Multiple Attractors in Juvenile-adult Single Species Models

The role of age-structure and the Allee effect in generating multiple attractors in juvenile-adult single species single patch discrete-time models without dispersal are studied. In the presence of the Allee effect juvenile-adult single patch models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. When the governing dynamics are compensatory, the boundaries of the basins of attraction have simple structure while complicated fractal basin boundaries are supported under overcompensatory dynamics.     

Low dimensional homeochaos in coevolving host–parasitoid dimorphic populations: Extinction thresholds under local noise

A discrete time model describing the population dynamics of coevolution between host and parasitoid haploid populations with a dimorphic matching allele coupling is investigated under both determinism and stochastic population disturbances. The role of the properties of the attractors governing the survival of both populations is analyzed considering equal mutation rates and focusing on host and parasitoid growth rates involving chaos. The purely deterministic model reveals a wide range of ordered and chaotic Red Queen dynamics causing cyclic and aperiodic fluctuations of haplotypes within each species. A Ruelle–Takens–Newhouse route to chaos is identified by increasing both host and parasitoid growth rates. From the bifurcation diagram structure and from numerical stability analysis, two different types of chaotic sets are roughly differentiated according to their size in phase space and to their largest Lyapunov exponent: the Confined and Expanded attractors. Under the presence of local population noise, these two types of attractors have a crucial role in the survival of both coevolving populations. The chaotic confined attractors, which have a low largest positive Lyapunov exponent, are shown to involve a very low extinction probability under the influence of local population noise. On the contrary, the expanded chaotic sets (with a higher largest positive Lyapunov exponent) involve higher host and parasitoid extinction probabilities under the presence of noise. The asynchronies between haplotypes in the chaotic regime combined with low dimensional homeochaos tied to the confined attractors is suggested to reinforce the long-term persistence of these coevolving populations under the influence of stochastic disturbances. These ideas are also discussed in the framework of spatially-distributed host–parasitoid populations.     

Dynamics with riddled basins of attraction in models of interacting populations

Recently it has been shown that when there are chaotic attractors whose basins are such that every point in the basin has pieces of another attractors's basin arbitrarily nearby, the basins are said to be riddled. A key requirement for the occurrence of a riddled basin is the loss of transverse stability of an invariant subspace, of dimension less than the full space, containing a chaotic attractor. This type of complex dynamics has been found in simple models of interacting populations for which the invariant subspace is defined by the extinction of one species. The characterizations and implications of these behaviors for population ecology are discussed.     

Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a₁=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.     

Discrete-time Metapopulation Dynamics and Unidirectional Dispersal

The effects of unidirectional dispersal on single pioneer species discrete-time metapopulations where the pre-dispersal local patch dynamics are of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are studied. Single-species unidirectional metapopulation models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and the dispersal rate is low. The pioneer species goes extinct in at least one patch when the dispersal rate is high, while it persists when the rate is low. Unidirectional dispersal can generate multiple attractors with fractal basin boundaries whenever the pre-dispersal local patch dynamics are overcompensatory, and is capable of altering the local patch dynamics in mixed systems from compensatory to overcompensatory dynamics and vice versa.     

Remarks on nontrivial pullback attractors of the 2-D Navier-Stokes equations with delays

This paper is concerned with some further research on the pullback dynamics for 2-D Navier-Stokes equations with delays. By some new definition of generalized Grashof numbers, we presented some sufficient conditions when the pullback attractors of the 2-D nonautonomous incompressible Navier-Stokes equations with differential continuous delays become a single trajectory, which is a preparation for the fractal dimension of pullback attractors for our problem with constant or variable delays.     

The dynamics of an epidemic model for pest control with impulsive effect

In pest control, there are only a few papers on mathematical models of the dynamics of microbial diseases. In this paper a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. The paper shows that there exists a globally stable susceptible pest eradication periodic solution when the impulsive period is less than some critical value. Further, the conditions for the permanence of the system are given. In addition, there exists a unique positive periodic solution via bifurcation theory, which implies both the susceptible pest and the infective pest populations oscillate with a positive amplitude. In this case, the susceptible pest population is infected to the maximum extent while the infective pest population has little effect on the crops. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamic, which implies that this model has more complex dynamics, including period-doubling bifurcation, chaos and strange attractors.     

Global multifractal relation between topological entropies and fractal dimensions

In one-dimensional chaotic dynamics, a global multifractal relation between topological entropies and fractal dimensions of arbitrary period-p-tupling attractors is analyzed on all critical (accumulation) points of transitions to chaos, where the Lyapunov characteristic exponent is zero. The global metric regularity of topological entropies versus fractal dimensions is well characterized by the self-similarity. By the fractal interpolation based on the iterated function system, the fractal dimensions of the curves of topological entropies versus capacity dimensions and versus information dimensions are both found to be 1.82.     

Endogenous chaotic dynamics in transitional economies

This paper examines a model of labor market dynamics in an economy undergoing transition from command socialism to market capitalism. State sector layoffs are modeled as a function of forecasts made by state planners of private sector wages where the laidoff workers are to be re-employed. The state switches between using a high information cost perfect forecast and a free naive forecast in a system that resembles a cobweb supply-demand model. Under certain specifications and parameter values chaotic dynamics are shown to endogenously emerge along with several other varieties of complex dynamics including strange attractors, coexistence of infinitely many stable cycles, cascades of infinitely many period doubling bifurcations and fractal basin boundaries between coexisting non-chaotic attractors.     

Density Delay and Organizational Survival: Computational Models and Empirical Comparisons

Research on the ecological dynamics oforganizational populations has demonstrated that competitiveconditions at the time of founding have enduring effects onorganizational survival. According to ecological theories,organizational life chances are systematically affected by density (the number of organizations in a population) at thetime of founding because the lower resource endowments thatcharacterize organizations appearing in periods of highpopulation density tend to become self-reinforcing, and—over time—amplify differences in mortality rates oforganizations founded under different conditions. However,credible arguments have been offered that could justify both positive and negative effects of the delayed effectsof population density on organizational mortality rates, andreceived empirical research in part reflects this ambiguity.To develop new insight into this issue and to explore theboundaries of received empirical results, in this study wepresent a computational model of organizational evolutionaccording to which the global dynamics of organizationalpopulations emerge from the iteration of simple rules oflocal interaction among individual organizations. We use the synthetic data produced by simulation to estimate eventhistory models of organizational mortality, and compare theparameter estimates with those reported in the most recentempirical studies of actual organizational populations. Theconclusions supported by the model qualify and extendreceived empirical results, and suggest that delayed effectsof density are highly sensitive the details of local structure of connections among members of organizationalpopulations.     

Crisis-limited chaotic dynamics in ecological systems

We review our recent efforts to understand why chaotic dynamics is rarely observed in natural populations. The study of two-model ecosystems considered in this paper suggests that chaos exists in narrow parameter ranges. This dynamical behaviour is caused by the crisis-induced sudden death of chaotic attractors. The computed bifurcation diagrams and basin boundary calculations reinforce our earlier conclusion [Chaos, Solitons & Fractals 8 (12) (1997) 1933; Int J Bifurc Chaos 8 (6) (1998) 1325] that the reason why chaos is rarely observed in natural populations is hidden within the mathematical structure of the ecological interactions and not with the problem associated with the data (insufficient length, precision, noise, etc.) and its analysis. We also argue that crisis-limited chaotic dynamics can be commonly found in model terrestrial ecosystems.     

Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.     

Optimal pulse fishing policy in stage-structured models with birth pulses

In this paper, we propose exploited models with stage structure for the dynamics in a fish population for which periodic birth pulse and pulse fishing occur at different fixed time. Using the stroboscopic map, we obtain an exact cycle of system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate (or pulse fishing time or harvesting effort) as the bifurcation parameter, and these are observed to display complex dynamic behaviors, including chaotic bands with period windows, period-doubling, multi-period-halving and incomplete period-doubling bifurcation, pitch-fork and tangent bifurcation, non-unique dynamics (meaning that several attractors or attractor and chaos coexist) and attractor crisis. This suggests that birth pulse and pulse fishing provide a natural period or cyclicity that make the dynamical behaviors more complex. Moreover, we show that the pulse fishing has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulse, the population can sustain much higher harvesting effort if the mature fish is removed as early as possible.     

Parallel simulation of individual-based, physiologically structured population models

A general scheme for parallel simulation of individual-based, structured population models is proposed. Algorithms are developed to simulate such models in a parallel computing environment. The simulation model consists of an individual model and a population model that incorporates the individual dynamics. The individual model is a continuous time representation of organism life history for growth with discrete allocations for reproductive processes. The population model is a continuous time simulation of a nonlinear partial differential equation of extended McKendrick-von Foerster-type.

As a prototypical example, we show that a specific individual-based, physiologically structured model for Daphnia populations is well suited for parallelization, and significant speed-ups can be obtained by using efficient algorithms developed along our general scheme. Because the parallel algorithms are applicable to generic structured populations which are the foundation for populations in a more complex community or food-web model, parallel computation appears to be a valuable tool for ecological modeling and simulation.     

Dynamic complexities in a single-species discrete population model with stage structure and birth pulses

Natural population, whose population numbers are small and generations are non-overlapping, can be modelled by difference equations that describe how the population evolve in discrete time-steps. This paper investigates a recent study on the dynamics complexities in a single-species discrete population model with stage structure and birth pulses. Using the stroboscopic map, we obtain an exact cycle of system, and obtain the threshold conditions for its stability. Above this, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that this the dynamical behaviors of the single-species discrete model with birth pulses are very complex, including (a) non-unique dynamics, meaning that several attractors and chaos coexist; (b) small-amplitude annual oscillations; (c) large-amplitude multi-annual cycles; (d) chaos. Some interesting results are obtained and they showed that pulsing provides a natural period or cyclicity that allows for a period-doubling route to chaos.     

Some advances on global analysis of nonlinear systems

Global analysis in nonlinear dynamics means the study of attractors and their basins of attraction; meanwhile a lot of complex dynamical behaviors and new phenomena are concerned such as fractal basin boundary, Wada basin boundary, infinite unstable periodic orbits embedded in chaotic attractor, chaotic saddle and transient chaos, crises, riddled basin of attractor, stochastic global dynamics, etc.To analyze the global dynamics analytically is difficult and interesting while the results are few. Then, the numerical analysis for global dynamics is usually the main approach.Global analysis captures both the interest and imagination of the wider communities in various fields, such as mathematics, physics, meteorology, life science, computational science, engineering, medicine, and others.Emphasis is put mainly on the development in this global dynamics field in China.     

Richness of chaotic dynamics in nonholonomic models of a celtic stone

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.     

Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall

Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated bubble and a coated bubble near an elastic wall.We demonstrate that complex dynamics can occur in these models. We are particularly interested in the multistability phenomenon of bubble dynamics. We show that coexisting attractors appear in all of these models, but for higher acoustic pressures for the models of an encapsulated bubble.We demonstrate how several tools can be used to localize the coexisting attractors. We provide some considerations why the multistability can be undesirable for applications.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

Attenuant cycles in periodically forced discrete-time age-structured population models

In discrete-time age-structured population models, a periodic environment is not always deleterious. We show that it is possible to have the average of the age class populations over an attracting cycle (in a periodic environment) not less than the average of the carrying capacities (in a corresponding constant environment). In our age-structured model, a periodic environment does not increase the average total biomass (no resonance). However, a periodic environment is disadvantageous for a population whenever there is no synchrony between the number of age classes and the period of the environment. As in periodically forced models without age-structure, we show that periodically forced age-structured population models support multiple attractors with complicated structures.