Competition in size-structured populations: mechanisms inducing cohort formation and population cycles

In this paper we investigate the consequences of size-dependent competition among the individuals of a consumer population by analyzing the dynamic properties of a physiologically structured population model. Only 2 size-classes of individuals are distinguished: juveniles and adults. Juveniles and adults both feed on one and the same resource and hence interact by means of exploitative competition. Juvenile individuals allocate all assimilated energy into development and mature on reaching a fixed developmental threshold. The combination of this fixed threshold and the resource-dependent developmental rate, implies that the juvenile delay between birth and the onset of reproduction may vary in time. Adult individuals allocate all assimilated energy to reproduction. Mortality of both juveniles and adults is assumed to be inversely proportional to the amount of energy assimilated. In this setting we study how the dynamics of the population are influenced by the relative foraging capabilities of juveniles and adults.In line with results that we previously obtained in size-structured consumer-resource models with pulsed reproduction, population cycles primarily occur when either juveniles or adults have a distinct competitive advantage. When adults have a larger per capita feeding rate and are hence competitively superior to juveniles, population oscillations occur that are primarily induced by the fact that the duration of the juvenile period changes with changing food conditions. These cycles do not occur when the juvenile delay is a fixed parameter. When juveniles are competitively superior, two different types of population fluctuations can occur: (1) rapid, low-amplitude fluctuations having a period of half the juvenile delay and (2) slow, large-amplitude fluctuations characterized by a period, which is roughly equal to the juvenile delay. The analysis of simplified versions of the structured model indicates that these two types of oscillations also occur if mortality and/or development is independent of food density, i.e. in a situation with a constant juvenile developmental delay and a constant, food-independent background mortality. Thus, the oscillations that occur when juveniles are more competitive are induced by the juvenile delay per se. When juveniles exert a larger foraging pressure on the shared resource, maturation implies an increase not only in adult density, but also in food density and consequently fecundity. Our analysis suggests that this correlation in time between adult density and fecundity is crucial for the occurrence of population cycles when juveniles are competitively superior.     

Demographic stochasticity and resource autocorrelation control biological invasions in heterogeneous landscapes

Mounting theoretical evidence suggests that demographic stochasticity, environmental heterogeneity and biased movement of organisms individually affect the dynamics of biological invasions and range expansions. Studies of species spread in heterogeneous landscapes have traditionally characterized invasion velocities as functions of the mean resource density throughout the landscape, thus neglecting higher‐order moments of the spatial resource distribution. Here, we show theoretically that different spatial arrangements of resources lead to different spread velocities even if the mean resource density throughout the landscape is kept constant. Specifically, we find that increasing the resource autocorrelation length causes a reduction in the speed of species spread. The model shows that demographic stochasticity plays a key role in the slowdown, which is strengthened when individuals can actively move towards resources. We then experimentally corroborated the theoretically predicted reduction in propagation speed in microcosm experiments with the protist Euglena gracilis by comparing spread in landscapes with different resource autocorrelation lengths. Our work identifies the resource autocorrelation length as a key modulator and a simple measure of landscape susceptibility to biological invasions, which needs to be considered for predicting invasion dynamics within naturally heterogeneous environmental corridors.     

Absolute stability in predator-prey models

An equilibrium of a time-lagged population model is said to be absolutely stable if it remains locally stable regardless of the length of the time delay, and it is argued that the criteria for absolute stability provide a valuable guide to the behavior of population models. For example, it is sometimes assumed that time delays have a limited impact until they exceed the natural time scale of a system; here it is stressed that under some conditions very short time delays can have a marked (and often maximal) destabilizing effect. Consequently it is important that our understanding of population dynamics is robust to the inclusion of the short time delays present in all biological systems. The absolute stability criteria are ideally suited for this role. Another important reason for using the criteria for absolute stability rather than using criteria which depend upon the details of a time delay is that biological time delays are unlikely to be constant. For example, a time delay due to maturation inevitably varies between individuals and the mean may itself vary over time. Here it is shown that the criteria for absolute stability are generally robust in the presence of distributed delays and of varying delays. The analysis presented is based upon a general predator-prey model and it is shown that absolute stability can be expected under a broad range of parameter values whenever the time delay is due to the maturation time of either the predator or the prey or of both. This stability occurs because of the interaction between delayed and undelayed dynamic features of the model. A time-delayed process, when viewed across all possible delays, always reduces stability and this effect occurs regardless of whether the process would act to stabilize or destabilize an undelayed system. Opposing the destabilization due to a time delay and making absolute stability a possibility are a number of processes which act without delay. Some of these processes can be identified as stabilizing from the analysis of undelayed models (for example, the type 3 functional response) but other cannot (for example, the nonreproductive numerical response of predators).     

一类具分段常数变量的捕食-食饵系统的Neimark-Sacker分支

讨论了具时滞与分段常数变量的捕食-食饵生态模型的稳定性及Neimark-Sacker分支;通过计算得到连续模型对应的差分模型,基于特征值理论和Schur-Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支理论和中心流形定理分析了Neimark-Sacker分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。     

Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

Simplifying a physiologically structured population model to a stage-structured biomass model

We formulate and analyze an archetypal consumer-resource model in terms of ordinary differential equations that consistently translates individual life history processes, in particular food-dependent growth in body size and stage-specific differences between juveniles and adults in resource use and mortality, to the population level. This stage-structured model is derived as an approximation to a physiologically structured population model, which accounts for a complete size-distribution of the consumer population and which is based on assumptions about the energy budget and size-dependent life history of individual consumers. The approximation ensures that under equilibrium conditions predictions of both models are completely identical. In addition we find that under non-equilibrium conditions the stage-structured model gives rise to dynamics that closely approximate the dynamics exhibited by the size-structured model, as long as adult consumers are superior foragers than juveniles with a higher mass-specific ingestion rate. When the mass-specific intake rate of juvenile consumers is higher, the size-structured model exhibits single-generation cycles, in which a single cohort of consumers dominates population dynamics throughout its life time and the population composition varies over time between a dominance by juveniles and adults, respectively. The stage-structured model does not capture these dynamics because it incorporates a distributed time delay between the birth and maturation of an individual organism in contrast to the size-structured model, in which maturation is a discrete event in individual life history. We investigate model dynamics with both semi-chemostat and logistic resource growth.     

Bioenergetic theory predicts infection dynamics of human schistosomes in intermediate host snails across ecological gradients

Epidemiological dynamics depend on the traits of hosts and parasites, but hosts and parasites are heterogeneous entities that exist in dynamic environments. Resource availability is a particularly dynamic and potent environmental driver of within‐host infection dynamics (temporal patterns of growth, reproduction, parasite production and survival). We developed, parameterised and validated a model for resource‐explicit infection dynamics by incorporating a parasitism module into dynamic energy budget theory. The model mechanistically explained the dynamic multivariate responses of the human parasite Schistosoma mansoni and its intermediate host snail to variation in resources and host density. At the population level, feedbacks mediated by resource competition could create a unimodal relationship between snail density and human risk of exposure to schistosomes. Consequently, weak snail control could backfire if reductions in snail density release remaining hosts from resource competition. If resource competition is strong and relevant to schistosome production in nature, it could inform control strategies.     

Adaptive plasticity in ontogenetic niche shifts stabilizes consumer-resource dynamics

Ontogenetic niche shifts, changes in the diet or habitats of organisms during their ontogeny, are widespread among various animal taxa. Ontogenetic niche shifts introduce stage structure in a population with different stages interacting with different communities and can substantially affect their dynamics. In this article, I use mathematical models to test the hypothesis that adaptive plasticity in the timing of ontogenetic niche shifts has a stabilizing effect on consumer-resource dynamics. Adaptive plasticity allows consumers in one ontogenetic niche to perform an early shift to the next ontogenetic niche if the resource density of the first niche is low. The early shift will reduce predation by the consumer on the scarce resource. On the other hand, adaptive plasticity allows consumers to delay their shift to the next niche if the resource density of the first niche is high. The delayed shift will increase the predation on the abundant resource. As a result, the scarce resource will tend to increase, and the abundant resource will tend to decrease. This causes density-dependent negative feedback in the resource dynamics, which stabilizes the consumer-resource dynamics. To test this hypothesis, I compare three consumer-resource models differing in terms of mechanisms controlling the timing of the ontogenetic niche shift: the fixed-age model assumes that the age at which the ontogenetic niche shift occurs is fixed; the fixed-size model assumes that the size at the shift is fixed; and the adaptive plasticity model assumes that the timing of the shift is such that the individual fitness of the consumer is maximized. I show that only the adaptive plasticity model has a locally stable equilibrium and that the stabilizing effect is due to the density-dependent negative feedback in the resource dynamics. I discuss the ontogenetic niche shifts of lake fish in light of the obtained result.     

Predator-prey models with delay and prey harvesting

It is known that predator-prey systems with constant rate harvesting exhibit very rich dynamics. On the other hand, incorporating time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of the harvesting rate and the time delay on the dynamics of the generalized Gause-type predator-prey models and the Wangersky-Cunningham model. It is shown that in these models the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities, while the harvesting rate has a stabilizing effect on the equilibrium if it is under the critical harvesting level. In particular, one of these models loses stability when the delay varies and then regains its stability when the harvesting rate is increased. Computer simulations are carried to explain the mathematical conclusions. Received: 1 March 2000 / Revised version: 7 September 2000 /?Published online: 21 August 2001     

Resource-dependent dispersal and the speed of biological invasions

Many mobile organisms exhibit resource-dependent movement in which movement rates adjust to changes in local resource densities through changes in either the probability of moving or the distance moved. Such changes may have important consequences for invasions because reductions in resources behind an invasion front may cause higher dispersal while simultaneously reducing population growth behind the front and thus lowering the number of dispersers. Intuiting how the interplay between population growth and dispersal affects invasions is difficult without mathematical models, yet most models assume dispersal rates are constant. Here we present spatial-spread models that allow for consumer-resource interactions and resource-dependent dispersal. Our results show that when resources affect the probability of dispersal, then the invasion dynamics are no different than if resources did not affect dispersal. When resources instead affect the distance dispersed, however, the invasion dynamics are strongly affected by the strength of the consumer-resource interaction, and population cycles behind the wave front lead to fluctuating rates of spread. Our results suggest that for actively dispersing invaders, invasion dynamics can be determined by species interactions. More practically, our work suggests that reducing invader densities behind the front may be a useful method of slowing an invader's rate of spread.     

Resource competition in stage-structured populations

Two models are made to account for the dynamics of a consumer-resource system in which the consumers are divided into juveniles and adults. The resource grows logistically and a type II functional response is assumed for consumers. Resource levels determine fecundity and maturation rates in one model, and mortality rates in the other. The analysis of the models shows that the condition for establishment of consumers is that the product of per capita fecundity rate and maturation rates is higher than the product of juvenile and adult per capita decay rates at a resource level equal to its carrying capacity. This result imposes a minimal abundance of resource able to maintain the consumers. A second result shows an equilibrium stage structure, with a small instability when juveniles and adults mean saturation constants are different. The implications of these results for community dynamics are discussed.     

Size-dependent predation and correlated life history traits alter eco-evolutionary dynamics and selection for faster individual growth

Age at maturation is a key life history trait influencing individual fitness, population age structure, and ecological interactions. We investigated the evolution of age at maturity through changes in the von Bertalanffy growth constant for organisms with a simple juvenile-adult life history. We used Gillespie eco-evolutionary models to uncover the role of predation in driving the evolution of the growth constant when eco-evolutionary dynamics are present. We incorporated both size-independent and size-dependent predation into our models to generate differences in selection and dynamics in the system. Our results generally support the idea that faster ontogenetic growth is beneficial when populations are growing but that predation tends to have little effect on age at maturity unless there are trade-offs with other life history traits. In particular, if faster ontogenetic growth comes at the cost of fecundity, our results suggest that predation selects for intermediate levels of growth and fecundity. Eco-evolutionary dynamics influenced the nature of selection only when growth was linked to fecundity. We also found that predators that increasingly consume larger prey tend to have higher population sizes due to the greater energy intake from larger prey, but the growth rate-fecundity trade-off reversed this pattern. Overall, our results suggest an important role for interactions between size-dependent foraging and life-history trade-offs in generating varying selection on age at maturity through underlying growth traits.     

Stage-structured ontogeny in resource populations generates non-additive stabilizing and de-stabilizing forces in populations and communities

Demographic heterogeneity influences how populations respond to density dependent intraspecific competition and trophic interactions. Distinct stages across an organism's development, or ontogeny, are an important example of demographic heterogeneity. In consumer populations, ontogenetic stage structure has been shown to produce categorical differences in population dynamics, community dynamics and even species coexistence compared to models lacking explicit ontogeny. The study of consumer–resource interactions must also consider the ontogenetic stage structure of the resource itself, particularly plants, given their fundamental role at the basis of terrestrial food webs. We incorporate distinct ontogenetic stages of plants into an adaptable multi-stage consumer–resource modeling framework that facilitates studying how stage specific consumers shape trophic dynamics at low trophic levels. We describe the role of density dependent demographic rates in mediating the dynamics of stage-structured plant populations. We then investigate how these demographic rates interact with consumer pressure to influence stability and coexistence in multiple stage-specific consumer–resource interactions. Results detail how density dependent effects across distinct ontogenetic stages in plant development produce non-additivity in the drivers of dynamic stability both in single populations and in consumer–resource settings, challenging the ubiquity of certain traditional ecological dynamic paradigms. We also find categorical differences in the population variability induced by herbivores consuming separate plant stages. Consumer–resource models, such as plant–herbivore interactions, often average out demographic heterogeneity in populations. Here, we show that explicitly including plant demographic heterogeneity through ontogeny yields distinct dynamic expectations for both plants and herbivores compared to traditional consumer–resource formulations. Our results indicate that efforts to understand the demographic effect of herbivores on plant populations may need to also consider the effects of plant demographics on herbivores and the reciprocal relationship between them.     

A community model of ciliateTetrahymena and bacteriaE. coli: Part I. Individual-based models ofTetrahymena andE. coli populations

The dynamics of a microbial community consisting of a eucaryotic ciliateTetrahymena pyriformis and procaryoticEscherichia coli in a batch culture is explored by employing an individual-based approach. In this portion of the article, Part I, population models are presented. Because both models are individual-based, models of individual organisms are developed prior to construction of the population models. The individual models use an energy budget method in which growth depends on energy gain from feeding and energy sinks such as maintenance and reproduction. These models are not limited by simplifying assumptions about constant yield, constant energy sinks and Monod growth kinetics as are traditional models of microbal organisms. Population models are generated from individual models by creating distinct individual types and assigning to each type the number of real individuals they represent. A population is a compilation of individual types that vary in a phase of cell cycle and physiological parameters such as filtering rate for ciliates and maximum anabolic rate for bacteria. An advantage of the developed models is that they realistically describe the growth of the individual cells feeding on resource which varies in density and composition. Part II, the core of the project, integrates models into a dynamic microbial community and provides model analysis based upon available data.     

Offspring size–number tradeoffs and food quality feedbacks impact population dynamics in a Daphnia–algae system

Population fluctuations can be affected by both extrinsic (e.g. weather patterns, food availability) and intrinsic (e.g. life‐history) factors. A key life‐history tradeoff is the production of offspring size versus number, ranging from many small offspring to few large offspring. Models show that this life‐history tradeoff in offspring size and number, through maturation time, can have significant impacts on population dynamics. However, few manipulative experiments have been conducted that can isolate life‐history effects from impacts of extrinsic factors in consumer–resource systems. We experimentally tested the effect of an offspring size–number tradeoff on population stability and food availability in a consumer–resource system. Using Daphnia pulex, we created a shift from many, small offspring being produced to fewer, larger offspring. Two sets of experiments were performed to examine the interaction of an extrinsic factor (light levels) and intrinsic population structure on dynamics, and we controlled for the ingestion pressure on algal prey at the time of the manipulation. We predicted that the tradeoff would impact internal consumer population characteristics, including biasing the stage structure towards adults, increasing adult size, and increasing average population‐level reproduction. This adult‐dominated stage structure was predicted to then lead to instability and a low quantity–high quality food state. Under all light levels, the manipulated populations became dominated by large adults. Contrary to predictions, the amplitudes of fluctuations in Daphnia biomass were lower in populations shifted to few–large offspring, representing higher stability in these populations. Furthermore, in high light conditions, a stable low Daphnia – high algae biomass (low food quality) state was observed in few–large offspring treatments but not in control (many–small offspring) treatments. Our results show a strong link between light levels as an extrinsic factor and the life‐history tradeoff of consumer offspring size versus number that impacts consumer–resource population dynamics through feedbacks with resource quality.     

Dynamic state-dependent modelling predicts optimal usage patterns of responsive defences

Chemical defences against predation often involve responses to specific predation events where the prey expels fluids, such as haemolymph or gut contents, which are aversive to the predator. The common link is that each predation attempt that is averted results in an energetic cost and a reduction in the chemical defences of the prey, which might leave the prey vulnerable if the next predation attempt occurs soon afterwards. Since prey appear to be able to control the magnitude of their responses, we should expect them to trade-off the need to repel the current threat against the need to preserve defences against future threats and conserve energy for other essential activities. Here we use dynamic state-dependent models to predict optimal strategies of defence deployment in the juvenile stage of an animal that has to survive to maturation. We explore the importance of resource level, predator density, and the costs of making defences on the magnitude of the responses and optimal age and size at maturation. We predict the patterns of investment and the magnitude of the deployment of defences to potentially multiple attacks over the juvenile period, and show that responses should be smaller when the costs of defences and/or predation risk are higher. The model enables us to predict that animals in which defences benefit the adult stage will employ different strategies than those that do not use the same defences as adults, and thereby experience a smaller reduction in body size as a result of repeated attacks. We also explore the effect of the importance of adult size, and find that the sex and mating system of the prey should also affect defensive strategies. Our work provides the first predictive theory of the adaptive use of responsive defences across taxa.     

Complex tactics in a dynamic large herbivore–carnivore spatiotemporal game

The spatiotemporal game between predators and prey is a fundamental process governing their distribution dynamics. Players may adopt different tactics as the associated costs and benefits change through time. Yet few studies have investigated the potentially simultaneous and dynamic nature of movement tactics used by both players. It is particularly unclear to what extent perceived predation risk mediates the fine‐scale distribution of large and dangerous prey, which are mostly driven by bottom–up, resource‐related processes. We built habitat use and movement models based on 10 years of monitoring GPS‐collared grey wolves Canis lupus and plains bison Bison bison bison in Prince Albert National Park, Canada, to investigate the predator–large prey game in a multi‐prey system. Bison did not underuse patches of high‐quality vegetation at any time during the seasonal cycle even though wolves were selectively patrolling these areas. Rather, in at least one season, bison engaged in complex tactics comprised of proactive responses to the long‐term distribution (risky places) and reactive responses to the immediate proximity (risky times) of their opponent. In summer–autumn, bison reduced the time spent in food‐rich patches as both the long‐term use and the immediate proximity of wolves increased. By demonstrating that wolf distribution triggers patch abandonment by bison, we provide a key element in support of the shell game hypothesis – where prey move constantly to avoid predators attempting to anticipate their location. In winter, a season of relatively high energetic stress, bison no longer abandoned food‐rich patches as predation risk increased, while no bison responses to wolves were observed in spring–summer. Our work demonstrates the highly dynamic and complex nature of the predator–large prey spatiotemporal game, a key trait‐mediated mechanism by which trophic interactions structure ecological communities.     

The onset of embryo maturation in Arabidopsis is determined by its developmental stage and does not depend on endosperm cellularization

Seeds are dormant and desiccated structures, filled with storage products to be used after germination. These properties are determined by the maturation program, which starts, in Arabidopsis thaliana, mid‐embryogenesis, at about the same time and developmental stage in all the seeds in a fruit. The two factors, chronological and developmental time, are closely entangled during seed development, so their relative contribution to the transition to maturation is not well understood. It is also unclear whether that transition is determined autonomously by each seed or whether it depends on signals from the fruit. The onset of maturation follows the cellularization of the endosperm, and it has been proposed that there exists a causal relationship between both processes. We explored all these issues by analyzing markers for maturation in Arabidopsis mutant seeds that develop at a slower pace, or where endosperm cellularization happens too early, too late, or not at all. Our data show that the developmental stage of the embryo is the key determinant of the initiation of maturation, and that each seed makes that transition autonomously. We also found that, in contrast with previous models, endosperm cellularization is not required for the onset of maturation, suggesting that this transition is independent of the hexose/sucrose ratio in the seed. Our observations indicate that the mechanisms that control endosperm cellularization, embryo growth, and embryo maturation act independently of each other.     

Resource supply and the evolution of public-goods cooperation in bacteria

Background  

Explaining public-goods cooperation is a challenge for evolutionary biology. However, cooperation is expected to more readily evolve if it imposes a smaller cost. Such costs of cooperation are expected to decline with increasing resource supply, an ecological parameter that varies widely in nature. We experimentally tested the effect of resource supply on the evolution of cooperation using two well-studied bacterial public-good traits: biofilm formation by Pseudomonas fluorescens and siderophore production by Pseudomonas aeruginosa.     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.