E. Bouse, D. Rachinskii
We consider systems where a cycle born via the Hopf bifurcation blows up to infinity as a parameter ranges over a finite interval. Two examples demonstrating this effect are presented: planar Lotka-Volterra type systems with a competition-cooperation term and quasi-linear higher order equations.
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Stable equilibria in the Lotka-Volterra equations
本文研究Lotka-Volterra系统，提供平衡点稳定的必要条件，补充了Adachi等人关于稳定平衡存在条件的研究。
Although social network analysis is a promising tool to study the structure and dynamics of wildlife communities, the current methods require costly and detailed network data, which often are not available over long time periods (e.g. decades). The present work aims to resolve this issue by developing a new methodology that requires much less detailed data and it relies on well-established Lotka-Volterra models. Using the long-term abundance data (1973-2003) of northeastern Kansas rodents, the changes in the magnitude and direction of interactions (e.g., changes from cooperative behavior to co…
Generalized Lotka-Volterra Systems and Complex Balanced Polyexponential Systems
本文研究广义Lotka-Volterra系统的全局稳定性，引入等价的多指数系统，证明复平衡条件下系统具有全局吸引状态，排除周期解和混沌，并给出基于图结构的简单充分条件。
On the impact of spatial heterogeneity and drift rate in a three-patch two-species Lotka-Volterra competition model over a stream
针对溪流网络中的三斑块两物种 Lotka-Volterra 竞争模型，研究环境异质性与漂移率对物种入侵及竞争结果的影响。
New stability criteria for periodic planar Lotka-Volterra systems
针对周期捕食者-猎物Lotka-Volterra模型，利用共存态平均值的边界，提出了周期解的新稳定性判据，补充了现有文献结果。
A Locally Stable Equilibria Criterion for the Generalized Lotka-Volterra
本文针对广义 Lotka-Volterra 模型的非退化情形，提出通过两个不动点对应群落矩阵的 Schur 补判断其局部稳定性的判据。
In the last years it has been shown that Lotka-Volterra mappings constitute a valuable tool from both the theoretical and the applied points of view, with developments in very diverse fields such as Physics, Population Dynamics, Chemistry and Economy. The purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism (originally conceived for the analysis of ordinary differential equations) can be extended into the mapping domain. The extension of the formalism into the discrete-time context is remar…
This paper deals with coexistence and extinction of time periodic Volterra-Lotka type competing systems with nonlocal dispersal. Such issues have already been studied for time independent systems with nonlocal dispersal and time periodic systems with random dispersal, but have not been studied yet for time periodic systems with nonlocal dispersal. In this paper, the relations between the coefficients representing Malthusian growths, self regulations and competitions of the two species have been obtained which ensure coexistence and extinction for the time periodic Volterra-Lotka type system wi…
A family of planar Lotka--Volterra systems with invariant algebraic curves of arbitrary degree
本文提出了一类平面Lotka-Volterra系统，该系统能够显式地具有任意高次的不变代数曲线。
We solve the analytic integrability problem for diferential systems in the plane whose origin is an isolated singularity and the first homogeneous component is a quadratic Lotka-Volterra type. As an application, we give the analytically integrable systems of a class of systems $\dot{x}= x(P1+P2)$; $\dot{y}= y(Q1+Q2)$; being Pi,Qi homogeneous polynomials of degree i.
The global asymptotic behavior of the classical diffusive Lotka-Volterra competition model with stage structure is studied. A complete classification of the global dynamics is given for the weak competition case. It is shown that under otherwise same conditions, the species with shorter maturation time prevails. The method is also applied to the global dynamics of another delayed competition models.
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.
In this paper, we study a Lotka-Volterra model which contains two prey and one predator with the Beddington-DeAngelis functional responses. First, we establish a set of sufficient conditions for existence of positive periodic solutions. Second, we investigate global asymptotic stability of boundary periodic solutions. Finally, we present some numerical examples.
In this paper, two competitive Lotka-Volterra populations in the two-patch-system with diffusion are considered. Each of the two spiecies can diffuse indepently and discretely between its in intrapatch and interpatch. By means of constructing Liapunov function, under moderate condition, the system has a unique almost periodic solution and which is asymptotically stable and globally attractive .
In this paper a Lotka Volterra type system is considered. For such a system, biHamiltonian formulation, symplectic realizations and symmetries are presented.
We carry out the generalization of the Lotka-Volterra embedding to flows not explicitly recognizable under the Generalized Lotka-Volterra format. The procedure introduces appropiate auxiliary variables, and it is shown how, to a great extent, the final Lotka-Volterra system is independent of their specific definition. Conservation of the topological equivalence during the process is also demonstrated.
For autonomous Lotka-Volterra systems of differential equations modelling the dynamics of n competing species, new criteria are established for the existence of a single point global attractor. Under the conditions of these criteria, some of the species will survive and stabilise at a steady state whereas the others, if any, will die out.
This paper is concerned with a nonlocal diffusion Lotka-Volterra type competition model that consisting of a native species and an invasive species in a one-dimensional habitat with free boundaries. We prove the well-posedness of the system and get a spreading-vanishing dichotomy for the invasive species. We also provide some sufficient conditions to ensure spreading success or spreading failure for the case that the invasive species is an inferior competitor or a superior competitor, respectively.
We prove that the Zarisky closure of the monodromy group of the polynomial $ x^2 y^2(1-x-y) $ is the symplectic group $Sp(4,\mathbb C)$. This shows that some previous results about this monodromy representation are wrong.
According to the long-memory principle appears in fractional-order dynamical systems, analysis of these systems is commonly more complicated than those described by nonlinear ordinary differential equations. Another difficulty is due to the fact that some classical tools such as the Lyapunov stability theorems cannot directly be applied to nonlinear fractional differential systems. The aim of this paper is to study the stability of a general form of the two-dimensional fractional-order Lotka-Volterra equation and numerically investigate the domain of attraction of its stable equilibrium points…
Global dynamical behaviors of the competitive Lotka-Volterra system even in 3-dimension are not fully understood. The Lyapunov function can provide us such knowledge once it is constructed. In this paper, we construct explicitly the Lyapunov function in three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-dimensional case; (2) a 3-dimensional model; (3) the model of May-Leonard. The dynamics of these examples include bistable case and cyclical behavior. The first two examples are the generalized gradient system defined in the Appendixes, while th…
Analysis of a class of Lotka--Volterra systems
该研究分析含无穷小参数的广义二维三次 Lotka–Volterra 模型，在一个非退化和两个退化情形下刻画局部行为，并给出相应分岔图及区域划分。
This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka-Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study the cycling behavior for a stochastic version, where a Brownian noise is introduced via an heterogeneous productivity factor. Sufficient conditions for existence of the system are provided. We prove that the system produces cycles around an equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techn…
In this paper we analyze the flow of a family of three dimensional Lotka-Volterra systems restricted to an invariant and bounded region. The behaviour of the flow in the interior of this region is simple: either every orbit is a periodic orbit or they move from one boundary to another. Nevertheless the complete study of the limit sets in the boundary allows to understand the bifurcations which take place in the region as a global bifurcation that we denote by focus--center--focus bifurcation.
We study a generalized system of ODE's modeling a finite number of biological populations in a competitive interaction. We adapt the techniques in two previous articles to prove the convergence to a unique stable equilibrium.
This paper is concerned with the existence of traveling wave solutions for diffusive two-species Lotka-Volterra systems with delay in both the reaction and diffusion terms without monotonicity. We extend the partial or cross monotone iteration method to systems that satisfy the partial quasi-monotone condition via construction appropriate upper and lower solutions. This is done by using Schauder's fixed point theorem.
Analytical approximations of Lotka-Volterra integrals
本文推导 $x-\ln x=y-\ln y$ 解的解析界，用于估计 Lotka-Volterra 型轨迹，并给出 Lambert $W$ 函数及 Rosenzweig-MacArthur 等捕食者-猎物系统的应用。
The classical Lotka-Volterra predator-prey system is often used in species competition modeling. An exact, closed-form solution is derived when the natural growth rate of the prey species and decay rate of the predators are equal in magnitude. A standard functional transformation yields a novel system of two partially uncoupled rst-order ODEs for \hybrid-species", with one being autonomous. New exact, closed-form time-dependent solutions are derived for each individual species. An analytical expression for the system's oscillation period valid for any value of the system's energy is derived in…
Lotka-Volterra model is one of the most popular in biochemistry. It is used to analyze cooperativity, autocatalysis, synchronization at large scale and especially oscillatory behavior in biomolecular interactions. These phenomena are in close relationship with the existence of first integrals in this model. In this paper we determine the independent first integrals of a family of $n$--dimensional Lotka-Volterra systems. We prove that when $n=3$ and $n=4$ the system is completely integrable. When $n\geq6$ is even, there are three independent first integrals, while when $n\geq5$ is odd there exi…
We examine the algebraic complete integrability of Lotka-Volterra equations in three dimensions. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painleve analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions for the algebraic integrability of the corresponding systems. We also show that the conditions are sufficient.
In the present paper, we study Lotka-Volterra (LV) type operators defined in finite dimensional simplex. We prove that any LV type operator is a surjection of the simplex. After, we introduce a new class of LV-type operators, called $M$LV type. We prove convergence of their trajectories and study certain its properties. Moreover, we show that such kind of operators have totaly different behavior than ${\mathbf{f}}$-monotone LV type operators.
Results for Global Attractivity of Interior Equilibrium Points for Lotka-Volterra Systems
本文研究了Lotka-Volterra系统内部平衡点的全局吸引性，提出了一般性结果并验证了三维案例的吸引性。
Four Limit Cycles in Three-Dimensional Competitive Lotka-Volterra Systems of Class 28 in Zeeman's Classification
本文构造了三维竞争Lotka-Volterra系统，包含四个极限环，属于Zeeman分类中的类28，结合已有结果表明26-29类中均存在至少四个极限环的系统。
Structural stability of invasion graphs for Lotka--Volterra systems
本文研究具有Volterra–Lyapunov稳定结构矩阵的Lotka–Volterra系统，证明入侵图的边刻画平衡点间全部异宿连接，并据此提出与经典数学概念一致的生态学结构稳定性定义。
It is known that Lotka - Volterra type differential equations with delays or distributed delays have an important role in modeling ecological systems. In this paper we study the effects of distributed delay on the dynamics of the harvested one predator - two prey model. Using the expectation of the distribution of the delay as a bifurcation parameter, we show that the equilibrium that was asymptotic stable becomes unstable and Hopf bifurcation can occur as the expectation crosses some critical values.
Predicting periodic solutions of cyclic Lotka-Volterra equations with dynamic mode decomposition
研究用线性隐式Kahan方法求解常微分方程和偏微分方程形式的循环Lotka-Volterra系统，通过动态模式分解预测解，扩展DMD和汉克尔DMD能高精度预测周期解，且能准确保持哈密顿量和卡西米尔量。
We construct a family of integrable deformations of the Bogoyavlenskij-Itoh systems and construct a Lax operator with spectral parameter for it. Our approach is based on the construction of a family of compatible Poisson structures for the undeformed system, whose Casimirs are shown to yield a generating function for the integrals in involution of the deformed systems. We show how these deformations are related to the Veselov-Shabat systems.
We revisit the bifurcation theory of the Lotka-Volterra quadratic system \begin{eqnarray} X_0 :\left\{\begin{aligned} \dot{x}=& - y -x^2+y^2 ,\\ \dot{y}= &\;\;\;\;x - 2xy \end{aligned} \right. \end{eqnarray} with respect to arbitrary quadratic deformations. The system $X_0$ has a double center, which is moreover isochronous. We show that the deformed system $X_0$ can have at most two limit cycles on the finite plane, with possible distribution $(i,j)$, where $i+j\leq2$. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterate…
For classic Lotka-Volterra systems governing many interacting species, we establish an exclusion principle that rules out the existence of linearly asymptotically stable steady states in subcommunitites of communities that admit a stable state which is internally D-stable. This type of stability is known to be ensured, e.g., by diagonal dominance or Volterra-Lyapunov stability conditions. By consequence, the number of stable steady states of this type is bounded by Sperner's lemma on anti-chains in a poset. The number of stable steady states can nevertheless be very large if there are many gro…
In this paper the problems of integrable and linearizable Lotka-Volterra equations with $(δ:-3 δ:δ)$-resonance are studied. The necessary conditions for both problems are obtained in the case when $δ=1$ and it's sufficiency are proved. It is also shown that non-standard analysis has an important role in proving the sufficient integrable conditions in some cases. The non-standard analysis approaches have been used for both perturbed and unperturbed cases for integral aspects of the given system.
e study the minimal wave speed and the asymptotics of the traveling wave solutions of a competitive Lotka Volterra system. The existence of the traveling wave solutions is derived by monotone iteration. The asymptotic behaviors of the wave solutions are derived by comparison argument and the exponential dichotomy, which seems to be the key to understand the geometry and the stability of the wave solutions. Also the uniqueness and the monotonicity of the waves are investigated via a generalized sliding domain method.
We introduce an algorithm to find possible constants of motion for a given replicator equation. The algorithm is inspired by Dirac geometry and a Hamiltonian description for the replicator equations with such constants of motion, up to a time re-parametrization, is provided using Dirac$\backslash$big-isotropic structures. Using the equivalence between replicator and Lotka-Volterra (LV) equations, the set of conservative LV equations is enlarged. Our approach generalizes the well-known use of gauge transformations to skew-symmetrize the interaction matrix of a LV system. In the case of predator…
We prove that the critical pulled front of Lotka-Volterra competition systems is nonlinearly asymptotically stable. More precisely, we show that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof relies on pointwise semigroup methods and utilizes in a crucial way that the faster decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.
This work derives sufficient conditions for the coexistence and exclusion of a stochastic competitive Lotka-Volterra model. The conditions obtained are close to the necessary conditions. In addition, convergence in distribution of positive solutions of the model is also established. A number of numerical examples are given to illustrate our results.
This work is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the paper is to reduce the computational complexity by using the two-time-scale systems. Because existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the genera…
Classification of permanence and impermanence for a Lotka-Volterra model of three competing species with seasonal succession
针对具有季节演替的三竞争物种 Lotka-Volterra 模型，论文利用承载单形按参数不等式完整划分永久持续与非永久持续，并数值揭示永久持续情形中存在不变闭曲线。
Integrability of a Family of Lotka--Volterra Three Species Biological System
本研究针对Lotka–Volterra三物种混沌生物系统的可积性问题，证明两参数为零时系统完全可积，且参数严格为正时不存在多项式、有理、形式级数及Darboux类首次积分。
Omnivory is defined as feeding on more than one trophic level. An example of this is the so-called intraguild predation (IG) which includes a predator and its prey that share a common resource. IG predation models are known to exhibit interesting dynamics including chaos. This work considers a three-species food web model with omnivory, where the interactions between the basal resource, the IG prey, and the IG predator are of Lotka-Volterra type. In the absence of predation, the basal resource follows a delayed logistic equation or popularly known as Hutchinson's equation. Conditions for the e…
We study some aspects of the global dynamics of an $n$-dimensional Lotka-Volterra system with infinite delay and patch structure, such as extinction, persistence, existence and global attractivity of a positive equilibrium. Both the cases of an irreducible and reducible linear community matrix are considered, and no restriction on the signs of the intra- and inter-specific delayed terms is imposed. Although the system is not cooperative, our approach often uses comparison results applied to an auxiliary cooperative system. Some models in recent literature are generalised, and results improved.
Optimal Harvesting of a Stochastic Lotka-Volterra Competition Model with Periodic Coefficients
本文研究了具有周期系数的随机Lotka-Volterra竞争模型的最优捕获问题，通过建立充分条件证明了模型的稳定性，并给出了最优捕获策略和最大可持续产量的显式表达式。
The main objective of this work is to investigate the integrability and linearizability problems around a singular point at the origin of the family of differential systems Particularly we are interested in the three-dimensional cubic Lotka-Volterra systems for (1:-1:1) resonant. We give a complete set of necessary conditions for both integrability and linearizability problems. The local existences of two independent first integrals guarantee that these conditions are sufficient as well. Furthermore, we use the Darboux method of integrability and some other techniques to show that the system a…
In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate t…
We investigate the traveling front solutions of a nonlocal Lotka Volterra system to illustrate the outcome of the competition between two species. The existence of the front solution is obtained through a new monotone iteration scheme, the uniqueness of the front solution corresponding to each propagation speed is proved by sliding domain method adapted to nonlocal systems, and the asymptotic decay rate of the fronts with critical and noncritical wave speeds is derived by a new method, which is different from the single equation case. The results demonstrate that in the long run, two weakly co…
In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using \emph{Chebyshev criterion}, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for $n=3$. The simultaneous bifurcation and distribution of limit cycles for the system with …
This paper is concerned with the traveling wave solutions of delayed reaction-diffusion systems. By using Schauder's fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka-Volterra competition systems with distributed delays are established…
Trees and superintegrable Lotka-Volterra families
本文为每棵 $n$ 顶点树构造含 $3n-2$ 个参数的 Lotka-Volterra 系统，证明其一般超可积，并给出可求积求解的 $(n-1)$ 维约化。
We consider two dimensional Lotka-Volterra systems in fluctuating environment. Relying on recent results on stochastic persistence and piecewise deterministic Markov processes, we show that random switching between two environments both favorable to the same species can lead to the extinction of this species or coexistence of the species.
In this paper we are concerned with the entire solutions for the classical competitive Lotka-Volterra system with diffusion in the weak competition. For this purpose we firstly analyze the asymptotic behavior of traveling front solutions for this system connecting the origin and the positive equilibrium. Then, by using two different ways to construct pairs of coupled super-sub solutions of this system, we obtain two different kinds of entire solutions. The construction of the first kind of entire solutions is based on these fronts, and their reflects as well as the solutions of the system with…
Modeling the Impact of NATO Policy Actions on Taliban Disinformation Campaigns with Lotka-Volterra Models
本研究用改进的Lotka-Volterra捕食者-猎物框架建模北约与塔利班虚假信息宣传的对抗关系，证实主动出击优于停滞，进而提出北约阿富汗青年变革计划以遏制虚假信息传播。
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,…
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