system

In this paper, we first consider an N -dimensional Lotka–Volterra system with fixed moments of impulsive perturbations. By means of piecewise continuous functions which are modifications of classical Lyapunov's functions we give sufficient conditions for asymptotic stability of the solutions.