Multiple attractors in stage-structured population models with birth pulses

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. A single species stage-structured model with density-dependent maturation rate and birth pulse is formulated. Using the discrete dynamical system determined by its Poincaré map, we report a detailed study of the various dynamics, including (a) existence and stability of nonnegative equilibria, (b) nonunique dynamics, meaning that several attractors coexist, (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of attractor), (d) supertransients, and (e) chaotic attractors. The occurrence of these complex dynamic behaviour is related to the fact that minor changes in parameter or initial values can strikingly change the dynamic behaviours of system. Further, it is shown that periodic birth pulse, in effect, provides a natural period or cyclicity that allows multiple oscillatory solutions in the continuous dynamical systems.     

On linear perturbations of the Ricker model

A class of linearly perturbed discrete-time single species scramble competition models, like the Ricker map, is considered. Perturbations can be of both recruitment and harvesting types. Stability (bistability) is considered for models, where parameters of the map do not depend on time. For models with recruitment, the result is in accordance with Levin and May conjecture [S.A. Levin, R.M. May, A note on difference delay equations, Theor. Pop. Biol. 9 (1976) 178]: the local stability of the positive equilibrium implies its global stability. For intrinsic growth rate r-->infinity the way to chaos is broken down to get extinction of population for the depletion case and to establish a stable two-cycle period for models with immigration. The latter behaviour is also studied for models with random discrete constant perturbations of recruitment type. Extinction, persistence and existence of periodic solutions are studied for the perturbed Ricker model with time-dependent parameters.     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

Dynamic reduction with applications to mathematical biology and other areas

In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the 'reduced' equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping Τ of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping Τ is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.     

Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

Approximate expressions of the bifurcating periodic solutions in a neuron model with delay-dependent parameters by perturbation approach

This paper is interested in gaining insights of approximate expressions of the bifurcating periodic solutions in a neuron model. This model shares the property of involving delay-dependent parameters. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work so harder. Most existing methods for studying the nonlinear dynamics fail when applied to such a class of delay models. Although Xu et al. (Phys Lett A 354:126–136, 2006) studied stability switches, Hopf bifurcation and chaos of the neuron model with delay-dependent parameters, the dynamics of this model are still largely undetermined. In this paper, a detailed analysis on approximation to the bifurcating periodic solutions is given by means of the perturbation approach. Moreover, some examples are provided for comparing approximations with numerical solutions of the bifurcating periodic solutions. It shows that the dynamics of the neuron model with delay-dependent parameters is quite different from that of systems with delay-independent parameters only.     

On the Population Dynamics of the Malaria Vector

A deterministic differential equation model for the population dynamics of the human malaria vector is derived and studied. Conditions for the existence and stability of a non-zero steady state vector population density are derived. These reveal that a threshold parameter, the vectorial basic reproduction number, exist and the vector can established itself in the community if and only if this parameter exceeds unity. When a non-zero steady state population density exists, it can be stable but it can also be driven to instability via a Hopf Bifurcation to periodic solutions, as a parameter is varied in parameter space. By considering a special case, an asymptotic perturbation analysis is used to derive the amplitude of the oscillating solutions for the full non-linear system. The present modelling exercise and results show that it is possible to study the population dynamics of disease vectors, and hence oscillatory behaviour as it is often observed in most indirectly transmitted infectious diseases of humans, without recourse to external seasonal forcing.     

Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

Viable controls in age-dependent population dynamics

The classical age-dependent population model is considered, where the mortality depends on total population and the fertility is age dependent. Conditions are determined under which such a system may be steered to a specific population level and held there.     

Persistent solutions for age-dependent pair-formation models

Conditions on the vital rates and the mating function are derived which imply existence or nonexistence of exponentially growing persistent age-distributions for age-dependent pair-formation models.     

Properties of small neural networks

Networks containing neuronal models of the type considered in the previous paper can be described by a set of first order differential equations. Steady-state solutions and the stability of these solutions to small perturbations can be obtained. Networks of physiological interest which give rise to second, third and fourth order linear equations are analysed in detail. Conditions are derived under which such networks can be condensed into a single neuron of similar order. Simple mechanisms for memory storage, for the generation of oscillatory activity and for decision making in neural systems are suggested.     

Two models for competition between age classes

We consider two lumped age-structured models of juvenile against adult competition, one of which is a compartmental ordinary differential equation (ODE) model and one of which is a system of ODEs with delay derived from the McKendrick partial differential equation (PDE) model. Existence and stability of positive equilibria and existence of oscillatory solutions of both models are studied. Using a competition parameter, which measures the intra-specific competition, we investigate the effects of the competitive interactions between juveniles and adults on the dynamics of the population for both models. We find that these effects are different for the two models and discuss the differences from the modeling perspective. The discussion reveals an interesting relation between the length of the maturation period and the vital rates' dependence on the competition parameter.     

Density-dependent birth rate, birth pulses and their population dynamic consequences

 In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Received: 13 June 2001 / Revised version: 7 September 2001 / Published online: 8 February 2002     

Intra-specific competition and density dependent juvenile growth

A difference equation model for the dynamics of a semelparous size-structured species consisting of juvenile and adult individuals is derived and studied. The adult population consists of two size classes, a smaller class and a larger more fertile class. Negative feedback occurs through slowed juvenile growth due to increased total population levels during the developmental period and consequently a smaller adult size at maturation. Intra-specific competition coefficients are size dependent and measure the strength of intra-specific competition between juveniles and adults. It is shown that equilibrium states in which adults and juveniles occur together at all times are in general destabilized by significantly increased juvenilevs adults competition with the result that stable periodic cycles appear, in which the generations alternate in time and hence avoid competition. This result supports the tenet that intra-specific competition between juveniles and adults is destabilizing. Exceptions to this destabilization principle are found, however, in which populations exhibiting non-equilibrium, aperiodic dynamics can be equilibrated by increase competition between juveniles and adults. This occurs, for example, when adult fertility and competition coefficients are significantly size class dependent. The author gratefully acknowledges the support of the Applied Mathematics Division and the Population Biology/Ecology Division of the National Science Foundation under NSF grant No. DMS-8902508. Research supported by the Department of Energy under contracts W-7405-ENG-36 and KC-07-01-01.     

Equilibria in an epistatic viability model under arbitrary strength of selection

A class of viability models that generalize the standard additive model for the case of pairwise additive by additive epistatic interactions is considered. Conditions for existence and stability of steady states in the corresponding two-locus model are analyzed. Using regular perturbation techniques, the case when selection is weaker than recombination and the case when selection is stronger than recombination are investigated. The results derived are used to make conclusions on the dependence of population characteristics on the relation between the strength of selection and the recombination rate.     

Small amplitude,long period outbreaks in seasonally driven epidemics

It is now documented that childhood diseases such as measles, mumps, and chickenpox exhibit a wide range of recurrent behavior (periodic as well as chaotic) in large population centers in the first world. Mathematical models used in the past (such as the SEIR model with seasonal forcing) have been able to predict the onset of both periodic and chaotic sustained epidemics using parameters of childhood diseases. Although these models possess stable solutions which appear to have the correct frequency content, the corresponding outbreaks require extremely large populations to support the epidemic. This paper shows that by relaxing the assumption of uniformity in the supply of susceptibles, simple models predict stable long period oscillatory epidemics having small amplitude. Both coupled and single population models are considered.     

具免疫时滞的HIV感染模型动力学性质分析

讨论了一类具免疫时滞的HIV感染模型.分析了未感染平衡点的全局渐近稳定性,给出了感染无免疫平衡点及感染免疫平衡点局部渐近稳定的充分条件.数值模拟结果表明,当易感细胞生成率的取值使得基本再生数满足平衡存在的条件且低于某一临界值时,时滞对平衡点的稳定性没有影响;若大于该临界值,随着时滞增大,稳定性开关发生,平衡点不稳定,出现一系列Hopf分支,最终表现为周期波动模式.     

The effect of seasonal harvesting on stage-structured population models

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose an exploited single-species model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Since birth pulse populations are often characterized with a discrete time dynamical system determined by its Poincaré map, we explore the consequences of harvest timing to equilibrium population sizes under seasonal dependence and obtain threshold conditions for their stability, and show that the timing of harvesting 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. Moreover, our results imply that the population can sustain much higher harvest rates if the mature fish is removed as early in the season (after the birth pulse) as possible. Further, the effects of harvesting effort and harvest timing on the dynamical complexity are also investigated. Bifurcation diagrams are constructed with the birth rate (or harvesting effort or harvest timing) as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, non-unique dynamics (meaning that several attractors coexist) and attractor crisis. This suggests that birth pulse, in effect, provides a natural period or cyclicity that makes the dynamical behavior more complex.This work is supported by National Natural Science Foundation of China (10171106)     

Bifurcation, stability, and cluster formation of multi-strain infection models

Clustering behaviours have been found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-states, periodic, or even chaotic motions can be self-organized into clusters. Such clustering behaviours are not a priori expected. It has been proposed that the cross-protection from multiple strains of pathogens is responsible for the clustering phenomenon. In this paper, we show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcation from the quotient networks of the models and the patterns which follow can be predicted through the stability analysis of the bifurcation. We calculate the stability criteria for the clustering patterns and show that some patterns are inherently unstable. Finally, the biological implications of these results are discussed.     

Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Summary For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.