Dispersal and transient dynamics in metapopulations

Abstract Recent studies of spatially explicit metapopulation models have shown the existence of complex transient behaviour (supertransients and mesotransients) characterized by spontaneous changes in the system's dynamics after thousands or hundreds of generations, respectively. Their detection in simple ecological models has been taken as evidence that transient dynamics may be common in nature. In this study, we explore the generality of these phenomena in a simple one‐dimensional spatially explicit metapopulation model. We investigate how frequently supertransient behaviour emerges in relation to the shape and type of the dispersal kernel used (normal and Laplace), system size, boundary conditions and how sensitive they are to initial conditions. Our results show that supertransients are rare, are heavily affected by initial conditions and occur for a small set of dispersal parameter values, which vary according to kernel type, system size, and boundary conditions. Similarly, mesotransients emerge over a very narrow range of dispersal parameter values and are rare under all circumstances. Thus, transient dynamics are not likely to be either common or widespread in simple models of ecological systems.     

Dynamic complexities in host-parasitoid interaction

In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of self-similarity and fractal basin boundaries), (c) intermittency, (d) supertransients, (e) chaotic attractors, and (f) "transient chaos". Because of these complexities minor changes in parameter or initial values may strikingly change the dynamic behavior of the system. All the phenomena presented in this paper should be kept in mind when examining and interpreting the dynamics of ecological systems. Copyright 1999 Academic Press.     

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.     

Maternal effects on offspring consumption can stabilize fluctuating predator–prey systems

Maternal effects, where the conditions experienced by mothers affect the phenotype of their offspring, are widespread in nature and have the potential to influence population dynamics. However, they are very rarely included in models of population dynamics. Here, we investigate a recently discovered maternal effect, where maternal food availability affects the feeding rate of offspring so that well-fed mothers produce fast-feeding offspring. To understand how this maternal effect influences population dynamics, we explore novel predator–prey models where the consumption rate of predators is modified by changes in maternal prey availability. We address the ‘paradox of enrichment'', a theoretical prediction that nutrient enrichment destabilizes populations, leading to cycling behaviour and an increased risk of extinction, which has proved difficult to confirm in the wild. Our models show that enriched populations can be stabilized by maternal effects on feeding rate, thus presenting an intriguing potential explanation for the general absence of ‘paradox of enrichment'' behaviour in natural populations. This stabilizing influence should also reduce a population''s risk of extinction and vulnerability to harvesting.     

Dynamics of a host–parasitoid model with Allee effect for the host and parasitoid aggregation

In this paper, a discrete-time host–parasitoid model is investigated. Two biological phenomena, the Allee effect of the host population and the aggregation of the parasitism, are considered in our mathematical model. Through extensive numerical simulations, we gain some interesting findings related to Allee effect from this research. Firstly, the ranges of parameter, in which the population dynamics is chaos, are compressed when Allee effect is added. Secondly, the sensitivity to initial conditions of the host–parasitoid system decreased after adding Allee effect. Thirdly, without Allee effect, we observed two complicated dynamics, intermittent chaos and supertransients. However, when Allee effect is included, these two phenomena are replaced by another kind of phenomenon-period alternation, where chaos is eliminated. From above three novel findings, it can be concluded that dynamic complexities are alleviated by Allee effect. This conclusion is crucial in resolving the discrepancy between real population dynamics and theoretical predictions. Furthermore, the importance of this research is to help us understand the mechanisms inducing the irregular fluctuations of the natural populations.     

具有年龄结构的食蚜蝇-蚜虫捕食模型

现有的具有年龄结构的捕食-食饵模型总是假设只有成年捕食者捕食猎物,这与实际情况不符。建立了一个幼年捕食者捕食食饵的具有年龄结构的食蚜蝇-蚜虫模型,应用微分方程定性理论,讨论了系统平衡点及其稳定性:其中平衡点E1(0,0,0)为不稳定的;满足一定条件时,边界平衡点E2(K,0,0)及正平衡点E3(x*,y1*,y2*)为局部渐近稳定的;且应用一致持续生存理论得到了系统永久持续生存的条件,为有害生物综合治理提供了理论依据。     

Stochastic hybrid delay population dynamics: well-posed models and extinction

Nonlinear differential equations have been used for decades for studying fluctuations in the populations of species, interactions of species with the environment, and competition and symbiosis between species. Over the years, the original non-linear models have been embellished with delay terms, stochastic terms and more recently discrete dynamics. In this paper, we investigate stochastic hybrid delay population dynamics (SHDPD), a very general class of population dynamics that comprises all of these phenomena. For this class of systems, we provide sufficient conditions to ensure that SHDPD have global positive, ultimately bounded solutions, a minimum requirement for a realistic, well-posed model. We then study the question of extinction and establish conditions under which an ecosystem modelled by SHDPD is doomed.     

Effects of asynchronism on evolutionary games

We analyze the influence of the update dynamics on symmetric 2-player evolutionary games, which are among the most used tools to study the emergence of cooperation in populations of interacting agents. A synchronous dynamics means that, at each time step, all the agents of the population update their strategies simultaneously. An extreme case of asynchronism is sequential dynamics, in which only one agent is updated each time. We first show that these two opposite update dynamics can lead to very different outcomes and that sequential dynamics is detrimental to the emergence of cooperation only when the probability of imitating the most successful neighbors is high. In this sense, we can say that, when the update dynamics has some influence, in general asynchronism is beneficial to the emergence of cooperation. We then explore the consequences of using intermediate levels of asynchronism, where only a fraction of the agents update their behavior each time. In general, the level of cooperation changes smoothly and monotonically as we gradually go from synchronous to sequential dynamics. However, there are some exceptions that should be taken into account. In addition, the results show that the possibility of agents taking irrational decisions has a key role in the sensitivity of these models to changes in the update dynamics. Explanations for the observed behaviors are advanced.     

Population and evolutionary dynamics in spatially structured seasonally varying environments

Increasingly imperative objectives in ecology are to understand and forecast population dynamic and evolutionary responses to seasonal environmental variation and change. Such population and evolutionary dynamics result from immediate and lagged responses of all key life‐history traits, and resulting demographic rates that affect population growth rate, to seasonal environmental conditions and population density. However, existing population dynamic and eco‐evolutionary theory and models have not yet fully encompassed within‐individual and among‐individual variation, covariation, structure and heterogeneity, and ongoing evolution, in a critical life‐history trait that allows individuals to respond to seasonal environmental conditions: seasonal migration. Meanwhile, empirical studies aided by new animal‐tracking technologies are increasingly demonstrating substantial within‐population variation in the occurrence and form of migration versus year‐round residence, generating diverse forms of ‘partial migration’ spanning diverse species, habitats and spatial scales. Such partially migratory systems form a continuum between the extreme scenarios of full migration and full year‐round residence, and are commonplace in nature. Here, we first review basic scenarios of partial migration and associated models designed to identify conditions that facilitate the maintenance of migratory polymorphism. We highlight that such models have been fundamental to the development of partial migration theory, but are spatially and demographically simplistic compared to the rich bodies of population dynamic theory and models that consider spatially structured populations with dispersal but no migration, or consider populations experiencing strong seasonality and full obligate migration. Second, to provide an overarching conceptual framework for spatio‐temporal population dynamics, we define a ‘partially migratory meta‐population’ system as a spatially structured set of locations that can be occupied by different sets of resident and migrant individuals in different seasons, and where locations that can support reproduction can also be linked by dispersal. We outline key forms of within‐individual and among‐individual variation and structure in migration that could arise within such systems and interact with variation in individual survival, reproduction and dispersal to create complex population dynamics and evolutionary responses across locations, seasons, years and generations. Third, we review approaches by which population dynamic and eco‐evolutionary models could be developed to test hypotheses regarding the dynamics and persistence of partially migratory meta‐populations given diverse forms of seasonal environmental variation and change, and to forecast system‐specific dynamics. To demonstrate one such approach, we use an evolutionary individual‐based model to illustrate that multiple forms of partial migration can readily co‐exist in a simple spatially structured landscape. Finally, we summarise recent empirical studies that demonstrate key components of demographic structure in partial migration, and demonstrate diverse associations with reproduction and survival. We thereby identify key theoretical and empirical knowledge gaps that remain, and consider multiple complementary approaches by which these gaps can be filled in order to elucidate population dynamic and eco‐evolutionary responses to spatio‐temporal seasonal environmental variation and change.     

On using integral projection models to generate demographically driven predictions of species' distributions: development and validation using sparse data

Knowledge of species' geographic distributions is critical for understanding and forecasting population dynamics, responses to environmental change, biodiversity patterns, and conservation planning. While many suggestive correlative occurrence models have been used to these ends, progress lies in understanding the underlying population biology that generates patterns of range dynamics. Here, we show how to use a limited quantity of demographic data to produce demographic distribution models (DDMs) using integral projection models for size‐structured populations. By modeling survival, growth, and fecundity using regression, integral projection models can interpolate across missing size data and environmental conditions to compensate for limited data. To accommodate the uncertainty associated with limited data and model assumptions, we use Bayesian models to propagate uncertainty through all stages of model development to predictions. DDMs have a number of strengths: 1) DDMs allow a mechanistic understanding of spatial occurrence patterns; 2) DDMs can predict spatial and temporal variation in local population dynamics; 3) DDMs can facilitate extrapolation under altered environmental conditions because one can evaluate the consequences for individual vital rates. To illustrate these features, we construct DDMs for an overstory perennial shrub in the Proteaceae family in the Cape Floristic Region of South Africa. We find that the species' population growth rate is limited most strongly by adult survival throughout the range and by individual growth in higher rainfall regions. While the models predict higher population growth rates in the core of the range under projected climates for 2050, they also suggest that the species faces a threat along arid range margins from the interaction of more frequent fire and drying climate. The results (and uncertainties) are helpful for prioritizing additional sampling of particular demographic parameters along these gradients to iteratively refine projections. In the appendices, we provide fully functional R code to perform all analyses.     

The role of weak selection and high mutation rates in nearly neutral evolution

Neutral dynamics occur in evolution if all types are ‘effectively equal’ in their reproductive success, where the definition of ‘effectively equal’ depends on the population size and the details of mutations. Empirically observed neutral genetic evolution in extremely large clonal populations can only be explained under current models if selection is completely absent. Such models typically consider the case where population dynamics occurs on a different timescale to evolution. However, this assumption is invalid when mutations are not rare in a whole population. We show that this has important consequences for the occurrence of neutral evolution in clonal populations. In highly connected type spaces, neutral dynamics can occur for all population sizes despite significant selective differences, via the forming of effectively neutral networks connecting rare neutral types. Biological implications include an explanation for the high diversity of rare types that survive in large clonal populations, and a theoretical justification for the use of neutral null models.     

Network-scale effects of invasive species on spatially-structured amphibian populations

Understanding the factors affecting the dynamics of spatially-structured populations (SSP) is a central topic of conservation and landscape ecology. Invasive alien species are increasingly important drivers of the dynamics of native species. However, the impacts of invasives are often assessed at the patch scale, while their effects on SSP dynamics are rarely considered. We used long-term abundance data to test whether the impact of invasive crayfish on subpopulations can also affect the whole SSP dynamics, through their influence on source populations. From 2010 to 2018, we surveyed a network of 58 ponds and recorded the abundance of Italian agile frog clutches, the occurrence of an invasive crayfish, and environmental features. Using Bayesian hierarchical models, we assessed relationhips between frog abundance in ponds and a) environmental features; b) connectivity within the SSP; c) occurrence of invasive species at both the patch- and the SSP-levels. If spatial relationships between ponds were overlooked, we did not detect effects of crayfish presence on frog abundance or trends. When we jointly considered habitat, subpopulation and SSP features, processes acting at all these levels affected frog abundance. At the subpopulation scale, frog abundance in a year was related to habitat features, but was unrelated to crayfish occurrence at that site during the previous year. However, when we considered the SSP level, we found a strong negative relationship between frog abundance in a given site and crayfish frequency in surrounding wetlands during the previous year. Hence, SSP-level analyses can identify effects that would remain unnoticed when focussing on single patches. Invasive species can affect population dynamics even in not invaded patches, through the degradation of subpopulation networks. Patch-scale assessments of the impact of invasive species can thus be insufficient: predicting the long-term interplay between invasive and native populations requires landscape-level approaches accounting for the complexity of spatial interactions.     

Linear discrete population models with two time scales in fast changing environments II: non-autonomous case

As the result of the complexity inherent in nature, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often deal with models for structured populations in which individuals are classified regarding their age, size, activity or location, and this structuring of the population leads to high dimensional systems. In many instances, the dynamics of the system is controlled by processes whose time scales are very different from each other. Aggregation techniques take advantage of this situation to build a low dimensional reduced system from which behavior we can approximate the dynamics of the complex original system.In this work we extend aggregation techniques to the case of time dependent discrete population models with two time scales where both the fast and the slow processes are allowed to change at their own characteristic time scale, generalizing the results of previous studies. We propose a non-autonomous model with two time scales, construct an aggregated model and give relationship between the variables governing the original and the reduced systems. We also explore how the properties of strong and weak ergodicity, regarding the capacity of the system to forget initial conditions, of the original system can be studied in terms of the reduced system.     

Simulating the maintenance of a rare fish morph experiencing negative frequency dependent selection

Empirical work assessing the maintenance of rare genotypes in natural populations is difficult over very long time scales. Skirting this problematic issue is possible with theory and simulations. Major theoretical constructs, including mutation-selection balance and balancing selection, explain the theoretical maintenance of rare genotypes, and the occurrence of multiple, rare genotypes over time. Additionally, numerical simulations are valuable tools for assessing evolving biological systems because they allow for monitoring systems over long time scales, as well as for controlling model parameters, thus contributing to the exploration of system dynamics that cannot be assessed in nature. Here we employed numerical simulations to explore the importance of several biological factors that contribute to the maintenance of a fish color-pattern polymorphism. We present a numerical model of a two-morph fish polymorphism that allowed us to test the sensitivity of the rare morph's persistence and the population's stability to multiple parameters. Our simulations ran over 10,000 years (where one year is approximately one generation) and demonstrated the maintenance of a stable polymorphism with a rare morph which persisted at a frequency of ∼10⁻², which is in-fact the frequency of the rare, mottled black mosquitofish morph in natural populations. This pigmentation polymorphism is stable, independent of changes in population size, but can be destabilized with very high predation when coupled with very low birth rates. Employing models with empirical fitness estimates is a valuable tool for monitoring rare vertebrate morphs in nature, however few studies exist that have accomplished this task. Our approach can be adapted for modeling rare morphs (particularly in additional live-bearing fishes like sailfin mollies) that also harbor rare, pigmentation morphs within large populations.     

Occupancy versus colonization–extinction models for projecting population trends at different spatial scales

Understanding spatiotemporal population trends and their drivers is a key aim in population ecology. We further need to be able to predict how the dynamics and sizes of populations are affected in the long term by changing landscapes and climate. However, predictions of future population trends are sensitive to a range of modeling assumptions. Deadwood‐dependent fungi are an excellent system for testing the performance of different predictive models of sessile species as these species have different rarity and spatial population dynamics, the populations are structured at different spatial scales, and they utilize distinct substrates. We tested how the projected large‐scale occupancies of species with differing landscape‐scale occupancies are affected over the coming century by different modeling assumptions. We compared projections based on occupancy models against colonization–extinction models, conducting the modeling at alternative spatial scales and using fine‐ or coarse‐resolution deadwood data. We also tested effects of key explanatory variables on species occurrence and colonization–extinction dynamics. The hierarchical Bayesian models applied were fitted to an extensive repeated survey of deadwood and fungi at 174 patches. We projected higher occurrence probabilities and more positive trends using the occupancy models compared to the colonization–extinction models, with greater difference for the species with lower occupancy, colonization rate, and colonization:extinction ratio than for the species with higher estimates of these statistics. The magnitude of future increase in occupancy depended strongly on the spatial modeling scale and resource resolution. We encourage using colonization–extinction models over occupancy models, modeling the process at the finest resource‐unit resolution that is utilizable by the species, and conducting projections for the same spatial scale and resource resolution at which the model fitting is conducted. Further, the models applied should include key variables driving the metapopulation dynamics, such as the availability of suitable resource units, habitat quality, and spatial connectivity.     

Spatial heterogeneity in three species,plant-parasite-hyperparasite,systems

This paper addresses the question of how heterogeneity may evolve due to interactions between the dynamics and movement of three-species systems involving hosts, parasites and hyperparasites in homogeneous environments. The models are motivated by the spread of soil-borne parasites within plant populations, where the hyperparasite is used as a biological control agent but where patchiness in the distribution of the parasite occurs, even when environmental conditions are apparently homogeneous. However, the models are introduced in generic form as three-species reaction-diffusion systems so that they have broad applicability to a range of ecological systems. We establish necessary criteria for the occurrence of population-driven patterning via diffusion-driven instability. Sufficient conditions are obtained for restricted cases with no host movement. The criteria are similar to those for the well-documented two-species reaction-diffusion system, although more possibilities arise for spatial patterning with three species. In particular, temporally varying patterns, that may be responsible for the apparent drifting of hot-spots of disease and periodic occurrence of disease at a given location, are possible when three species interact. We propose that the criteria can be used to screen population interactions, to distinguish those that cannot cause patterning from those that may give rise to population-driven patterning. This establishes a basic dynamical ''landscape'' against which other perturbations, including environmentally driven variations, can be analysed and distinguished from population-driven patterns. By applying the theory to a specific model example for host-parasite-hyperparasite interactions both with and without host movement, we show directly how the evolution of spatial pattern is related to biologically meaningful parameters. In particular, we demonstrate that when there is strong density dependence limiting host growth, the pattern is stable over time, whereas with less stable underlying host growth, the pattern varies with time.     

HIV-1 infection and low steady state viral loads

Highly active antiretroviral therapy (HAART) reduces the viral burden in human immunodeficiency virus type 1 (HIV-1) infected patients below the threshold of detectability. However, substantial evidence indicates that viral replication persists in these individuals. In this paper we examine the ability of several biologically motivated models of HIV-1 dynamics to explain sustained low viral loads. At or near drug efficacies that result in steady state viral loads below detectability, most models are extremely sensitive to small changes in drug efficacy. We argue that if these models reflect reality many patients should have cleared the virus, contrary to observation. We find that a model in which the infected cell death rate is dependent on the infected cell density does not suffer this shortcoming. The shortcoming is also overcome in two more conventional models that include small populations of cells in which the drug is less effective than in the main population, suggesting that difficulties with drug penetrance and maintenance of effective intracellular drug concentrations in all cells susceptible to HIV infection may underlie ongoing viral replication.     

When can noise induce chaos and why does it matter: a critique

Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ₀ that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.     

Mass mortality in marine mammals: Its implications for population dynamics and genetics

The death of tens of thousands of common seals (Phoca vitulina) around the coast of Europe in 1988 provoked wide interest in the popular press, but it also raised questions about the importance of mass mortality in the dynamics of marine mammal populations. Here, we summarize published information on the occurrence of mass mortalities among marine mammals and review the mathematical models that have been developed to investigate the role of such disasters in population dynamics. We conclude that mass mortalities may play a more important role than density-dependent factors in the dynamics of some marine mammal populations. This, in combination with recent improvements in our understanding of the structure of these populations, has important implications for their genetics and evolution.     

A replicated climate change field experiment reveals rapid evolutionary response in an ecologically important soil invertebrate

Whether species can respond evolutionarily to current climate change is crucial for the persistence of many species. Yet, very few studies have examined genetic responses to climate change in manipulated experiments carried out in natural field conditions. We examined the evolutionary response to climate change in a common annelid worm using a controlled replicated experiment where climatic conditions were manipulated in a natural setting. Analyzing the transcribed genome of 15 local populations, we found that about 12% of the genetic polymorphisms exhibit differences in allele frequencies associated to changes in soil temperature and soil moisture. This shows an evolutionary response to realistic climate change happening over short‐time scale, and calls for incorporating evolution into models predicting future response of species to climate change. It also shows that designed climate change experiments coupled with genome sequencing offer great potential to test for the occurrence (or lack) of an evolutionary response.