Existence and stability of equilibria in age-structured population dynamics

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results.     

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.     

Perfect harmony: the discrete dynamics of cooperation

We study the discrete model for cooperation as expressed through the dynamics of the family of noninvertible planar maps (x, y) (x exp(r(1 – x) + sy), y exp(r(1 – y) + sx)), with parameters r, s > 0. We prove that the map is proper in the open positive quadrant and describe its various stretching and folding actions. We determine conditions for a Hopf bifurcation — probably one of a cascade of double, quadruple, ... limit cycles, as a curve is followed in parameter space. For r > s an approximating version of the map is dissipative and permanent in the positive quadrant. We include the results of an extensive computer simulation, including a bifurcation diagram (y vs. r, with s fixed) through which is cut a number of x–y phase-plane plots; (an r–y curve penetrates each plot like a thread through cards). These indicate a complex dynamical evolution for cooperation, from stable cycle to strange attractor. A general conclusion is that the benefit of cooperation can be relatively high average values at the cost of oscillations of high amplitude.     

Periodic and quasi-periodic behavior in resource-dependent age structured population models

To describe the dynamics of a resource-dependent age structured population, a general non-linear Leslie type model is derived. The dependence on the resources is introduced through the death rates of the reproductive age classes. The conditions assumed in the derivation of the model are regularity and plausible limiting behaviors of the functions in the model. It is shown that the model dynamics restricted to its ω-limit sets is a diffeomorphism of a compact set, and the period-1 fixed points of the model are structurally stable. The loss of stability of the non-zero steady state occurs by a discrete Hopf bifurcation. Under general conditions, and after the loss of stability of the structurally stable steady states, the time evolution of population numbers is periodic or quasi-periodic. Numerical analysis with prototype functions has been performed, and the conditions leading to chaotic behavior in time are discussed.     

Oscillatory population growth in periodic environments

A general first-order nonlinear differential equation is derived for the dynamics of a population in such a way that the inherent growth rate r and the equilibrium “carrying capacity” K appear explicitly as parameters. By means of standard regular perturbation techniques, properties of the periodic asymptotic state of the population are studied under the assumption that r and K suffer periodic perturbations of small amplitude. Specific examples are studied analytically and numerically.     

Modeling the dynamics of natural rotifer populations: Phase-parametric analysis

A model of the dynamics of natural rotifer populations is described as a discrete non-linear map depending on three parameters, which reflect characteristics of the population and environment. Model dynamics and their change by variation of these parameters were investigated by methods of bifurcation theory. A phase-parametric portrait of the model was constructed and domains of population persistence (stable equilibrium, periodic and a-periodic oscillations of population size) as well as population extinction were identified and investigated. The criteria for population persistence and approaches to determining critical parameter values are described. The results identify parameter values that lead to population extinction under various environmental conditions. They further illustrate that the likelihood of extinction can be substantially increased by small changes in environmental quality, which shifts populations into new dynamical regimes.     

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 systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS)

In this study, we investigate systematically the role played by the reproductive number (the number of secondary infections generated by an infectious individual in a population of susceptibles) on single group populations models of the spread of HIV/AIDS. Our results for a single group model show that if R 1, the disease will die out, and strongly suggest that if R > 1 the disease will persist regardless of initial conditions. Our extensive (but incomplete) mathematical analysis and the numerical simulations of various research groups support the conclusion that the reproductive number R is a global bifurcation parameter. The bifurcation that takes place as R is varied is a transcritical bifurcation; in other words, when R crosses 1 there is a global transfer of stability from the infection-free state to the endemic equilibrium, and vice versa. These results do not depend on the distribution of times spent in the infectious categories (the survivorship functions). Furthermore, by keeping all the key statistics fixed, we can compare two extremes: exponential survivorship versus piecewise constant survivorship (individuals remain infectious for a fixed length of time). By choosing some realistic parameters we can see (at least in these cases) that the reproductive numbers corresponding to these two extreme cases do not differ significantly whenever the two distributions have the same mean. At any rate a formula is provided that allows us to estimate the role played by the survivorship function (and hence the incubation period) in the global dynamics of HIV. These results support the conclusion that single population models of this type are robust and hence are good building blocks for the construction of multiple group models. Our understanding of the dynamics of HIV in the context of mathematical models for multiple groups is critical to our understanding of the dynamics of HIV in a highly heterogeneous population.     

Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations

In nonlinear matrix models, strong Allee effects typically arise when the fundamental bifurcation of positive equilibria from the extinction equilibrium at r=1 (or R₀=1) is backward. This occurs when positive feedback (component Allee) effects are dominant at low densities and negative feedback effects are dominant at high densities. This scenario allows population survival when r (or equivalently R₀) is less than 1, provided population densities are sufficiently high. For r>1 (or equivalently R₀>1) the extinction equilibrium is unstable and a strong Allee effect cannot occur. We give criteria sufficient for a strong Allee effect to occur in a general nonlinear matrix model. A juvenile–adult example model illustrates the criteria as well as some other possible phenomena concerning strong Allee effects (such as positive cycles instead of equilibria).     

Three stage semelparous Leslie models

Nonlinear Leslie matrix models have a long history of use for modeling the dynamics of semelparous species. Semelparous models, as do nonlinear matrix models in general, undergo a transcritical equilibrium bifurcation at inherent net reproductive number R ₀ = 1 where the extinction equilibrium loses stability. Semelparous models however do not fall under the purview of the general theory because this bifurcation is of higher co-dimension. This mathematical fact has biological implications that relate to a dichotomy of dynamic possibilities, namely, an equilibration with over lapping age classes as opposed to an oscillation in which age classes are periodically missing. The latter possibility makes these models of particular interest, for example, in application to the well known outbreaks of periodical insects. While the nature of the bifurcation at R ₀ = 1 is known for two-dimensional semelparous Leslie models, only limited results are available for higher dimensional models. In this paper I give a thorough accounting of the bifurcation at R ₀ = 1 in the three-dimensional case, under some monotonicity assumptions on the nonlinearities. In addition to the bifurcation of positive equilibria, there occurs a bifurcation of invariant loops that lie on the boundary of the positive cone. I describe the geometry of these loops, classify them into three distinct types, and show that they consist of either one or two three-cycles and heteroclinic orbits connecting (the phases of) these cycles. Furthermore, I determine stability and instability properties of these loops, in terms of model parameters, as well as those of the positive equilibria. The analysis also provides the global dynamics on the boundary of the cone. The stability and instability conditions are expressed in terms of certain measures of the strength and the symmetry/asymmetry of the inter-age class competitive interactions. Roughly speaking, strong inter-age class competitive interactions promote oscillations (not necessarily periodic) with separated life-cycle stages, while weak interactions promote stable equilibration with overlapping life-cycle stages. Methods used include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques.      

Interspecific competition counteracts negative effects of dispersal on adaptation of an arthropod herbivore to a new host

Dispersal and competition have both been suggested to drive variation in adaptability to a new environment, either positively or negatively. A simultaneous experimental test of both mechanisms is however lacking. Here, we experimentally investigate how population dynamics and local adaptation to a new host plant in a model species, the two‐spotted spider mite (Tetranychus urticae), are affected by dispersal from a stock population (no‐adapted) and competition with an already adapted spider mite species (Tetranychus evansi). For the population dynamics, we find that competition generally reduces population size and increases the risk of population extinction. However, these negative effects are counteracted by dispersal. For local adaptation, the roles of competition and dispersal are reversed. Without competition, dispersal exerts a negative effect on adaptation (measured as fecundity) to a novel host and females receiving the highest number of immigrants performed similarly to the stock population females. By contrast, with competition, adding more immigrants did not result in a lower fecundity. Females from populations with competition receiving the highest number of immigrants had a significantly higher fecundity than females from populations without competition (same dispersal treatment) and than the stock population females. We suggest that by exerting a stronger selection on the adapting populations, competition can counteract the migration load effect of dispersal. Interestingly, adaptation to the new host does not significantly reduce performance on the ancestral host, regardless of dispersal rate or competition. Our results highlight that assessments of how species can adapt to changing conditions need to jointly consider connectivity and the community context.     

Reproductive interference in laboratory experiments of interspecific competition

Many studies that have researched interspecific competition in Callosobruchus (bean beetles), Drosophila (fruit flies), and Tribolium (flour beetles) have considered the major drivers of interspecific competition to be interspecific resource competition and intraguild cannibalism. These competition drivers have a density-dependent effect on the population dynamics. However, some studies have also detected a relative-frequency-dependent effect in the observed population dynamics. The most likely causal mechanism of this relative frequency dependence is reproductive interference, defined as any interspecific sexual interaction that damages female reproductive success. Reproductive interference has been overlooked by most laboratory studies in spite of the critical effect on the competition outcome. In this paper, I review laboratory studies of these insect genera from the perspective of reproductive interference and show that the reported results can be more reasonably interpreted by the joint action of reproductive interference and resource competition, including intraguild cannibalism. In addition, on the basis of results reported by a small number of related studies, I discuss the behavioral and evolutionary changes induced in those genera by reproductive interference.     

Critical slowing down as an indicator of transitions in two-species models

Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.     

大兴安岭次生林区优势种落叶松分布格局及竞争作用

竞争是形成特定群落结构、构成分布格局的基本驱动力之一,树种的空间分布和大小并不是相互独立的,而是广泛受到竞争过程影响。在大兴安岭设置一块具有代表性的天然次生林恢复样地,采用点格局分析中的最近邻体距离G（r）函数、双关联函数g（r）和基于个体胸径标记的点格局分析方法同时结合4种点格局零模型（完全随机模型、均值托马斯模型、随机标签模型、先决条件零模型）研究分析大兴安岭次生林区优势种落叶松不同生活史阶段的分布格局、关联性及竞争作用。研究结果显示：落叶松不同发育阶段分布格局不同、同一发育阶段不同研究尺度分布格局也不同;落叶松种群内存在潜在的竞争、且竞争作用与龄级和研究尺度有关;落叶松幼龄个体在小尺度（r<5m）下的聚集分布是由扩散限制作用和竞争作用所导致;落叶松种群内成龄与幼龄个体间有较强的正关联性、成龄个体对幼龄个体在小尺度下有明显的庇护作用;标记点格局分析方法在检测植物群落中的竞争作用时敏感性较高。     

Metapopulation structure,population trends,and status of Florida Grasshopper Sparrows

ABSTRACT The federally Endangered Florida Grasshopper Sparrow (FGSP, Ammodramus savannarum floridanus) is endemic to dry prairie habitat in central Florida and is currently only found at three public management areas (Avon Park Air Force Range, APAFR; Kissimmee Prairie Preserve State Park, KPPSP; and Three Lakes Wildlife Management Area, TLWMA). We analyzed long‐term (1991–2008) point‐count data to compare population trends of FGSPs at these management areas to determine if they function as independent populations by using Spearman's rank correlation to test for independence between annual trends. We also examined banding and resighting data to infer metapopulation structure. Populations fluctuated across years at all three sites, declining significantly at APAFR (r_s=?0.89, P < 0.001, N= 13) and KPPSP (r_s=?0.78, P= 0.004, N= 11) and remaining stable at TLWMA (r_s= 0.19, P= 0.46, N= 17). Population trends among the three management areas appeared independent (absolute value of r_s≤ 0.50, P≥ 0.12). Previous studies indicated that sparrows in the three areas were not genetically differentiated, and two cases of dispersal between APAFR and KPPSP have been documented. However, dispersal rates among areas appear to be too low to influence demographic dynamics within individual areas. Within APAFR, FGSPs are aggregated into three spatially distinct habitat patches previously considered separate populations, but dispersal among these patches is more frequent than previously reported and population trends among these patches are correlated (r_s≥ 0.91, P < 0.001). These patterns suggest that a single metapopulation of FGSPs exists consisting of three distinct populations (APAFR, KPPSP, and TLWMA), and the spatially distinct aggregations at APAFR constitute a single population using several habitat patches. The population at APAFR is at risk of extirpation, and immediate action is needed if that population is to recover. Taking broader metapopulation dynamics into account will be useful for guiding management efforts aimed at conserving the FGSP in the broader central Florida landscape.     

Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis

To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.     

Spatial effects in discrete generation population models

A framework is developed for constructing a large class of discrete generation, continuous space models of evolving single species populations and finding their bifurcating patterned spatial distributions. Our models involve, in separate stages, the spatial redistribution (through movement laws) and local regulation of the population; and the fundamental properties of these events in a homogeneous environment are found. Emphasis is placed on the interaction of migrating individuals with the existing population through conspecific attraction (or repulsion), as well as on random dispersion. The nature of the competition of these two effects in a linearized scenario is clarified. The bifurcation of stationary spatially patterned population distributions is studied, with special attention given to the role played by that competition.Acknowledgement We gratefully received valuable help through discussions with Hiroshi Matano, Davar Khosnevisan, and Nacho Barradas. Khosnevisan provided us with the background information for Sections 3.3.1 and 3.3.2. Matano provided us with a proof of Lemma 4.4 similar to the one given here. Barradas drew our attention to the relation (2.1).     

Optimal age of maturity in fluctuating environments under r‐ and K‐selection

Optimality models for evolution of life histories have shown that increased environmental stochasticity promotes early age of maturity. Here we argue that if r‐selection for early maturation implies a tradeoff making those phenotypes more sensitive to a change in population size than phenotypes maturing at older ages, K‐selection can favor delayed onset of maturation. We analyze a general stochastic Leslie‐matrix model with a simplified density regulation affecting all survivals equally through a function of the population vector, often called the ‘critical age class’. We show that the outcome of such an age‐dependent r‐ and K‐selection is that the expected value of the ‘critical age class’ is maximized by evolution, a strategy strongly influenced by the magnitude of the environmental stochasticity. We also demonstrate that evolution caused by such density‐dependent selection influences the population dynamics, showing a possible reciprocal effect between ecology and evolution in age‐structured populations. This modeling approach reveals that changes in population size affecting the fitness of phenotypes with different age of maturity may be an important selective agent for variation in onset of reproduction in fluctuating environments. This provides a testable hypothesis for how patterns in the population dynamics should affect life history variation.     

Linked exploitation and interference competition drives the variable behavior of a classic predator–prey system

The potential connection between exploitation and interference competition was recognized long ago but has not been evaluated. We measured the levels of both forms of competition for the protist Didinium preying upon Paramecium. Across populations, exploitation intensity was tightly linked to interference intensity, and the form of this relationship follows from a simple model of interaction speeds. The variation in interference competition was as large across populations of Didinium as has been observed previously across species from a variety of taxa including birds, mammals, insects, crustaceans, flatworms and protists. The link between exploitation and interference competition alters our understanding of how interference competition influences population dynamics. Instead of simply stabilizing systems, variation in interference levels can shift population dynamics through qualitatively different regimes because of its association with exploitation competition. Strong interference competition pushes a system to a regime of deterministic extinction, but intermediate interference generates a system that is stable with a high competitive ability. This may help to explain why the distribution of interference values is unimodal and mostly intermediate in intensity. Synthesis Exploitation and interference competition are typically viewed as separate processes. Exploitation is described with a functional response in which the inclusion of interference competition – the effect of predator density on foraging rates – is optional. Although recent work indicates that interference competition is widespread, there is little work linking the two forms of competition. In this article we present evidence that exploitation and interference competition are linked mechanistically through movement patterns that simultaneously generate beneficial interactions of consumers with resources and detrimental interactions with other consumers. This connection alters our view of the role that interference plays in ecological dynamics.