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In this paper, we study the dynamics of a discrete Kolmogorov predator-prey model with Ricker-type prey growth. We give the sufficient and necessary condition to guarantee the existence and uniqueness of the positive fixed point. Using the center manifold theory, we prove that the period-doubling bifurcations can occur at the positive fixed point. Furthermore, our numerical simulations reveal that the model can exhibit cascades of period-doubling bifurcations leading to chaos, which is a significant difference from the behavior of continuous predator-prey models. Despite the complexities of the model dynamics, we are able to provide a criterion for the global stability of the positive fixed point by using a geometric analysis of the nullclines.

In this paper we study the permanence and impermanence for continuous-time competitive Kolmogorov systems via the carrying simplex. We first give an extension to attractors of V. Hutson's results on the existence of repellors in continuous-time dynamical systems that have found wide use in the study of permanence via average Liapunov functions. We then give a general criterion for the stability of the boundary of carrying simplex for competitive Kolmogorov systems of differential equations, which determines the permanence and impermanence of such systems. Based on the criterion, we present a complete classification of the permanence and impermanence in terms of inequalities on parameters for all three-dimensional competitive systems which have linearly determined nullclines. The results are applied to many classical models in population dynamics including the Lotka-Volterrra system, Gompertz system, Leslie-Gower system and Ricker system., Comment: arXiv admin note: text overlap with arXiv:1905.00385

In this paper, we are concerned with the permanence of a Lotka-Volterra model of three competing species with seasonal succession. Based on the existence of a carrying simplex, that is a globally attracting hypersurface of codimension one, we provide a complete classification of the permanence and impermanence in terms of inequalities on the parameters of this model. Moreover, we numerically show that invariant closed curves can occur in the permanent classes, which means that the positive fixed point of the associated Poincare map in the permanent classes is not always globally asymptotically stable.

The current series of two papers focus on a 3-dimensional Lotka-Volterra competition model of differential equations with seasonal succession, which exhibits that populations experience an external periodically forced environment. We are devoted to providing a delicate global dynamical description for the model. In the first part of the series, we first use a novel technique to construct an index formula for the associated Poincare map, by which we thoroughly classify the dynamics of the model into 33 classes via the equivalence relation relative to boundary dynamics. More precisely, we show that in classes 1--18, there is no positive fixed point and every orbit tends to certain boundary fixed point. While, for classes 19--33, there exists at least one (but not necessarily unique) positive fixed point, that is, a positive harmonic time-periodic solution of the model. Among them, the dynamics is trivial in classes 19--25 and 33, provided that the positive fixed point is unique. We emphasize that, unlike the corresponding $2$-dimensional system, a major significant difference and difficulty for the analysis of the global dynamics for 3-dimensional system is that it may not possess the uniqueness of the positive fixed point. In the forthcoming second part of the series, we shall address the issues of (non-)uniqueness of the positive fixed points for the associated Poincare map.

In this paper, we study the Cauchy problem to the 3D fractional compressible isentropic generalized Navier-Stokes equations for viscous compressible fluid with one Levy diffusion process. We obtain the existence and uniqueness of global strong solution for small initial data by providing several commutators via the Littlewood-Paley theory. Moreover, we derive $L^2$-decay rate for the highest derivative of the strong solution without decay loss by using a cancellation of a low-medium-frequency quantity. Our results improve the results provided in [J. Differential Equations 377(2023): 369-417].

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 33 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 1-25 and 33; there is always a heteroclinic cycle in class 27; Neimark-Sacker bifurcations may occur in classes 26-31 but cannot occur in class 32. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes 29,31,33 and those in class 27 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.

A folklore result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In particular, the classical Grobman-Hartman theorem can not be used on the carrying simplex. In this paper, we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. On the other hand, our results also give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics. Although we focus on monotone maps, many results of the paper can be applied to maps that admit a non-smooth center-manifold., Comment: 32 pages, 9 figures; minor revisions; an accepted manuscript of a paper to appear in Journal of Differential Equations

Motivated by the work of Busse et al. [6] on turbulent convection in a rotating layer, we exploit the long-run behavior for stochastic Lotka-Volterra (LV) systems both in pull-back trajectory and in stationary measure. It is proved stochastic decomposition formula describing the relation between solutions of stochastic and deterministic LV systems and stochastic Logistic equation. By virtue of this formula, it is verified that every pull-back omega limit set is an omega limit set for deterministic LV systems multiplied by the random equilibrium of the stochastic Logistic equation. This formula is used to derive the existence of a stationary measure, its support and ergodicity. We prove the tightness for the set of stationary measures and the invariance for their weak limits as the noise intensity vanishes, whose supports are contained in the Birkhoff center. The developed theory is successfully utilized to completely classify three dimensional competitive stochastic LV systems into $37$ classes. Time average probability measures weakly converge to an ergodic stationary measure on the attracting domain of an omega limit set in all classes except class 27 c). Among them there are two classes possessing a continuum of random closed orbits and ergodic stationary measures supported in cone surfaces, which weakly converge to the Haar measures of periodic orbits as the noise intensity vanishes. In the exceptional class, almost every pull-back trajectory cyclically oscillates around the boundary of the stochastic carrying simplex characterized by three unstable stationary solutions. The limit for the time average probability measures is neither unique nor ergodic. These are subject to turbulent characteristics.