Extinction in a two dimensional Lotka–Volterra system with infinite delay

A nonautonomous two dimensional Lotka–Volterra system with infinite delay is considered. An extension of the principle of competitive exclusion is obtained.     

Global attractor in competitive Lotka–Volterra systems

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 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniform persistence for Lotka–Volterra-type delay differential systems

Consider the uniform persistence (permanence) of models governed by the following Lotka–Volterra-type delay differential system:

where each r_i(t) is a nonnegative continuous function on [0,+∞), r_i(t)0, each a_i0 and τ_ij^k(t)t, 1i,jn, 0km.In this paper, we establish sufficient conditions of the uniform persistence and contractivity for solutions (and global asymptotic stability). In particular, we extend the results in Wang and Ma (J. Math. Anal. Appl. 158 (1991) 256) for a predator–prey system and Lu and Takeuchi (Nonlinear Anal. TMA 22 (1994) 847) for a competitive system in the case n=2, to the above system with n2.     

The Criteria for Globally Stable Equilibrium in n-Dimensional Lotka–Volterra Systems

In this paper, a set of sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium point in various submodels of the classic n-dimensional Lotka–Volterra system. The submodels are the following systems: competition (cooperative or predator–prey) chain system and competition (cooperative or predator–prey) model between one and multispecies. The criteria in this paper are in explicit forms of the parameters and thus are easily verifiable.     

Multi‐dimensional Lotka–Volterra systems for carcinogenesis mutations

In the paper we consider three classes of models describing carcinogenesis mutations. Every considered model is described by the system of (n+1) equations, and in each class three models are studied: the first is expressed as a system of ordinary differential equations (ODEs), the second—as a system of reaction–diffusion equations (RDEs) with the same kinetics as the first one and with the Neumann boundary conditions, while the third is also described by the system of RDEs but with the Dirichlet boundary conditions. The models are formulated on the basis of the Lotka–Volterra systems (food chains and competition systems) and in the case of RDEs the linear diffusion is considered. The differences between studied classes of models are expressed by the kinetic functions, namely by the form of kinetic function for the last variable, which reflects the dynamics of malignant cells (that is the last stage of mutations). In the first class the models are described by the typical food chain with favourable unbounded environment for the last stage, in the second one—the last equation expresses competition between the pre‐malignant and malignant cells and the environment is also unbounded, while for the third one—it is expressed by predation term but the environment is unfavourable. The properties of the systems in each class are studied and compared. It occurs that the behaviour of solutions to the systems of ODEs and RDEs with the Neumann boundary conditions is similar in each class; i.e. it does not depend on diffusion coefficients, but strongly depends on the class of models. On the other hand, in the case of the Dirichlet boundary conditions this behaviour is related to the magnitude of diffusion coefficients. For sufficiently large diffusion coefficients it is similar independently of the class of models, i.e. the trivial solution that is unstable for zero diffusion gains stability. Copyright © 2009 John Wiley & Sons, Ltd.     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

Singularity of Lotka‐Volterra models under unfoldings

In this paper, we classify the singularity of a Lotka‐Volterra competitive model with a Gaussian competition function and non‐Gaussian carrying capacity functions. These functions need not be completely different to affect adaptive dynamics of the model. For instance, it will be seen how ostensibly similar models can actually give rise to quite different behaviors due to their properties under unfolding. The use of Gaussian‐like carrying capacity functions can also show the sensitivity of the model to assumptions on the carrying capacity function's shapes. The classification is achieved using singularity theory of fitness functions under dimorphism equivalence. We also investigate the effect of the presence of unfolding and bifurcation parameters on the evolution of the system near its singular points. Particularly, we study the adaptive dynamics of the system near the singularity by focusing on ESS and CvSS types, and dimorphisms. Mutual invasibility plots are used to show regions of coexistence.     

Survival of three species in a nonautonomous Lotka–Volterra system

In Ahmad and Stamova (2004) [1], the author considers a competitive Lotka–Volterra system of three species with constant interaction coefficients. In this paper, we study a nonautonomous Lotka–Volterra model with one predator and two preys. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution.     

Almost periodic solution for n-species Lotka–Volterra competitive system with delay and feedback controls

In this paper, a n-species Lotka–Volterra competition system with delay and feedback controls is investigated. By means of the theory of comparison theorem and suitable Lyapunov functional, some sufficient conditions for the existence and uniqueness of almost positive periodic solutions of this system is obtained.     

Dynamical behavior of Lotka–Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise

This article is concerned with the study of trajectory behavior of Lotka–Volterra competition bistable systems and systems with telegraph noises. We proved that for bistable systems, there exists a unique solution, bounded above and below by positive constants. The oscillatory situation of systems with telegraph noises is pointed out.     

Global asymptotic stability in n-species nonautonomous Lotka–Volterra competitive systems with infinite delays

By means of Lyapunov functional, we have succeeded in establishing the global asymptotic stability of the positive solutions of a delayed n-species nonautonomous Lotka–Volterra type competitive system without dominating instantaneous negative feedbacks. As a corollary, we show that the global asymptotic stability of the positive solution is maintained provided that the delayed negative feedbacks dominate other interspecific interaction effects with delays and the mean delays are sufficiently small.     

0–1 laws for maps

A class of finite structures has a 0–1 law with respect to a logic if every property expressible in the logic has a probability approaching a limit of 0 or 1 as the structure size grows. To formulate 0–1 laws for maps (i.e., embeddings of graphs in a surface), it is necessary to represent maps as logical structures. Three such representations are given, the most general being the full cross representation based on Tutte's theory of combinatorial maps. The main result says that if a class of maps has two properties, richness and large representativity, then the corresponding class of full cross representations has a 0–1 law with respect to first‐order logic. As a corollary the following classes of maps on a surface of fixed type have a first‐order 0–1 law: all maps, smooth maps, 2‐connected maps, 3‐connected maps, triangular maps, 2‐connected triangular maps, and 3‐connected triangular maps. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 215–237, 1999     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

Conservation laws for the Black–Scholes equation

In the search for solutions to the important partial differential equation due to Black, Scholes and Merton potential symmetries are very useful as new solutions of the equation can be obtained as a result. These potential symmetries require that the equation be written in conserved form, ie. we need to determine conservation laws for the equation. We calculate the conservation laws utilizing the point symmetries of the equation following the method of Kara and Mahomed [A.H. Kara, F.M. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40].     

Survival and coexistence in stochastic spatial Lotka–Volterra models

A spatially explicit, stochastic Lotka–Volterra model was introduced by Neuhauser and Pacala in Neuhauser and Pacala (Ann. Appl. Probab. 9, 1226–1259, 1999). A low density limit theorem for this process was proved by the authors in Cox and Perkins (Ann. Probab. 33, 904–947, 2005), showing that certain generalized rescaled Lotka–Volterra models converge to super-Brownian motion with drift. Here we use this convergence result to extend what is known about the parameter regions for the Lotka–Volterra process where (i) survival of one type holds, and (ii) coexistence holds. Supported in part by an NSERC Research grant.     

Conservation laws and double reduction of (2+1) dimensional Calogero–Bogoyavlenskii–Schiff equation

In this paper, we obtain conservation laws of (2+1) dimensional Calogero–Bogoyavlenskii–Schiff equation by non‐local conservation theorem method. Besides, exact solutions are obtained by the aid of the symmetries associated with conservation laws. Double reduction is used to obtain these exact solution of Calogero–Bogoyavlenskii–Schiff equation. Copyright © 2016 John Wiley & Sons, Ltd.     

Transient and asymptotic dynamics of a prey–predator system with diffusion

In this paper, we study a prey–predator system associated with the classical Lotka–Volterra nonlinearity. We show that the dynamics of the system are controlled by the ODE part. First, we show that the solution becomes spatially homogeneous and is subject to the ODE part as t → ∞ . Next, we take the shadow system to approximate the original system as D → ∞ . The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as D → ∞ . Although the asymptotic dynamics of the original system are also controlled by the ODE, the time periods of these ODE solutions may be different. Concerning this property, we have that any δ > 0 admits D₀ > 0 such that if , the time period of the ODE, satisfies , then the solution to the original system with D ≥ D₀ cannot approach the stationary state. Copyright © 2012 John Wiley & Sons, Ltd.     

Traveling wave solutions of Lotka–Volterra type two predators‐one prey model

In this work, we consider a model with one basal resource and two species of predators feeding by the same resource. There are three non‐trivial boundary equilibria. One is the saturated state E_K of the prey without any predator. Other two equilibria, E₁ and E₂, are the coexistence states of the prey with only one species of predators. Using a high‐dimensional shooting method, the Wazewski' principle, we establish the conditions for the existence of traveling wave solutions from E_K to E₂ and from E₁ to E₂. These results show that the advantageous species v₂ always win in the competition and exclude species v₁ eventually. Finally, some numerical simulations are presented, and biological interpretations are given. Copyright © 2016 John Wiley & Sons, Ltd.     

非线性对流扩散方程的守恒律

利用直接方法研究了非线性对流扩散方程的守恒律,得到了关于非线性对流扩散方程的守恒律乘子性质的一个定理.利用这个定理,可以简化守恒律乘子的确定方程.随后通过对确定方程中的变量函数进行分析,发现在四种情况下乘子的确定方程是可解的.最后解出这些守恒律乘子,利用积分公式法分别得到了四种情况下对应于各个守恒律乘子的守恒律.     

Computation of periodic orbits in three‐dimensional Lotka‐Volterra systems

This paper deals with an adaptation of the Poincaré‐Lindstedt method for the determination of periodic orbits in three‐dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three‐dimensional Lotka‐Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.