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We study the occurrence of chaos in the Atkinson–Allen model of four competing species, which plays the role as a discrete-time Lotka–Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark–Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson–Allen dynamics can lead to chaos.

This paper is devoted to a stochastic retarded reaction-diffusion equation on all d-dimensional space with additive white noise. We first show that the stochastic retarded reaction-diffusion equation generates a random dynamical system by transforming this stochastic equation into a random one through a tempered stationary random homeomorphism. Then, we establish the existence of a random attractor for the random equation. And the existence of a random attractor for the stochastic equation follows from the conjugation relation between two random dynamical systems. The pullback asymptotic compactness is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.

This paper is devoted to a stochastic retarded lattice dynamical system with additive white noise. We extend the method of tail estimates to stochastic retarded lattice dynamical systems and prove the existence of a compact global random attractor within the set of tempered random bounded sets.

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.

This paper studies the existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response. We derive an existence criterion of periodic orbit in terms of inequalities of system's nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation theory and hypersurface theory. We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value, or between its Hopf bifurcating values. Combining theoretic analysis and numerical simulations, we investigate global bifurcations of limit cycles for a varying parameter system, which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles. Similar results are obtained for the system without mutual interference. [ABSTRACT FROM AUTHOR]

Stochastic discrete-time SIS and SIR models of endemic diseases are introduced and analyzed. For the deterministic, mean-field model, the basic reproductive number R0 determines their global dynamics. If R0≤1, then the frequency of infected individuals asymptotically converges to zero. If R0>1, then the infectious class uniformly persists for all time; conditions for a globally stable, endemic equilibrium are given. In contrast, the infection goes extinct in finite time with probability one in the stochastic models for all R0 values. To understand the length of the transient prior to extinction as well as the behavior of the transients, the quasi-stationary distributions and the associated mean time to extinction are analyzed using large deviation methods. When R0>1, these mean times to extinction are shown to increase exponentially with the population size N. Moreover, as N approaches ∞, the quasi-stationary distributions are supported by a compact set bounded away from extinction; sufficient conditions for convergence to a Dirac measure at the endemic equilibrium of the deterministic model are also given. In contrast, when R0<1, the mean times to extinction are bounded above 1/(1-α) where α<1 is the geometric rate of decrease of the infection when rare; as N approaches ∞, the quasi-stationary distributions converge to a Dirac measure at the disease-free equilibrium for the deterministic model. For several special cases, explicit formulas for approximating the quasi-stationary distribution and the associated mean extinction are given. These formulas illustrate how for arbitrarily small R0 values, the mean time to extinction can be arbitrarily large, and how for arbitrarily large R0 values, the mean time to extinction can be arbitrarily large. [ABSTRACT FROM AUTHOR]

We investigate the dynamical behavior of pull-back trajectories for feedback systems with multiplicative noise and prove that there exists a globally stable positive random equilibrium in the nonnegative orthant Rd , where the global stability means that all pull-back trajectories originating from nonnegative orthant converge to this positive random equilibrium almost surely. The output functions (feedback functions) are assumed to either possess bounded derivatives or be uniformly bounded away from zero. In the first case, we first prove the joint measurability of the metric dynamical system θ with respect to the σ-algebra B(R-) ⊗ F-, where F- = σ{ω 1→ Wt(ω) : t ≤ 0} is the past σ-algebra and Wt(ω) is an Rd-valued two-sided Wiener process, and then combine the L1-integrability of the tempered random variable coming from the definition of the top Lyapunov exponent and the independence between the past σ-algebra and the future σ-algebra F+ = σ{ω 1→ Wt(ω) : t ≥ 0} to obtain a globally stable random equilibrium by constructing the contraction mapping on an F--measurable, L1-integrable, and complete metric input space; in the second case, the sublinearity of output functions (feedback functions) and the part metric play the main roles in the existence and uniqueness of globally attracting positive fixed point in the part of a normal, solid cone. Our results can be applied to a well-known stochastic Goodwin negative feedback system, Othmer-Tyson positive feedback system, and Griffith positive feedback system as well as other stochastic cooperative, competitive, and predator-prey systems. [ABSTRACT FROM AUTHOR]

This paper studies a small-gain theorem for nonlinear stochastic equations driven by additive white noise in both trajectories and stationary distribution. Motivated by the most recent work of Marcondes de Freitas and Sontag [SIAM J. Control Optim., 53 (2015), pp. 2657-2695], we first define the input-to-state characteristic operator K(u) of the system in a suitably chosen input space via a backward Itô integral, and then for a given output function h we define the gain operator as the composition of output function h and the input-to-state characteristic operator K(u) on the input space. Suppose that the output function is either order-preserving or anti-order-preserving in the usual vector order and the global Lipschitz constant of the output function is less than the absolute of the negative principal eigenvalue of the linear matrix. Then we prove the so-called small-gain theorem: the gain operator has a unique fixed point; the image for the input-to-state characteristic operator at the fixed point is a globally attracting stochastic equilibrium for the random dynamical system generated by the stochastic system. Under the same assumption for the relation between the Lipschitz constant of the output function and the maximal real part of the stable linear matrix, we prove that the stochastic system has a unique stationary distribution, which is regarded as a stationary distribution version of the small-gain theorem. These results can be applied to stochastic cooperative, competitive, and predator-prey systems, or even others. [ABSTRACT FROM AUTHOR]

We investigate the long-run behavior of time-dependent Lotka-Volterra equations (Eφ) : dxi/dt = xi(1 + φ(t) + Σnj=1aijxj), i = 1, . . .,n, on the positive orthant. It is proved that the system (Eφ) is decomposed into (E0) and the logistic equation (L) : dq/dt = g(1 + φ(t) - g) in the sense that (D) : Φ(t) = g(t)ψ(∫t0g(s)ds), where Φ(·), ψ(·), and g(·) are the solutions of (Eφ), (E0), and (L), respectively. Suppose that φ(t) is periodic with the mean value M{φ} > -1. Then the existence of stable equilibrium, periodic solution, and chaotic motion for (E0) implies the existence of stable periodic solution, quasiperiodic solution, and chaotic motion for (Eφ), respectively. The complete dynamical classification for 3-dimensional competitive system (E0) is provided in terms of competitive coefficients. There are 37 topological classes, in 34 of which any trajectory converges to an equilibrium. Among the remaining three classes, the first has a heteroclinic cycle attracting all positive points except the ray joining the origin and the positive equilibrium; the second possesses a family of periodic orbits attracting all positive points; the third class, where a family of periodic orbits, an asymptotically stable equilibrium, a heteroclinic cycle, and an invariant noncoordinate plane coexist, is a new type we find for the first time. It is shown that for 3-dimensional periodic competitive system (Eφ), any trajectory for Poincaré mapping tends to a fixed point, a periodic orbit, an invariant minimal closed curve, or a heteroclinic cycle. All their attracting domains, which are all cone-like, are exactly described via (D) and carrying simplex theory. The dynamics can be completely classified into 37 classes corresponding to the autonomous systems. All counterparts hold when φ(t) is another minimal function, such as a quasiperiodic, almost periodic, or almost automorphic function, via (D), the classification above, and skew-product flow theory. [ABSTRACT FROM AUTHOR]

A competitive system on the n-rectangle: {x ∈ Rn: 0 ⩽xi ⩽li, i = 1,... ,n} was con- sidered, each species of which, in isolation, admits logistic growth with the hyperbolic structure saturation. It has an (n - 1)-dimensional invariant surface called carrying simplex Σ as a globe attractor, hence the long term dynamics of the system is completely determined by the dynamics on Σ. For the three-dimensional system, the whole dynamical behavior was presented. It has a unique positive equilibrium point and any limit set is either an equilibrium point or a limit cycle. The system is permanent and it is proved that the number of periodic orbits is finite and non-periodic oscillation the May-Leonard phenomenon does not exist. A criterion for the positive equilibrium to be globally asymptotically stable is provided. Whether there exist limit cycles or not remains open. [ABSTRACT FROM AUTHOR]

This paper studies almost periodic/automorphic monotone systems with positive translation invariance via skew-product flows. It is proved that every bounded solution of such systems is asymptotically almost periodic/automorphic. Applications are made to a chemical reaction network, especially to enzymatic futile cycles with almost periodic/automorphic reaction coefficients. [ABSTRACT FROM AUTHOR]

Suppose that $(theta ,varphi )$ is a monotone (order-preserving) random dynamical system (RDS for short) with state space $V$, where $V$ is a real separable Banach space with a normal solid minihedral cone $V_{+}$. It is proved that the unique equilibrium of $(theta ,varphi )$ is globally attractive if every pull-back trajectory has compact closure in $V$. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability and saddle-node bifurcation of a model for the West Nile virus transmission dynamics. The existence and classification of the equilibria are presented. By the theory of K-competitive dynamical systems and index theory of dynamical systems on a surface, sufficient and necessary conditions for local stability of equilibria are obtained. We also study the saddle-node bifurcation of the system. Explicit subthreshold conditions in terms of parameters are obtained beyond the basic reproduction number which provides further guidelines for accessing control of the spread of the West Nile virus. Our results suggest that the basic reproductive number itself is not enough to describe whether West Nile virus will prevail or not and suggest that we should pay more attention to the initial state of West Nile virus. The results also partially explained the mechanism of the recurrence of the small scale endemic of the virus in North America. [ABSTRACT FROM AUTHOR]

Bowman et al. [Bull. Math. Biol., 67 (2005), pp. 1107-1133] proposed a ninedimensional system of ordinary differential equations modelling West Nile virus in a mosquito-birdhuman community and presented some mathematical analysis and its biological explanation. Jiang et al. [Bull. Math. Biol., 2008, DOI 10.1007/s11538-008-9374-6] continued to study the existence and classification of all equilibria and all their local stability and dealt with saddle-node bifurcation of the system. The previous investigation shows that the unique positive equilibrium is globally asymptotically stable if the basic reproduction number is greater than one and the bird death rate is suitably small, but numerical simulations suggest that the unique endemic equilibrium is globally asymptotically stable even if for a large value of the bird death rate. So, they all leave it an open problem. The present paper is to provide a thorough classification of dynamics for this system. In particular, if the reproduction number is greater than one, then a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the feasible region, and the disease persists at an endemic equilibrium if it initially exists, which completely solves the open problem above. Besides, the sufficient and necessary conditions for switch phenomena of the model are obtained if the reproduction number is smaller than one. The results show that the reproduction number alone is not enough to determine whether West Nile virus can prevail or not. Meanwhile, the dynamics of the model for the critical case where the reproduction number is one is also analyzed. [ABSTRACT FROM AUTHOR]

Abstract  This paper provides a complete classification on the global phase portraits of quadratic Lotka–Volterra system $${\dot{x}=x(a_{0}+a_{1}x+a_{2}y),\, \dot{y}=y(b_{0}+b_{1}x+b_{2}y)}$$ . There are 143 different global phase portraits in the Poincaré disc up to a reversal of sense of all orbits. Firstly the sufficient and necessary conditions of the system with closed orbits are given. Then the properties on infinite and finite critical points are discussed. All global phase portraits are finally classified by combining all local information. [ABSTRACT FROM AUTHOR]

Abstract  A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population was proposed by Li et al. in (J Math Biol 10:1037–1052, 1986). The sufficient and necessary conditions for coexistence and the sufficient conditions for stability of the boundary equilibria were provided. This paper will present a thorough classification of dynamics for this model in terms of the first and second so called reproductive numbers of infection in strains I and J. This classification not only solves a conjecture proposed in (Li et al., J Math Biol 10:1037–1052, 1986) but also gives the sufficient and necessary conditions for the competitive exclusion. [ABSTRACT FROM AUTHOR]

In this paper, dynamics of the n-species competitive system with migration is studied. It is proved that if the Jacobian matrix of the system is irreducible at every point in Int ℝ+2n, then there is a defined countable family of invariant (2n - 1)-cells which attract all nonconvergent persistent trajectories. Moreover, it is proved that the Poincaré-Bendixson theorem holds for 2-species competitive systems with migration. [ABSTRACT FROM AUTHOR]

This article gives some sufficient conditions for the existence and nonexistence of nonzero periodic solutions of the general autonomous system of Liénard type [This symbol cannot be presented in ASCII format] The main purpose of this article is to study the problem of how small the extent for the function f(x,y) should be to guarantee the existence of nonzero periodic solutions of this system. With some standard additional assumptions, we prove that if [This symbol cannot be presented in ASCII format] then the system has at least one nonzero periodic solution, otherwise, the system has no nonzero periodic solution. [ABSTRACT FROM AUTHOR]

The authors study the ω-limit set dichotomy of the Kolmogorov systems \/math>i=xifi(x),\quad xi≥0,\;\;1≤i≤n with the cooperative and irreducible hypotheses and obtain the quasiconvergence almost everywhere when n=3, which gives an affirmative answer to the open problem by Smith [9, p.72] in the case of n=3. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of dynamical systems with limited competition is investigated. We study index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to. A nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits. Applications are made to time-periodic ordinary differential equations and reaction-diffusion equations. [ABSTRACT FROM AUTHOR]

It is proved that every solution of the Neumann initial-boundary problem { ∂ui/∂t = diΔui + Fi(u) t > 0, x ∈ Ω, ∂ui/∂n(t, x) = 0 t > 0, x ∈ ∂Ω, i = 1,2,...,n, ui(x, 0) = ui,0(x) ≥ 0 x ∈ Ω..., converges to some equilibrium, if the system satisfies (i) ∂Fi/∂uj ≥ 0 for all 1 ≤ i ≠ j ≤ n, (ii) F(u*g(s)) ≥ h(s) * F(u) whenever u ∈ IR+n and 0 ≤ s ≤ 1, where x * y = (x1y1,..., xnyn) and g, h : [0, 1] → [0, 1]n are continuous functions satisfying gi(0) = hi(0) = 0, gi(1) = hi(1) = 1, 0 < gi(s); hi(s) < 1 for all s ∈ (0, 1) and i = 1,2,...,n, and (iii) the solution of the corresponding ordinary differential equation system is bounded in IR+n. We also study the convergence of the solution of the Lotka-Volterra system { ∂ui/∂t = Δui + ui(ri + ...aijuj t > 0, x ∈ Ω, ∂ui/∂n + αui = 0 t > 0, x ∈ ∂Ω, i = 1,2,...,n, ui(x, 0) = ui,0(x) ≥ 0 x ∈ Ω..., where ri > 0, α ≥ 0, and aij ≥ 0 for i ≠ j. [ABSTRACT FROM PUBLISHER]