Equilibria in structured populations

The existence of a stable positive equilibrium state for the density of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.This research was done while the author was on leave at the Lehrstuhl für Biomathematik, Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen 1, Federal Republic of Germany     

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.     

Moments of the probability distribution of moving molecules undergoing reversible isomerization,with applications to experiments

Exact expressions are obtained for the mean position and the variance about the mean of macromolecules which are moving in an electrostatic or centrifugal field and which, at the same time, are switching back and forth between two isomeric states. Comparison with experiment then yields the forward and backward switching rates. The following special cases are considered: (a) only one species present initially; (b) both species present initially but not in their equilibrium proportions; (c) both species present initially in their equilibrium proportions. It is shown that in the first two cases we need only measure the mean position of all the molecules in order to measure the absolute switching rates k₁ and k₂. In the third case, however, we must measure the variance (mean-square deviation) of the position in order to obtain k₁ and k₂. The first two situations arise when “jumps” (e.g., in temperature or pressure) are made, while the third situation is obtained if the experiment is conducted with the species in chemical equilibrium throughout the experiment.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

Temporally auto-correlated predator attacks structure ecological communities

For species primarily regulated by a common predator, the P* rule of Holt & Lawton (Holt & Lawton, 1993. Am. Nat. 142, 623–645. (doi:10.1086/285561)) predicts that the prey species that supports the highest mean predator density (P*) excludes the other prey species. This prediction is re-examined in the presence of temporal fluctuations in the vital rates of the interacting species including predator attack rates. When the fluctuations in predator attack rates are temporally uncorrelated, the P* rule still holds even when the other vital rates are temporally auto-correlated. However, when temporal auto-correlations in attack rates are positive but not too strong, the prey species can coexist due to the emergence of a positive covariance between predator density and prey vulnerability. This coexistence mechanism is similar to the storage effect for species regulated by a common resource. Negative or strongly positive auto-correlations in attack rates generate a negative covariance between predator density and prey vulnerability and a stochastic priority effect can emerge: with non-zero probability either prey species is excluded. These results highlight how temporally auto-correlated species’ interaction rates impact the structure and dynamics of ecological communities.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

Asymptotic behaviour of a model of hierarchically structured population dynamics

 A hierarchically structured population model with a dependence of the vital rates on a function of the population density (environment) is considered. The existence, uniqueness and the asymptotic behaviour of the solutions is obtained transforming the original non-local PDE of the model into a local one. Under natural conditions, the global asymptotical stability of a nontrivial equilibrium is proved. Finally, if the environment is a function of the biomass distribution, the existence of a positive total biomass equilibrium without a nontrivial population equilibrium is shown. Received 16 February 1996; received in revised form 16 September 1996     

A quantitative test of the size efficiency hypothesis by means of a physiologically structured model

According to the size‐efficiency hypothesis (SEH) larger bodied cladocerans are better competitors for food than small bodied species. In environments with fish, however, the higher losses of the large bodied species due to size‐selective predation may shift the balance in favor of the small bodied species. Here we present a theoretical framework for the analysis of the competitive abilities of zooplankton species that takes both competition and predation into account in one coherent analysis. By applying the conceptually well‐understood framework of physiologically structured population models we were able to predict the relative difference in predation rates necessary to cause a shift in dominance of the large‐bodied species (Daphnia pulicaria) to the small‐bodied species (D. galeata). These predictions depend only on seven easily interpretable parameters per species: size at birth, size at maturity and maximum size, age at maturity, maximal clutch size, egg development time and finally the half‐saturation constant for food. The critical equilibrium mortality of D. pulicaria was 0.16 d^?1 at food concentrations close to the critical food concentration of D. galeata, i.e. D. pulicaria will win the competition as long as its mortality rate is below 0.16 d^?1. At higher food concentrations the differential mortality curve (plotting equilibrium mortalities of both species against each other) approached a linear function with a slope of one and an intercept equal to the difference in maximal population birth rates. The prediction of critical predation rates was independent of the ingestion rate of the cladocerans and the algal carrying capacity and food regeneration rate of the environment although the mechanism works through competition for a shared algal food resource. We interpret these findings in terms of the relative predation risk large and small‐bodied cladocerans will face in various freshwater ecosystems.     

Phenological responses in a sycamore–aphid–parasitoid system and consequences for aphid population dynamics: A 20 year case study

Species interactions have a spatiotemporal component driven by environmental cues, which if altered by climate change can drive shifts in community dynamics. There is insufficient understanding of the precise time windows during which inter‐annual variation in weather drives phenological shifts and the consequences for mismatches between interacting species and resultant population dynamics—particularly for insects. We use a 20 year study on a tri‐trophic system: sycamore Acer pseudoplatanus, two associated aphid species Drepanosiphum platanoidis and Periphyllus testudinaceus and their hymenopteran parasitoids. Using a sliding window approach, we assess climatic drivers of phenology in all three trophic levels. We quantify the magnitude of resultant trophic mismatches between aphids and their plant hosts and parasitoids, and then model the impacts of these mismatches, direct weather effects and density dependence on local‐scale aphid population dynamics. Warmer temperatures in mid‐March to late‐April were associated with advanced sycamore budburst, parasitoid attack and (marginally) D. platanoidis emergence. The precise time window during which spring weather advances phenology varies considerably across each species. Crucially, warmer temperatures in late winter delayed the emergence of both aphid species. Seasonal variation in warming rates thus generates marked shifts in the relative timing of spring events across trophic levels and mismatches in the phenology of interacting species. Despite this, we found no evidence that aphid population growth rates were adversely impacted by the magnitude of mismatch with their host plants or parasitoids, or direct impacts of temperature and precipitation. Strong density dependence effects occurred in both aphid species and probably buffered populations, through density‐dependent compensation, from adverse impacts of the marked inter‐annual climatic variation that occurred during the study period. These findings explain the resilience of aphid populations to climate change and uncover a key mechanism, warmer winter temperatures delaying insect phenology, by which climate change drives asynchronous shifts between interacting species.     

Above-ground forest biomass is not consistently related to wood density in tropical forests

Aim  It is increasingly accepted that the mean wood density of trees within a forest is tightly coupled to above-ground forest biomass. It is unknown, however, if a positive relationship between forest biomass and mean community wood density is a general phenomenon across forests. Understanding spatial variation in biomass as a function of wood density both within and among forests is important for predicting changes in stored carbon in response to global change, and here we evaluated the generality of a positive biomass–wood density relationship within and among six tropical forests.
Location  Costa Rica, Panama, Puerto Rico and Ecuador.
Methods  Individual stem data, including diameter at breast height and spatial position, for six forest dynamics plots were merged with an extensive wood density database. Individual stem biomass values were calculated from these data using published statistical models. Total above ground biomass, total basal area and mean community wood density were also quantified across a range of subcommunity plot sizes within each forest.
Results  Among forests, biomass did not vary with mean community wood density. The relationship between subcommunity biomass and mean wood density within a forest varied from negative to null to positive depending on the size of subcommunities and forest identity. The direction of correlation was determined by the associated total basal area–mean wood density correlation, the slope of which increased strongly with whole forest mean wood density.
Main conclusions  There is no general relationship between forest biomass and wood density, and in some forests, stored carbon is highest where wood density is lowest. Our results suggest that declining wood density, due to global change, will result in decreased or increased stored carbon in forests with high or low mean wood density, respectively.     

Backward bifurcation,equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R ₀ < 1, where R ₀ is the basic reproductive number, and a unique supercritical endemic equilibrium for R ₀ > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b.      

Numerical solutions of the Lamm equation. II. Equilibrium sedimentation

The Lamm equation has been solved numerically for conditions corresponding to equilibrium runs for a nonlinear concentration dependence of the form s/s₀ = (1 + kc)^?1. It is shown that the approach to equilibrium is very close to being exponential (in time) as in the case k = 0. We also compare results for the nonlinear case given above with results obtained for linear c-dependence of the form s/s₀ = 1 – kc. For relatively high speeds the time required to attain equilibrium may be greatly underestimated by use of the latter approximation. Finally, we present analytic approximations for the concentration distribution at equilibrium and as a function of time.     

Heterospecific courtship,minority effects and niche separation between cryptic butterfly species

Species interacting in varied ecological conditions often evolve in different directions in different local populations. The butterflies of the cryptic Leptidea complex are sympatrically distributed in different combinations across their Eurasian range. Interestingly, the same species is a habitat generalist in some regions and a habitat specialist in others, where a sibling species has the habitat generalist role. Previous studies suggest that this geographically variable niche divergence is generated by local processes in different contact zones. By varying the absolute and relative densities of Leptidea sinapis and Leptidea juvernica in large outdoor cages, we show that female mating success is unaffected by conspecific density, but strongly negatively affected by the density of the other species. Whereas 80% of the females mated when a conspecific couple was alone in a cage, less than 10% mated when the single couple shared the cage with five pairs of the other species. The heterospecific courtships can thus affect the population fitness, and for the species in the local minority, the suitability of a habitat is likely to depend on the presence or absence of the locally interacting species. If the local relative abundance of the different species depends on the colonization order, priority effects might determine the ecological roles of interacting species in this system.     

Evolutionary tracking is determined by differential selection on demographic rates and density dependence

Recent ecological forecasts predict that ~25% of species worldwide will go extinct by 2050. However, these estimates are primarily based on environmental changes alone and fail to incorporate important biological mechanisms such as genetic adaptation via evolution. Thus, environmental change can affect population dynamics in ways that classical frameworks can neither describe nor predict. Furthermore, often due to a lack of data, forecasting models commonly describe changes in population demography by summarizing changes in fecundity and survival concurrently with the intrinsic growth rate (r). This has been shown to be an oversimplification as the environment may impose selective pressure on specific demographic rates (birth and death) rather than directly on r (the difference between the birth and death rates). This differential pressure may alter population response to density, in each demographic rate, further diluting the information combined to produce r. Thus, when we consider the potential for persistence via adaptive evolution, populations with the same r can have different abilities to persist amidst environmental change. Therefore, we cannot adequately forecast population response to climate change without accounting for demography and selection on density dependence. Using a continuous‐time Markov chain model to describe the stochastic dynamics of the logistic model of population growth and allow for trait evolution via mutations arising during birth events, we find persistence via evolutionary tracking more likely when environmental change alters birth rather than the death rate. Furthermore, species that evolve responses to changes in the strength of density dependence due to environmental change are less vulnerable to extinction than species that undergo selection independent of population density. By incorporating these key demographic considerations into our predictive models, we can better understand how species will respond to climate change.     

Ecological Equivalence: A Realistic Assumption for Niche Theory as a Testable Alternative to Neutral Theory

Background

Hubbell''s 2001 neutral theory unifies biodiversity and biogeography by modelling steady-state distributions of species richness and abundances across spatio-temporal scales. Accurate predictions have issued from its core premise that all species have identical vital rates. Yet no ecologist believes that species are identical in reality. Here I explain this paradox in terms of the ecological equivalence that species must achieve at their coexistence equilibrium, defined by zero net fitness for all regardless of intrinsic differences between them. I show that the distinction of realised from intrinsic vital rates is crucial to evaluating community resilience.

Principal Findings

An analysis of competitive interactions reveals how zero-sum patterns of abundance emerge for species with contrasting life-history traits as for identical species. I develop a stochastic model to simulate community assembly from a random drift of invasions sustaining the dynamics of recruitment following deaths and extinctions. Species are allocated identical intrinsic vital rates for neutral dynamics, or random intrinsic vital rates and competitive abilities for niche dynamics either on a continuous scale or between dominant-fugitive extremes. Resulting communities have steady-state distributions of the same type for more or less extremely differentiated species as for identical species. All produce negatively skewed log-normal distributions of species abundance, zero-sum relationships of total abundance to area, and Arrhenius relationships of species to area. Intrinsically identical species nevertheless support fewer total individuals, because their densities impact as strongly on each other as on themselves. Truly neutral communities have measurably lower abundance/area and higher species/abundance ratios.

Conclusions

Neutral scenarios can be parameterized as null hypotheses for testing competitive release, which is a sure signal of niche dynamics. Ignoring the true strength of interactions between and within species risks a substantial misrepresentation of community resilience to habitat loss.     

Classification of South Swedish Isoetid vegetation with the help of numerical methods

Relevés of Isoetid vegetation from 60 lakes in southern Sweden have been classified with the help on numerical methods. A community system is constructed, at the variant level by clustering with the TABORD program, and at the community and subcommunity levels by reference to traditional floristic characteristics. The diagnostic species were selected with the help of PCA. Isoëtes lacustris, Lobelia dortmanna and Littorella uniflora determined the community composition together with Eleocharis acicularis. The differentation of the syntaxa along the first three ordination axes of a PCA was clear. The complex water depth factor determines clearly the position of the small syntaxa (variants) in the PCA ordination space. At high levels of similarity the relevés were grouped effectively with the TABORD program, so that the clusters are floristically homogeneous and easy to identify on the basis of floristicsociological criteria. At higher syntaxon levels a selective use of diagnostic species was made.     

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.      

Persistence in fluctuating environments for interacting structured populations

Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov exponents of random matrix products. Our coexistence criterion requires that a convex combination of these long-term growth rates is positive with probability one whenever one or more species are at low density. When this condition holds, the community is stochastically persistent: the fraction of time that a species density goes below \(\delta >0\) approaches zero as \(\delta \) approaches zero. Applications to predator-prey interactions in an autocorrelated environment, a stochastic LPA model, and spatial lottery models are provided. These applications demonstrate that positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence, fluctuations in log-fecundity can facilitate persistence in structured populations, and long-lived, relatively sedentary competing populations are likely to coexist in spatially and temporally heterogenous environments.     

A model-independent test for the presence of regulatory equilibrium and non-random structure in island species trajectories

Aims To test a key prevision of the dynamic equilibrium theory of island biogeography, namely that changes in species numbers on islands over time (hereafter, species trajectories) are equilibrial, and to characterize aspects of the dynamical properties of species change over time using a model‐independent test. Methods We tested for regulatory equilibrium and non‐random structure in species numbers through time by comparing observed correlation coefficients at lag‐k for species trajectories from four true islands and two habitat islands. First, we estimated the shape of the autocorrelation function for each observed species trajectory by calculating correlation coefficients of the observed data between pairs of values N_t?k and N_t separated by lag‐k (k = 1, 2, …, N ? 1). Second, we tested the observed correlation coefficients at each lag against a distribution of correlation coefficients generated by randomly ordering observed numbers in the species trajectories. Results The patterns of autocorrelation functions for all but one of the observed species trajectories did not exhibit evidence of regulatory equilibrium, and, in fact, closely matched what would be expected from a non‐stationary or ‘random walk’ process. The majority of the correlation coefficients generated from the observed species trajectories did not deviate significantly from correlation coefficients produced by the randomized trajectories. However, there was strong evidence of unusual positive autocorrelation at small time lags for birds on islands measured annually (2‐ to 4‐year lags) and for arthropods on islands measured weekly (7‐ to 8‐week lags), suggesting some degree of structure in change in species richness over time. Main conclusions The autocorrelation function patterns for all but one of the observed species trajectories showed various forms of non‐stationarity. These types of patterns suggest that the numbers of species through time gradually wandered away from their initial sizes. Our model‐independent test of individual correlation coefficients revealed significant structure in the observed species trajectories. These trajectories appear to be non‐random at relatively short lag intervals, indicating a process with short memory.     

Ecosystem functioning in relation to species identity,density, and biomass in two tunneller dung beetles

1. Species abundance, biomass, and identity are the main factors that influence ecosystem functioning. Previous studies have shown that community attributes and species identity help to maintain natural ecosystem functioning. 2. This study examined how species identity, biomass, and abundance in dung pats (i.e. density) of dung beetles affect multiple ecological functions: dung removal, seed dispersal, and germination. Specifically, two species of tunnellers were targeted: Onthophagus illyricus (Scopoli, 1763) and Copris lunaris (Linnaeus, 1758). In accordance with their natural abundance, densities ranging from 10 to 80 individuals were considered for O. illyricus, and those from two to eight were considered for C. lunaris, spanning the total biomass per treatment from 0.22 to 1.76 g. 3. Results showed that, even at higher abundance, O. illyricus is not as efficient as C. lunaris. These results show that species identity, biomass, and density are crucial factors for maintaining ecosystem functioning. The combined effect of species identity and density/biomass facilitated dung removal and seed dispersal. Conversely, species identity is the only relevant factor for germination. Moreover, relationships among functions depend on the species investigated: C. lunaris showed a positive correlation between dung removal and seed dispersal, whereas O. illyricus showed a positive correlation between germination and dung removal. 4. In conclusion, optimal ecosystem functioning depends on multiple factors, such as density and species identity, and thus also on body size, nesting strategies and ecological functions investigated. Moreover, the loss of larger and efficient species cannot be compensated by higher abundances of small species.