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In this paper, we first construct a microfluidic chemostat model for the growth of biofilms and planktonic populations with random dilution ratios and then investigate its dynamical behavior. Using the theory of monotone dynamical systems and the Multiplicative Ergodic Theorem, we show the existence of random attractors and stationary measures, and present Lyapunov exponents for the linearized cocycle with respect to the random model. Further on, if the top Lyapunov exponent is negative, we give the extinction of microbial populations, including the forward and pull-back trajectories.

This paper provides the first Liapunov coefficient formula of 3D Lotka-Volterra equations. This formula gives applications to stability of positive equilibrium and to detecting sub/super criticality of Hopf bifurcation. For 3D competitive Lotka-Volterra equations, combining this formula with the Poincare-Bendixson theorem, we obtain criteria on multiplicity of limit cycles among Zeeman's classes 27-31, and present a series of examples to admit at least two limit cycles, which are rigorously proved by the first Liapunov coefficient formula, rather than by symbolic computation using Maple. A new Hopf bifurcation that all 2 × 2 principal minors of the community matrix are positive is found, and numerical simulation reveals its global limit cycle bifurcations are plenty.

This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a null set of any limit measure. Using this criterion, we first provide a series of nontrivial concrete examples to show that their repelling limit cycles or quasi-periodic orbits are null sets for all limit measures, which deduces that all their limit measures are concentrated on stable equilibria and stable limit cycles or quasi-periodic orbits, and saddles. Interesting open questions on exact supports of limit measures are proposed.

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.

We study the dynamics of the Ricker model (map) T. It is known that under mild conditions, T admits a carrying simplex , which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all 3D Ricker models admitting carrying simplices. There are a total of 33 stable equivalence classes. We list them in terms of simple inequalities on the parameters, and draw each one's phase portrait on Σ. Classes 1-18 have trivial dynamics, i.e. every orbit converges to some fixed point. Each map from classes 19-25 admits a unique positive fixed point with index -1, and Neimark-Sacker bifurcations do not occur in these 7 classes. In classes 26-33, there exists a unique positive fixed point with index 1. Within each of classes 26 to 31, there do exist Neimark-Sacker bifurcations, while in class 32 Neimark-Sacker bifurcations can not occur. Whether there is a Neimark-Sacker bifurcation in class 33 or not is still an open problem. Class 29 can admit Chenciner bifurcations, so two isolated closed invariant curves can coexist on the carrying simplex in this class. Each map in class 27 admits a heteroclinic cycle, i.e. a cyclic arrangement of saddle fixed points and heteroclinic connections. As the growth rate increases the carrying simplex will break, and chaos can occur for large growth rate. We also numerically show that the 4D Ricker map can admit a carrying simplex containing a chaotic attractor, which is found in competitive mappings for the first time.

In the recent paper [E. C. Balreira, S. Elaydi, and R. Luis, J. Differ. Equ. Appl. 23 (2017), pp. 2037-2071], Balreira, Elaydi and Luis established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luis' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application.

We study the occurrence of the chaotic attractor in the four-dimensional classical Leslie-Gower competition model. We find that chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark-Sacker bifurcation of the positive fixed point in this model. The chaotic attractor is contained in the three-dimensional carrying simplex, that is a globally attracting invariant manifold. Biologically, the result implies that the invasion attempts by an invader into a trimorphic population under the Leslie-Gower dynamics can lead to chaos. (C) 2019 Elsevier B.V. All rights reserved.

Stochastic discrete-time SIS and SIR models of endemic diseases are introduced and analyzed. For the deterministic, mean-field model, the basic reproductive number $R_0$ determines their global dynamics. If $R_0\le 1$, then the frequency of infected individuals asymptotically converges to zero. If $R_0>1$, 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 $R_0$ 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 $R_0>1$, these mean times to extinction are shown to increase exponentially with the population size $N$. Moreover, as $N$ approaches $\infty$, 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 $R_0

This paper mainly aims to study the influence of individuals' different heterogeneous contact patterns on the spread of the disease. For this purpose, an SIS epidemic model with a general form of heterogeneous infection rate is investigated on complex heterogeneous networks. A qualitative analysis of this model reveals that, depending on the epidemic threshold \begin{document}$R_0$ \end{document} , either the disease-free equilibrium or the endemic equilibrium is globally asymptotically stable. Interestingly, no matter what functional form the heterogeneous infection rate is, whether the disease will disappear or not is completely determined by the value of \begin{document}$R_0$ \end{document} , but the heterogeneous infection rate has close relation with the epidemic threshold \begin{document}$R_0$ \end{document} . Especially, the heterogeneous infection rate can directly affect the final number of infected nodes when the disease is endemic. The obtained results improve and generalize some known results. Finally, based on the heterogeneity of contact patterns, the effects of different immunization schemes are discussed and compared. Meanwhile, we explore the relation between the immunization rate and the recovery rate, which are the two important parameters that can be improved. To illustrate our theoretical results, the corresponding numerical simulations are also included.

We study three dimensional competitive differential equations with linearly determined nullclines and prove that they always have 33 stable nullcline classes in total. Each class is given in terms of inequalities on the intrinsic growth rates and competitive coefficients and is independent of generating functions. The common characteristics are that every trajectory converges to an equilibrium in classes 1–25, that Hopf bifurcations do not occur within class 32, and that there is always a heteroclinic cycle in class 27. Nontrivial dynamical behaviors, such as the existence and multiplicity of limit cycles, only may occur in classes 26–33, but these nontrivial dynamical behaviors depend on generating functions. We show that Hopf bifurcation can occur within each of classes 26–31 for continuous-time Leslie/Gower system and Ricker system, the same as Lotka–Volterra system; but it only occurs in classes 26 and 27 for continuous-time Atkinson/Allen system and Gompertz system. There is an apparent distinction between Lotka–Volterra system and Leslie/Gower system, Ricker system, Atkinson/Allen system, and Gompertz system with the identical growth rate. Lotka–Volterra system with the identical growth rate has no limit cycle, but admits a center on the carrying simplex in classes 26 and 27. But Leslie/Gower system, Ricker system, Atkinson/Allen system, and Gompertz system with the identical growth rate do possess limit cycles. At last, we provide examples to show that Leslie/Gower system and Ricker system can also admit two limit cycles. This general classification greatly widens applications of Zeeman's method and makes it possible to investigate the existence and multiplicity of limit cycles, centers and stability of heteroclinic cycles for three dimensional competitive systems with linearly determined nullclines, as done in planar systems.

Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly $24$ hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411-2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.

We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of \begin{document}$ 33 $\end{document} stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes \begin{document}$ 1-25 $\end{document} and \begin{document}$ 33 $\end{document} ; there is always a heteroclinic cycle in class \begin{document}$ 27 $\end{document} ; Neimark-Sacker bifurcations may occur in classes \begin{document}$ 26-31 $\end{document} but cannot occur in class \begin{document}$ 32 $\end{document} . Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes \begin{document}$ 29, 31, 33 $\end{document} and those in class \begin{document}$ 27 $\end{document} with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.

Motivated by the remarkable works of Busse and his collaborators in the 1980s on turbulent convection in a rotating layer, we explore the long-run behavior of stochastic Lotka-Volterra (LV) systems both in pull-back trajectories and in stationary measures. A decomposition formula is established to describe the relationship between the solutions of stochastic and deterministic LV systems and the stochastic logistic equation. By virtue of this formula, it can be verified that every pull-back omega limit set is an omega limit set of the deterministic LV system multiplied by the random equilibrium of the stochastic logistic equation. The formula is also used to derive the existence of a stationary measure, its support and ergodicity. We prove the tightness of stationary measures and that their weak limits are invariant with respect to the corresponding deterministic system and supported on the Birkhoff center. The developed theory is successfully utilized to completely classify three dimensional competitive stochastic LV systems into 37 classes. The expected occupation measures weakly converge to a strongly mixing measure and all stationary measures are obtained for each class except class 27 c). Among them there are two classes possessing a continuum of random closed orbits and strongly mixing measures supported on the cone surfaces, which weakly converge to the Haar measures of periodic orbits as the noise intensity vanishes. The class 27 c) is an exception, almost every pull-back trajectory cyclically oscillates around the boundary of the stochastic carrying simplex characterized by three unstable stationary solutions. The limit of the expected occupation measures is neither unique nor ergodic. These are consistent with symptoms of turbulence. (C) 2019 Elsevier Masson SAS. All rights reserved.

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold R0 which completely governs the disease dynamics: when R01, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when R01, the disease is permanent. It is interesting that the threshold value R0 bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.

The comparison theorems under Wu and Freedman's order are proved for neutral stochastic functional differential equations with finite or infinite delay whose drift terms satisfy the quasimonotone condition and diffusion term is the same.

In this paper we propose the stochastic epidemic model that relates directly to the deterministic counterpart and reveal close connections between these two models. Under the classic assumptions, the sample path of process eventually converges to the disease-free equilibrium, even though the corresponding deterministic flow converges to an endemic equilibrium. From the fact that disease can occur sporadically, we adjust the stochastic model slightly by introducing a stochastic incidence and establish precise connections between the long-run behavior of the discrete stochastic process and its deterministic flow approximation for large populations.

We propose the generalized competitive Atkinson-Allen map \begin{document}$T_i(x)=\frac{(1+r_i)(1-c_i)x_i}{1+\sum_{j=1}^nb_{ij}x_j}+c_ix_i, 0 0, i, j=1, ···, n, $ \end{document} which is the classical Atkson-Allen map when \begin{document}$r_i=1$\end{document} and \begin{document}$c_i=c$\end{document} for all \begin{document}$i=1, ..., n$\end{document} and a discretized system of the competitive Lotka-Volterra equations. It is proved that every \begin{document}$n$\end{document} -dimensional map \begin{document}$T$\end{document} of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all such three-dimensional maps. In the three-dimensional case we list a total of 33 stable equivalence classes and draw the corresponding phase portraits on each Σ. The dynamics of the generalized competitive Atkinson-Allen map differs from the dynamics of the standard one in that Neimark-Sacker bifurcations occur in two classes for which no such bifurcations were possible for the standard competitive Atkinson-Allen map. We also found Chenciner bifurcations by numerical examples which implies that two invariant closed curves can coexist for this model, whereas those have not yet been found for all other three-dimensional competitive mappings via the carrying simplex. In one class every map admits a heteroclinic cycle; we provide a stability criterion for heteroclinic cycles. Besides, the generalized Atkinson-Allen model is not dynamically consistent with the Lotka-Volterra system.

We study the fixed point index on the carrying simplex of the competitive map. The sum of the indices of the fixed points on the carrying simplex for the three-dimensional competitive map is unit. Based on that, we analyze the asymptotic behavior of the three-dimensional competitive Atkinson/Allen model. We present all the equivalence classes relative to the boundary of the carrying simplex of the low-dimensional (two or three) map, depending upon relationship among the model coefficients. For the two-dimensional case, there are only three dynamic scenarios, and every orbit converges to some fixed point. For the three-dimensional case, there are total $33$ stable equivalence classes, and in $18$ of them all the compact limit sets are fixed points. Further, we focus on the analysis of the dynamics of the other $15$ cases. Hopf bifurcation is studied and a necessary condition for it occurring is given, which implies that the classes $19$-$25$, $28$, $30$ and $32$ do not have any Hopf bifurcation. However, the class $26$ and class $27$ do admit Hopf bifurcations, which means that these two classes may have isolated invariant closed curves in their carrying simplex, and such an invariant closed curve corresponds to either a subharmonic or a quasiperiodic solution in continuous time systems. Each system in class $27$ has a heteroclinic cycle and the numerical simulation also reveals that there exist systems having May-Leonard phenomenon: the existence of nonquasiperiodic oscillation.

We investigate a competitive model of three species, each of which, in isolation, admits Gompertz growth. A well-known theorem by M.W. Hirsch guarantees the existence of carrying simplex. Based on this, we compare three dimensional competitive Gompertz models with three dimensional competitive Lotka–Volterra models, and we find that each Gompertz model has a corresponding Lotka–Volterra model with identical nullclines. We then present the complete classification of nullcline stable models and arrive at a total of 33 stable nullcline classes, and show that in 27 of these classes all the compact limit sets are equilibria. Despite the common results, we go on to show that the behavior on the carrying simplex of Gompertz systems is subtly different from that on Lotka–Volterra systems. The number of limit cycles is finite in 5 of the remaining 6 classes, and that only the classes 26 and 27 admit Hopf bifurcations and the other 4 do not. The class 27 , which has a heteroclinic cycle, contains a system having May–Leonard phenomenon: the existence of nonperiodic oscillation, and still admitting at least two limit cycles. The numerical simulation reveals that there are some systems in class 28 with two limit cycles.

The limiting behavior of stochastic evolution processes with small noise intensity $\epsilon$ is investigated in distribution-based approach. Let $\mu^{\epsilon}$ be stationary measure for stochastic process $X^{\epsilon}$ with small $\epsilon$ and $X^{0}$ be a semiflow on a Polish space. Assume that $\{\mu^{\epsilon}: 0

Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy.

Under some local conditions on W ( t , u ) , the existence of homoclinic solutions is obtained for the nonperiodic second-order Hamiltonian systems u ( t ) − L ( t ) u ( t ) + ∇ W ( t , u ( t ) ) = f ( t ) as a limit of periodic solutions of a certain sequence of boundary-value problems which are obtained by a new critical point theorem.

A new result for the existence of homoclinic orbits is obtained for the second-order Hamiltonian systems u ( t ) + ∇ V ( t , u ( t ) ) = f ( t ) , where t ∈ R , u ∈ R n and V ∈ C 1 ( R × R n , R ) , V ( t , x ) = − K ( t , x ) + W ( t , x ) is T -periodic with respect to t , T > 0 and f : R → R n is a continuous and bounded function. This result generalizes and improves some existing results in the known literature.

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 \cite{FS3}, we firstly define the {\it "input-to-state characteristic operator"} $\mathcal{K}(u)$ of the system in a suitably chosen input space via backward It\^o integral, and then for a given output function $h$, define the $"gain\ operator"$ as the composition of output function $h$ and the input-to-state characteristic operator $\mathcal{K}(u)$ 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 linear matrix. Then we prove the so-called {\it "small-gain theorem"}: the gain operator has a unique fixed point, the image for 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 maximal real part of stable linear matrix, we prove that the stochastic system has a unique stationary distribution, which is regarded as a stationary distribution version of small-gain theorem. These results can be applied to stochastic cooperative, competitive and predator-prey systems, or even others.

This paper examines a system of reaction–diffusion equations arising from a mathematical model of two microbial species competing for two complementary resources with internal storage in an unstirred chemostat. The governing system can be reduced to a limiting system based on two uncoupled conservation principles. One of main technical difficulties in our analysis is the singularities in the reaction terms. Conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems, maximum principle and the theory of monotone dynamical systems.

This paper studies autonomous/time-periodic translation-invariant monotone systems without stronger notion. It is proved that every solution is asymptotically periodic if the Poincare mapping has at least one fixed point. The result gets rid of the “strong” assumption in [1] . Applications are made to chemical reaction networks, especially to enzymatic futile cycles.

In this paper, we investigate a generalized two-species semi-ratio-dependent predator-prey system with the prey dispersal in a two-patch environment. With the help of a continuation theorem based on coincidence degree theory, we establish sufficient conditions for the existence of positive periodic solutions. Our results show that for any monotonic prey growth and the most celebrated functional responses such as the Holling type, the sigmoidal type, the Ivlev type, the Monod-Haldane type, and the Beddington-DeAngelis type, the system always has at least one positive periodic solution.

In this paper we study a system of reaction–diffusion equations arising from competition of two microbial populations for a single-limited nutrient with internal storage in an unstirred chemostat. The conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient. The reduced system (limiting system) generates a strongly monotone dynamical system in its feasible domain under a partial order. We construct suitable upper, lower solutions to establish the existence of positive steady-state solutions. Given the parameters of the reduced system, we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors. The primary conclusion is that the survival of species depends on species's intrinsic biological characteristics, the external environment forces and the principal eigenvalues of some scalar partial differential equations. We also lift the dynamics of the limiting system to the full system.