Volterra System Research Articles

Limit cycles for competitor–competitor–mutualist Lotka–Volterra systems

It is known that a limit cycle (or periodic coexistence) can occur in a competitor–competitor–mutualist Lotka–Volterra system { x ̇ 1 = x 1 ( r 1 − a 11 x 1 − a 12 x 2 + a 13 x 3 ) , x ̇ 2 = x 2 ( r 2 − a 21 x 1 − a 22 x 2 + a 23 x 3 ) , x 3 ̇ = x 3 ( r 3 + a 31 x 1 + a 32 x 2 − a 33 x 3 ) , where r i , a i j are positive real constants [X. Liang, J. Jiang, The dynamical behavior of type- K competitive Kolmogorov systems and its applications to 3-dimensional type- K competitive Lotka–Volterra systems, Nonlinearity 16 (2003) 785–801]. In this paper, we shall construct an example with at least two limit cycles, and furthermore, we will show that the number of periodic orbits (and hence a fortiori of limit cycles) is finite. It is also shown that, contrary to three-dimensional competitive Lotka–Volterra systems, nontrivial periodic coexistence does occur even if none of the three species can resist invasion from either of the others. In this case, new amenable conditions are given on the coefficients under which the system has no nontrivial periodic orbits. These conditions imply that the positive equilibrium, if it exists, is globally asymptotically stable.

Linearizability of the polynomial differential systems with a resonant singular point

Integrability and linearizability of polynomial differential systems are studied. The computation of generalized period constants is a way to find necessary conditions for linearizable systems for any rational resonance ratio. A method to compute generalized period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As the application, we discuss linearizable conditions for several Lotka–Volterra systems, and where this is the first time that the linearizability is considered for 3 : − 4 and 3 : − 5 resonances.

Persistence and global stability in discrete models of Lotka–Volterra type

In this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka–Volterra type: { N i ( p + 1 ) = N i ( p ) exp { c i − a i N i ( p ) − ∑ j = 1 n a i j N j ( p − k i j ) } , p ⩾ 0 , 1 ⩽ i ⩽ n , N i ( p ) = N i p ⩾ 0 , p ⩽ 0 , and N i 0 > 0 , 1 ⩽ i ⩽ n , where each N i p for p ⩽ 0 , each c i , a i and a i j are finite and { a i > 0 , a i + a i i > 0 , 1 ⩽ i ⩽ n , and k i j ⩾ 0 , 1 ⩽ i , j ⩽ n . Applying the former results [Y. Muroya, Persistence and global stability for discrete models of nonautonomous Lotka–Volterra type, J. Math. Anal. Appl. 273 (2002) 492–511] on sufficient conditions for the persistence of nonautonomous discrete Lotka–Volterra systems, we first obtain conditions for the persistence of the above autonomous system, and extending a similar technique to use a nonnegative Lyapunov-like function offered by Y. Saito, T. Hara and W. Ma [Y. Saito, T. Hara, W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka–Volterra system with two delays, J. Math. Anal. Appl. 236 (1999) 534–556] for n = 2 to the above system for n ⩾ 2 , we establish new conditions for global asymptotic stability of the positive equilibrium. In some special cases that k i j = k j j , 1 ⩽ i , j ⩽ n , and ∑ j = 1 n a j i a j k = 0 , i ≠ k , these conditions become a i > ∑ j = 1 n a j i 2 , 1 ⩽ i ⩽ n , and improve the well-known stability conditions a i > ∑ j = 1 n | a j i | , 1 ⩽ i ⩽ n , obtained by K. Gopalsamy [K. Gopalsamy, Global asymptotic stability in Volterra's population systems, J. Math. Biol. 19 (1984) 157–168].

A computational method for system of Volterra–Fredholm integral equations

Adomian decomposition method has been applied for solving Volterra–Fredholm equation by Maleknejad [K. Maleknejad, M. Hadizadeh, A new computational method for Volterra–Fredholm integral equations, Comput. Math. Appl. 37 (1999) 1–8] in 1999. In this paper, we solve the system of nonlinear Volterra–Fredholm integral equations by Adomian decomposition method. Some examples are solved to illustrate the accuracy of this method.

Global attractivity of positive periodic solution to periodic Lotka–Volterra competition systems with pure delay

We consider a periodic Lotka–Volterra competition system without instantaneous negative feedbacks (i.e., pure-delay systems) (*) x ˙ i ( t ) = x i ( t ) [ r i ( t ) − ∑ j = 1 n a i j ( t ) x j ( t − τ i j ( t ) ) ] , i = 1 , 2 , … , n . We establish some 3/2-type criteria for global attractivity of a positive periodic solution of the system, which generalize the well-known Wright's 3/2 criteria for the autonomous delay logistic equation, and thereby, address the open problem proposed by both Kuang [Y. Kuang, Global stability in delayed nonautonomous Lotka–Volterra type systems without saturated equilibria, Differential Integral Equations 9 (1996) 557–567] and Teng [Z. Teng, Nonautonomous Lotka–Volterra systems with delays, J. Differential Equations 179 (2002) 538–561].

Almost periodic solution of non-autonomous Lotka–Volterra predator–prey dispersal system with delays

This paper studies a non-autonomous Lotka–Volterra almost periodic predator–prey dispersal system with discrete and continuous time delays which consists of n-patches, the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are “profitless”. Finally, conclusions and two particular cases are given. These results are basically an extension of the known results for non-autonomous Lotka–Volterra systems.

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

We study the general properties of stochastic two-species models for predator-prey competition and coexistence with Lotka–Volterra type interactions defined on a d-dimensional lattice. Introducing spatial degrees of freedom and allowing for stochastic fluctuations generically invalidates the classical, deterministic mean-field picture. Already within mean-field theory, however, spatial constraints, modeling locally limited resources, lead to the emergence of a continuous active-to-absorbing state phase transition. Field-theoretic arguments, supported by Monte Carlo simulation results, indicate that this transition, which represents an extinction threshold for the predator population, is governed by the directed percolation universality class. In the active state, where predators and prey coexist, the classical center singularities with associated population cycles are replaced by either nodes or foci. In the vicinity of the stable nodes, the system is characterized by essentially stationary localized clusters of predators in a sea of prey. Near the stable foci, however, the stochastic lattice Lotka–Volterra system displays complex, correlated spatio-temporal patterns of competing activity fronts. Correspondingly, the population densities in our numerical simulations turn out to oscillate irregularly in time, with amplitudes that tend to zero in the thermodynamic limit. Yet in finite systems these oscillatory fluctuations are quite persistent, and their features are determined by the intrinsic interaction rates rather than the initial conditions. We emphasize the robustness of this scenario with respect to various model perturbations.

Extinction in two dimensional nonautonomous Lotka–Volterra systems with the effect of toxic substances

In this paper, we consider a nonautonomous competitive Lotka–Volterra system of two species with the effect of toxic substances. It is shown that toxic substances play an important role in the extinction of species. We prove that one of components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation under some conditions.

Influence of stochastic perturbation on prey–predator systems

We analyse the influence of various stochastic perturbations on prey–predator systems. The prey–predator model is described by stochastic versions of a deterministic Lotka–Volterra system. We study long-time behaviour of both trajectories and distributions of the solutions. We indicate the differences between the deterministic and stochastic models.

Global stability for separable nonlinear delay differential systems

In this paper, we consider separable nonlinear delay differential systems and we establish conditions for global asymptotic stability of the zero solution. Applying these, we offer improved 3/2-type criteria for global asymptotic stability of nonautonomous Lotka–Volterra systems with delays.

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.

Revisiting spurious-free dynamic range of communication receivers

The spurious free dynamic range (SFDR) is commonly used as a measure of dynamic range for the radio frequency and microwave front-end receivers. Although well defined in narrow-band systems, the definition becomes less clear in wide-band systems, when the nonlinearity is memoryless and the the noise figure is frequency dependent. To generalize the SFDR to wide-band systems, a meaningful physical interpretation of the conventional two-tone test is first developed. Based on this interpretation, the upper bound of the wide-band SFDR is obtained by applying a multitone test, while the lower bound is computed using the effective noise figure. The multitone test in both the memoryless and memory nonlinear Volterra systems is considered. A practical measurement technique to characterize the Volterra kernel is also provided. A realistic example based on a low noise amplifier shows a significant difference between the conventional and wide-band SFDR values. In this example, our results suggest that the use of two tones widely separated in frequency to model the interferers provides sufficiently accurate results compared to a multitone approximation.

System of nonlinear Volterra’s integral equations with polar kernel and singularities

We study an existence-uniqueness theorem for a system of nonlinear Volterra-type integral equations with polar kernel and singularities. To do so, we use a method of regularization of fractional derivatives with a delta sequence to avoid singularities over diagonal { y | x − y = 0 } . The other singularities we regularize with delta sequences with different growth. As a framework, we use Colombeau special algebra.

Stability of disease free sets in epidemic models

In this paper the attractivity properties of disease free subsets are considered in the context of disease transmission models. Sufficient conditions are derived for the existence of stable disease free subsets in a general compartmental disease transmission model. The conditions are stated in terms of the system linearized along the trajectories limited to a subset of disease free states. The proof is in the framework of the classical direct method of Lyapunov. As illustrations of the result a multigroup SIRS vaccination model and a Lotka–Volterra system with prey epidemic interaction are presented.

Stability analysis for Lotka–Volterra systems based on an algorithm of real root isolation

Based on the theory of monotone flows of solutions of systems of differential equations, the Routh–Hurwitz theorem and a real root isolation algorithm of multivariate polynomials are applied to a class of Lotka–Volterra diffusion systems. An algorithm to determine the location of equilibria and the stability of the nonlinear dynamics systems is implemented in Maple.

Periodic solutions for a two-species nonautonomous competition system with diffusion and impulses

Abstract By re-estimating the upper bound of ∫ 0 ω e u i ( t ) d t (i = 1, 2), we generalize a result about the existence of a positive periodic solution for a two-species nonautonomous patchy competition system with time delay. Based on that system, we consider the impulsive harvesting and stocking, and establish a two-species nonautonomous competition Lotka–Volterra system with diffusion and impulsive effects. With the continuation theorem of coincidence degree theory, we obtain the existence of a positive periodic solution for such a system. At last, two examples are given to demonstrate our results.

Tom-Based Blind Identification of Nonlinear Volterra Systems

This paper extends blind single-input single-output (SISO) Volterra-system identification from the second-order statistics (SOSs) domain into the third-order statistics domain. For the full-sized Volterra system with finite order and memory, which is excited by unobservable independent identically distributed (i.i.d.) stationary random sequences, it is known that blind identifiability is not possible in the SOS domain. Although this conclusion is also true in the higher order statistics (HOSs) domain, it will be shown that under some sufficient conditions, a larger set of sparse Volterra systems can be identified blindly in third-order moment (TOM) domain than in the SOS counterpart. This is due to the fact that (n+1)(3n+2)/2 terms of different statistical quantities can be used in the third-order-statistics domain while only (n+2) terms of statistical information are nonredundant for SOS-based blind identification, where n is the memory length of the system. The validity and usefulness of the approach are demonstrated in numerical simulations as well as experiments applied to blindly identify the primary path of active-noise-control (ANC) systems in a practical scenario.

Effects of predation pressure on species packing on a resource gradient: insights from nonlinear dynamics

The classical case of three competitors arranged on a resource gradient such that the central competitor will be excluded due to competition from the other two is studied from the point of view of the effects of added predators. The basic formulation is motivated by a desire to understand the effects of asymmetries in multidimensional Lotka–Volterra systems. We first study the effects of perfectly specialist predators and find a rich collection of possible behaviors of the system including (1) extinction of all predators and subsequent extinction of the subordinate competitor, (2) dominant competitors and their predators coexist but the subdominant competitor goes extinct, (3) all species except the predator of the subordinate competitor coexist in coordinated phase-reversed chaos, (4) exclusion of one or more species occurs through an expanding heteroclinic cycle, and (5) all species coexist in an uncoordinated chaos. We then study the effects of five qualitatively distinct forms of polyphagy. In one case, corresponding to the well-known vulnerability to predation versus competitive ability trade-off, it is possible to have the subordinate competitor be the only survivor in the system. The other three cases of polyphagy lead to distortions in the basic pattern seen in the previously analyzed specialist case. Studying this case of ecologically motivated asymmetries in the basic Lotka–Volterra formulation is a step in the direction of fully understanding interacting populations.

Asymptotic behaviour in periodic three species predator–prey systems

For a three-dimensional periodic Lotka–Volterra system, the asymptotic behaviour of its positive solutions is investigated. More specifically, we give suitable average conditions which lead to extinction of the competitively inferior species and global stable coexistence of the two remaining predator and prey species.

Periodic solution of neutral Lotka–Volterra system with periodic delays

A nonautonomous n-species Lotka–Volterra system with neutral delays is investigated. A set of verifiable sufficient conditions is derived for the existence of at least one strictly positive periodic solution of this Lotka–Volterra system by applying an existence theorem and some analysis techniques, where the assumptions of the existence theorem are different from that of Gaines and Mawhin's continuation theorem [R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977] and that of abstract continuation theory for k-set contraction [W. Petryshyn, Z. Yu, Existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems, Nonlinear Anal. 6 (1982) 943–969]. Moreover, a problem proposed by Freedman and Wu [H.I. Freedman, J. Wu, Periodic solution of single species models with periodic delay, SIAM J. Math. Anal. 23 (1992) 689–701] is answered.