Environmental variability and fisheries: what can models do?

This review is based on 58 climate-fisheries models published over the last 28 years that describe the impacts of fishery pressure and environmental variability on populations and ecosystems and include basic principles of population dynamics. It points out that the incorporation of environmental factors in fishery models has already been done and is of great importance for future models used in the assessment of marine resources. The work is guided by the questions to what extent have these models a) enhanced our understanding of the interrelationships between the environment, the fishery and the state of the exploited resources and b) helped to improve the prediction of population dynamics and the assessment of marine resources. For each of the six most commonly used model categories a case study is critically analyzed. The problems of “breaking relationships” between environmental factors and the biological response used in models, the trade-off between model complexity (realism) and simplicity (data availability) and the potential of multivariate climate indices for forecasting ecosystem states and for use as proxies for combined models are discussed, as are novel non-linear and spatially explicit modeling approaches. Approaches differ in terms of model complexity, use of linear or non-linear equations, number of parameters, forecast time horizon and type of resource modelled. A majority of the models were constructed for fish and invertebrate stocks of the northeast Pacific and the epicontinental seas of the Atlantic, reflecting the advancement of fisheries science in these regions. New, in parts highly complex models and sophisticated approaches were identified. The reviewed studies demonstrate that the performance of fished stocks can better be described if environmental or climatic variability is incorporated into the fisheries models. We conclude that due to the already available knowledge, the greatly enhanced computer power, new methods and recent findings of large-scale climatic/oceanographic cycles, fisheries modeling should progress greatly in the coming years.     

Alternative stable states in host–phage dynamics

Bacteriophage are ubiquitous in nature, yet many central aspects of host–phage biology have not been integrated into mathematical models. We propose a novel model of host–phage population dynamics that accounts for the decreased ability of phages to lyse hosts as hosts approach their carrying capacity. In contrast to existing predator–prey-like models, we find a parameter regime in which phages cannot invade a host-only system but, nonetheless, can stably coexist with hosts at lower densities. The finding of alternative stable states suggests clear linkages with observed life history strategies of phages. In addition, we solve a limiting case of the proposed model and show that conservative predator-prey like systems do not inevitably exhibit population cycles. Finally, we discuss possible extensions of the present model and scenarios for experimental testing.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Population size and survival of the Brazilian Torrent Frog Hylodes heyeri (Anura,Hylodidae)

Population dynamics are influenced by environmental variability and understanding the abundance and persistence of individuals and populations is a fundamental goal of population ecology. Thus, estimating demographic parameters to identify the factors important for population variability is required to understand temporal and spatial dynamics. The stream-living diurnal frog Hylodes heyeri is endemic to the Atlantic Forest of Brazil in the states of Paraná, São Paulo and Santa Catarina. Here we use capture-mark-recapture methods to estimate survival rates and population size of this Brazilian Torrent Frog in Pico do Marumbi State Park, Paraná. We used CJS models for an open population to estimate apparent survival, capturability and population size in two streams. The number of captures during each session was positively correlated with the minimum weekly temperature. Despite that correlation, the most parsimonious model of survival and capturability was the constant model for both parameters, resulting in a monthly survival rate of 0.38 (95% CI = 0.30–0.46). Thus, only the abundance of the frog differed in the two streams (79 vs. 36), with the population size estimate of 187 individuals. Reproduction is seasonal in the Brazilian Torrent Frog and so the low monthly survival rate suggests that animals move over time rather than die, because 38% month⁻¹ survival should result in <1% of the population remaining after 5 months. Thus, researchers must recognize that populations are labile and individuals often move or are washed downstream during heavy rainfall, generating apparently rapid local turnover that is unlikely to reflect true mortality.     

Host–parasitoid spatial models: the interplay of demographic stochasticity and dynamics

Host–parasitoid metapopulation models have typically been deterministic models formulated with population numbers as a continuous variable. Spatial heterogeneity in local population abundance is a typical (and often essential) feature of these models and means that, even when average population density is high, some patches have small population sizes. In addition, large temporal population fluctuations are characteristic of many of these models, and this also results in periodically small local population sizes. Whenever population abundances are small, demographic stochasticity can become important in several ways. To investigate this problem, we have reformulated a deterministic, host–parasitoid metapopulation as an integer-based model in which encounters between hosts and parasitoids, and the fecundity of individuals are modelled as stochastic processes. This has a number of important consequences: (1) stochastic fluctuations at small population sizes tend to be amplified by the dynamics to cause massive population variability, i.e. the demographic stochasticity has a destabilizing effect; (2) the spatial patterns of local abundance observed in the deterministic counterpart are largely maintained (although the area of ''spatial chaos'' is extended); (3) at small population sizes, dispersal by discrete individuals leads to a smaller fraction of new patches being colonized, so that parasitoids with small dispersal rates have a greater tendency for extinction and higher dispersal rates have a larger competitive advantage; and (4) competing parasitoids that could coexist in the deterministic model due to spatial segregation cannot now coexist for any combination of parameters.     

Structured population dynamics: continuous size and discontinuous stage structures

A nonlinear stochastic model for the dynamics of a population with either a continuous size structure or a discontinuous stage structure is formulated in the Eulerian formalism. It takes into account dispersion effects due to stochastic variability of the development process of the individuals. The discrete equations of the numerical approximation are derived, and an analysis of the existence and stability of the equilibrium states is performed. An application to a copepod population is illustrated; numerical results of Eulerian and Lagrangian models are compared.      

Correlated extinctions, colonizations and population fluctuations in a highly connected ringlet butterfly metapopulation

 The persistence of metapopulations is likely to be highly dependent on whether population dynamics are correlated among habitat patches as a result of migration between patches and spatially-correlated environmental stochasticity (weather effects). We examined whether population dynamics of the ringlet butterfly, Aphantopus hyperantus, were synchronous in an area of approximately 0.5 km², with respect to extinction, colonization and population fluctuations. Monks Wood Butterfly Monitoring Scheme transect count data from 1973 to 1995, revealed (A) a major environmental perturbation, the drought of 1976, which caused synchronized extinctions of A. hyperantus in subsequent years, (B) synchronized recolonization in years following the large number of apparent extinctions, and (C) population changes by A. hyperantus were highly correlated in many of the 14 sections of the transect, presumably reflecting similar responses to environmental stochasticity, and the exchange of individuals among sections. However, extinction and population synchrony depended on habitat type. Following the 1976 drought, A. hyperantus apparently became extinct from the most open and most shady habitats it occupied, with some persistence in habitats of intermediate shading, thus showing retraction to core populations in central parts of an environmental gradient, albeit with an average shift to relatively open habitat. Populations at extreme ends of the environmental gradient occupied by A. hyperantus fluctuated least synchronously, suggesting a potential buffering effect of habitat heterogeneity, but this was not crucial to survival after the 1976 drought. Thus, not all habitats are equally important to persistence. Correlated temporal dynamics, variation in habitat quality and the interaction between habitat quality and temporal environmental stochasticity are important determinants of metapopulation persistence and should be incorporated in metapopulation models. Received: 26 April 1996 / Accepted: 17 July 1996     

A two-sex demographic model with single-dependent divorce rate

Divorce appears to be one of the least studied demographic processes, both empirically and in two-sex demographic models. In this paper, we study mathematical as well as biological implications of the assumption that the divorce rate is positively affected by the amount of single (i.e., unmarried/unpaired) individuals in the population. We do that by modifying the classical exponential two-sex model accounting for pair formation and separation. We model the divorce rate as an increasing function of the single population size and show that the single population pressure on the established couples alters the exponential behavior of the classical model in which the divorce rate is assumed constant. In particular, the total population size becomes bounded and a unique positive equilibrium exists. In addition, a Hopf bifurcation analysis around the positive equilibrium shows that the modified model may exhibit sustained oscillations.     

Intraguild predation: fiction or reality?

Intraguild predation has become a major research topic in biological control. Quantification of multipredator interactions and an understanding of the consequences on target prey populations are needed, which only highlights the importance of population dynamics models in this field. However, intraguild predation models are usually based on Lotka–Volterra equations, which have been shown not to be adequate for modeling population dynamics of aphidophagous insects and their prey. Here we use a simple model developed for simulation of population dynamics of aphidophagous insects, which is based on the type of egg distribution made by predatory females, to estimate the real strength of intraguild predation in the aphidophagous insects. The model consists of two components: random egg distribution among aphid colonies, and between-season population dynamics of the predatory species. The model is used to estimate the proportion of predatory individuals that face a conflict with a heterospecific competitor at least once during their life. Based on this, predictions are made on the population dynamics of both predatory species. The predictions are confronted with our data on intraguild predation in ladybirds.     

The evolution of intermittent breeding

A central issue in life history theory is how organisms trade off current and future reproduction. A variety of organisms exhibit intermittent breeding, meaning sexually mature adults will skip breeding opportunities between reproduction attempts. It’s thought that intermittent breeding occurs when reproduction incurs an extra cost in terms of survival, energy, or recovery time. We have developed a matrix population model for intermittent breeding, and use adaptive dynamics to determine under what conditions individuals should breed at every opportunity, and under what conditions they should skip some breeding opportunities (and if so, how many). We also examine the effect of environmental stochasticity on breeding behavior. We find that the evolutionarily stable strategy (ESS) for breeding behavior depends on an individual’s expected growth and mortality, and that the conditions for skipped breeding depend on the type of reproductive cost incurred (survival, energy, recovery time). In constant environments there is always a pure ESS, however environmental stochasticity and deterministic population fluctuations can both select for a mixed ESS. Finally, we compare our model results to patterns of intermittent breeding in species from a range of taxonomic groups.     

Stochastic analogues of deterministic single-species population models

Although single-species deterministic difference equations have long been used in modeling the dynamics of animal populations, little attention has been paid to how stochasticity should be incorporated into these models. By deriving stochastic analogues to difference equations from first principles, we show that the form of these models depends on whether noise in the population process is demographic or environmental. When noise is demographic, we argue that variance around the expectation is proportional to the expectation. When noise is environmental the variance depends in a non-trivial way on how variation enters into model parameters, but we argue that if the environment affects the population multiplicatively then variance is proportional to the square of the expectation. We compare various stochastic analogues of the Ricker map model by fitting them, using maximum likelihood estimation, to data generated from an individual-based model and the weevil data of Utida. Our demographic models are significantly better than our environmental models at fitting noise generated by population processes where noise is mainly demographic. However, the traditionally chosen stochastic analogues to deterministic models--additive normally distributed noise and multiplicative lognormally distributed noise--generally fit all data sets well. Thus, the form of the variance does play a role in the fitting of models to ecological time series, but may not be important in practice as first supposed.     

Neighbourhood size,dispersal distance and the complex dynamics of the spatial Ricker model

Amongst the most frequently made assumptions in simple population models are that individuals interact equally with every other individual and that dispersal occurs with equal likelihood to any location. This is especially true for models of a single population (as opposed to a patchy population or metapopulation). For many species of animals and probably for all plant species these assumptions are unlikely to hold true. Here one much-studied population model—the Ricker model—is reformulated such that interactions occur only between individuals located within a certain distance of each other and dispersal distance is finite. Two alternative reformulations are presented. Results demonstrate that both limiting the interaction neighbourhood and reducing dispersal distance tend to stabilise the global population dynamics, although the extent to which this occurs depends upon the reformulation used. Spatial pattern formation is a feature of the simulated population. At lower intrinsic rates of growth (r) these patterns tend to be static, while for higher r, they are dynamic. Both the stabilisation of global dynamics and spatial pattern formation are well-described features of metapopulation models. Here, similar effects are shown to occur on a single contiguous patch of habitat.     

Synergistic effects of food shortage and an insecticide on a <Emphasis Type="Italic">Daphnia</Emphasis> population: rapid decline of food density at the peak of population density reduces tolerance to the chemical and induces a large population crash

Laboratory populations of cloned Daphnia magna were exposed at different population phases (growing phase, density peak, stable phase) to the insecticide carbaryl at 15 μg 1⁻¹, which was harmful to juveniles but not to adults, and their population dynamics were analyzed. The population declined most at the density peak, when not only juveniles but also many adult individuals died. To analyze the factors affecting population vulnerability to carbaryl, acute toxicity tests were conducted using Daphnia individuals of different body sizes under different food conditions. The test revealed that daphnid sensitivity to carbaryl increased greatly when food density was changed from a high food level to a low level. This food condition, of low availability, might be the condition to which the Daphnia populations were exposed at their density peak. The synergism of the negative impacts of anthropogenic and natural stresses such as insecticides and food shortage may control aquatic populations.     

A model for the control of malaria using genetically modified vectors

Recent works have considered the problem of using transgenic mosquitoes to control a malaria epidemic. These insects have been genetically engineered to reduce their capacity to infect humans with malaria parasites. We analyze a model of the mosquito population dynamics when genetically modified individuals are introduced into a wild type population so that the effect of their introduction can be assessed. The model describes the dynamics of gene selection under sexual reproduction in a closed vector population. Our results show that the fitness of the resulting heterozygous population is the key parameter for the success of the invasion, independently of the fitness of homozygous vectors. The vector population dynamics model is then combined with an epidemiological model to study the feasibility of controlling a malaria epidemic. Basic reproductive numbers are calculated for both models, and conditions are obtained for preventing reappearance of the epidemic. Simulations on this model show that it may be possible to reduce or even eradicate the epidemic only if the heterozygous population is better adapted than the wild type. They also show that this can be achieved without completely eliminating the wild type mosquitoes.     

Density-dependent cooperation as a mechanism for persistence and coexistence

To overcome stress, such as resource limitation, an organism often needs to successfully mediate competition with other members of its own species. This may favor the evolution of defective traits that are harmful to the species population as a whole, and that may lead to its dilution or even to its extinction (the tragedy of the commons). Here, we show that this phenomenon can be circumvented by cooperation plasticity, in which an individual decides, based on environmental conditions, whether to cooperate or to defect. Specifically, we analyze the evolution of density-dependent cooperation. In our model, the population is spatially subdivided, periodically remixed, and comprises several species. We find that evolution pushes individuals to be more cooperative when their own species is at lower densities, and we show that not only could this cooperation prevent the tragedy of the commons, but it could also facilitate coexistence between many species that compete for the same resource.     

ECOLOGICAL CONSTRAINTS INFLUENCE THE EMERGENCE OF COOPERATIVE BREEDING WHEN POPULATION DYNAMICS DETERMINE THE FITNESS OF HELPERS

Cooperative breeding is a system in which certain individuals facilitate the production of offspring by others. The ecological constraints hypothesis states that ecological conditions deter individuals from breeding independently, and so individuals breed cooperatively to make the best of a bad situation. Current theoretical support for the ecological constraints hypothesis is lacking. We formulate a mathematical model that emphasizes the underlying ecology of cooperative breeders. Our goal is to derive theoretical support for the ecological constraints hypothesis using an ecological model of population dynamics. We consider a population composed of two kinds of individuals, nonbreeders (auxiliaries) and breeders. We suppose that help provided by an auxiliary increases breeder fecundity, but reduces the probability with which the auxiliary becomes a breeder. Our main result is a condition that guarantees success of auxiliary help. We predict that increasing the cost of dispersal promotes helping, in agreement with verbal theory. We also predict that increasing breeder mortality can either hinder helping (at high population densities), or promote it (at low population densities). We conclude that ecological constraints can exert influence over the evolution of auxiliary help when population dynamics are considered; moreover, that influence need not coincide with direct fitness benefits as previously found.     

Genetic differentiation in populations of the yellow-necked mouse,Apodemus flavicollis,harbouring B chromosomes in different frequencies

Two alternative models are used to explain maintenance of polymorphism of B chromosomes (Bs) in populations of a great number of species. The parasitic model suggests deleterious effects of Bs on fitness of carriers, while the heterotic model assumes that, in the absence of drive, equilibrium is produced by beneficial effects of Bs at low numbers. In order to determine the potential contribution of Bs to genetic differentiation and diversity, four populations of Apodemus flavicollis, differing in frequency of Bs (from 0.23 to 0.38) and settled in ecologically different habitats, were analyzed by 471 AFLP markers. Although numerous loci were demonstrated to be population specific, none of them was associated with individuals with Bs. AMOVA showed that the presence of Bs does not affect population differentiation, pointing to greater genetic similarity of Bs to A chromosomes. The greatest genetic diversity (0.241) was found in the population settled in optimal conditions for this species featured by the lowest frequency of animals with Bs (0.23). We found that the majority of loci marked as loci under directional selection, are characteristic of either a population with lower or one with a higher frequency of Bs. Several loci detected as outliers were associated with environmental variables that could directly and/or indirectly influence population dynamics of A. flavicollis. Thus, we suggest that the different frequency of Bs carriers in populations is related to adaptive differentiation to diverse habitats, which is in accordance with the heterotic model of Bs maintenance.     

Patchy patterns due to group dispersal

The presence of multiple foci in population patterns may be due to various processes arising in the population dynamics. Group dispersal, which has been lightly investigated for airborne species, is one of these processes.We built a stochastic model generating the dispersal of groups of particles. This model may be viewed as an extension of classical dispersal models based on parametric kernels. It has a hierarchical structure: at the first stage group centers are drawn under a classical dispersal kernel; at the second stage the particles are diffused around their group centers. Analytic and simulation results show that group dispersal is a sufficient condition to generate patterns with multiple foci, i.e. patchy patterns, even if the population can remain particularly concentrated.     

Establishment success and extinction risk in autocorrelated environments

We consider establishment success (and extinction risk of small populations) in fluctuating environments, by means of an inhomogeneous branching process model. In this model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period individuals reproduce independently and have random numbers of offspring. Expected numbers of offspring vary over reproduction periods due to random environmental changes. Previous simulation results indicated that there is a positive autocorrelation between the establishment probabilities of invaders in successive reproduction periods when environmental states are independently distributed. This result was never formally proved. In this paper we prove that this is indeed true, regardless of the form of the distribution of environmental states or the offspring distribution (under a monotonicity condition, which holds for biologically realistic models). Furthermore, we prove that it is also true for positively autocorrelated environmental states. We show by a counterexample that in environments with a strong negative autocorrelation establishment probabilities can be negatively autocorrelated. This was further examined through simulations. Our results imply that in independent, positively autocorrelated and weakly negatively autocorrelated environments the probability of success of invasion in different independently varying sites is the highest, followed by sequential invasion. For environments with a strong negative autocorrelation, sequential invasion has the highest probability of success. Effects of autocorrelation were further examined with simulations. From the results it appears that the expected length of 'runs of bad luck' is the most crucial factor for establishment success.