A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

An alternative formulation for a delayed logistic equation

We derive an alternative expression for a delayed logistic equation, assuming that the rate of change of the population depends on three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the model. We provide a complete global analysis, showing that, unlike the dynamics of the classical logistic delay differential equation (DDE) model, no sustained oscillations are possible. Just as for the classical logistic ordinary differential equation (ODE) growth model, all solutions approach a globally asymptotically stable equilibrium. However, unlike both the logistic ODE and DDE growth models, the value of this equilibrium depends on all of the parameters, including the delay, and there is a threshold that determines whether the population survives or dies out. In particular, if the delay is too long, the population dies out. When the population survives, i.e., the attracting equilibrium has a positive value, we explore how this value depends on the parameters. When this value is positive, solutions of our DDE model seem to be well approximated by solutions of the logistic ODE growth model with this carrying capacity and an appropriate choice for the intrinsic growth rate that is independent of the initial conditions.     

Lotka's Game in Predator–Prey Theory: Linking Populations to Individuals

This paper further examines an individual-based model of a spatially distributed predator–prey population that demonstrates strong spatial structuring in contrast with predictions from its representative analytic formulation. Examination of a small, localized population reveals that extinctions due to demographic stochasticity dominate the dynamics. Local extinction dynamics produce wave pulses and the interactions of these wave pulses constitute global dynamics. The results motivate a population-level cell-based model with each cell representing a local population and parameterized by local extinction probabilities, rather than individual-based interaction rates. A detailed comparison of spatiotemporal plots from the two modelling frameworks shows that the population-level model captures the broad range of dynamics exhibited by the individual-based model. The agreement between these two complementary theoretical frameworks, one formulated at the level of individuals, the other at the level of populations, provides a mechanistic understanding of the dynamics.     

Scaling up: how do exogenous fluctuations in individual-based resource competition models re-emerge in aggregated stochastic population models?

In applied population dynamics the choice of stochastic per capita growth function has implications for population viability analyses, management recommendations, and pest control. This model choice is often based on statistical criteria, mathematical tractability or personal preferences, and general ecological guidelines are either too vague or entirely missing. To identify such guidelines, it is important to understand how exogenous and endogenous factors interact at the individual level and re-emerge at the aggregated population level. We therefore study different types of resource competition (contest vs. scramble competition) and different types of exogenous fluctuations (food and weather fluctuations) at the individual level in a simple individual-based simulation model. We statistically fit the resulting time series to find out (1) which functional form of the growth function (‘hyperbolic’ or ‘exponential’) better describes contest and scramble competition and (2) whether the pattern of population fluctuations resulting from the simulations can be assigned to vertical, lateral or nonlinear perturbations in the stochastic growth function (a classification scheme suggested by Royama 1992, Analytical Population Dynamics, Chapman and Hall, London). We found that the same type of competition can result in ‘hyperbolic’ or ‘exponential’ functional forms, depending on the type of exogenous fluctuations. So it is the interplay between exogenous variability and endogenous resource competition that affects model performance. In contrast to the widespread assumption of vertical (additive) perturbations, our findings highlight the importance of (non-additive) lateral and nonlinear perturbations and their combinations with vertical perturbations. The choice of the stochastic growth function should therefore consider not only statistical criteria but also ecological guidelines. We derive such ecological guidelines from our analysis.     

Reconciling classical and individual-based approaches in theoretical population ecology: a protocol for extracting population parameters from individual-based models

The two main approaches in theoretical population ecology-the classical approach using differential equations and the approach using individual-based modeling-seem to be incompatible. Linked to these two approaches are two different timescales: population dynamics and behavior or physiology. Thus, the question of the relationship between classical and individual-based approaches is related to the question of the mutual relationship between processes on the population and the behavioral timescales. We present a simple protocol that allows the two different approaches to be reconciled by making explicit use of the fact that processes operating on two different timescales can be treated separately. Using an individual-based model of nomadic birds as an example, we extract the population growth rate by deactivating all demographic processes-in other words, the individuals behave but do not age, die, or reproduce. The growth rate closely matches the logistic growth rate for a wide range of parameters. The implications of this result and the conditions for applying the protocol to other individual-based models are discussed. Since in physics the technique of separating timescales is linked to some concepts of self-organization, we believe that the protocol will also help to develop concepts of self-organization in ecology.     

Individual variability and population regulation: a model of the significance of within-generation density dependence

Most models of theoretical population ecology consider population density as a state variable and thus ignore the fact that populations are composed not of identical average individuals but of individuals which are usually different. However, this individual variability may be important for population regulation. We therefore analysed an individual-based population model which explicitly describes within-generation processes, i.e. individual growth, starvation, and resource dynamics. The results show that if population dynamics are dominated by slow changes in resource level, the population size in the model undergoes wide oscillation, often leading to extinction. If, on the other hand, fast within-generation processes predominate, such as starvation and sudden drops in resource levels, the population fluctuates to a limited extent around an average. Within-generation density dependence may thus be an important mechanism which is largely ignored in classic time-discrete state-variable models. We conclude that the individual-based approach provides important insights into the hierarchical organization of population dynamics, i.e. the relationship between fast processes at the individual level and slower processes at the population level.     

Validation of an individual-based model of forest dynamics using self-thinning relationships

Self-thinning relationships link mean plant size and plant density for even-aged populations subjected to density-dependent mortality. Because the relationships are expressed at the population or the community level, they constitute a validation test for individual-based models of plant population dynamics. The model proposed here stems from forest gap models of the JABOWA/FORET-type and succeeds the validation test. This validates the growth and mortality individual-based local rules used in most gap models. The result arises in the model because some basic assumptions make density a negative exponential function of time and mean individual size a sigmoid function of time.     

Coexistence of cycling and dispersing consumer species: Armstrong and McGehee in space

Two competing consumer species may coexist using a single homogeneous resource when the more efficient consumer--the one having the lowest equilibrium resource density--has a more nonlinear functional response that generates consumer-resource cycles. We extend this model of nonequilibrium coexistence, as proposed by Armstrong and McGehee, by putting the interaction into a spatial context using two frameworks: a spatially explicit individual-based model and a spatially implicit metapopulation model. We find that Armstrong and McGehee's mechanism of coexistence can operate in a spatial context. However, individual-based simulations suggest that decreased dispersal restricts coexistence in most cases, whereas differential equation models of metapopulations suggest that a low rate of dispersal between subpopulations often increases the coexistence region. This difference arises in part because of two potentially opposing effects on coexistence due to the asynchrony in the temporal dynamics at different locations. Asynchrony implies that the less efficient species is more likely to be favored in some spatial locations at any given time, which broadens the conditions for coexistence. On the other hand, asynchrony and dispersal can also reduce the amplitude of local population cycles, which restricts coexistence. The relative influence of these two effects depends on details of the population dynamics and the representation of space. Our results also demonstrate that coexistence via the Armstrong-McGehee mechanism can occur even when there is little variation in the global densities of either the consumers or the resource, suggesting that empirical studies of the mechanisms should measure densities on several spatial scales.     

On the probability model for asthma attacks

In environmental epidemiology, the impact of environmental agents on symptoms or health status is of interest. This influence is described quantitatively in the theory of Whittemore & Keller (1979). They formulated a logistic model for individuals that is useful in evaluation of panel studies in which each participant protocols whether he does or does not have a certain symptom each day. In the present paper an equation for the prevalence of symptoms in the study population that is defined as the fraction of symptomatic subjects is deduced from the model for individuals. The model for the aggregated quantity depends on the individuals' parameters in a nonlinear manner. The relationship between the individual-based model and the corresponding population-based model is illustrated by means of a simulated panel. Bayesian estimates of the parameters are calculated and compared for both approaches. Bayesian inference enables to apply the prevalence model to a population of non-identical individuals. For such a heterogeneous population, we observe an attenuation of environmental effects on the aggregated symptom prevalence in comparison to the individual-based approach. The presented theory is applicable not only to panel studies but also in time-series analysis of prevalences and incidences.     

Small-scale processes in desert locust swarm formation: how vegetation patterns influence gregarization

Desert locusts ( Schistocerca gregaria ) change phase in response to population density: 'solitarious' insects avoid one another, but when crowded they shift to the gregarious phase and aggregate. This individual-level process is the basis for population-level responses that may ultimately include swarm formation. We have recently developed an individual-based model of locust behavior in which contagious resource distribution leads to phase change. This model shows how population gregarization can result from simple processes operating at the individual level. In the present study, we performed a series of laboratory experiments in which vegetation pattern and locust phase state were assigned quantitative, measurable indices. The pattern of distribution of the resource was represented via fractal dimension; the phase state was evaluated using a behavioral assay based on logistic regression analysis. Locusts were exposed to different patterns of food resource in an artificial arena, after which their behavioral phase state was assayed. These experiments showed that when the distribution of the vegetation was patchy, locusts were more active, experienced higher levels of crowding, and became more gregarious. These results are consistent with simulation predictions and field observations, and demonstrate that small-scale vegetation distribution influences individual behavior and phase state and plays a role in population-level responses.     

A rigorous approach to investigating common assumptions about disease transmission

Changing scale, for example, the ability to move seamlessly from an individual-based model to a population-based model, is an important problem in many fields. In this paper, we introduce process algebra as a novel solution to this problem in the context of models of infectious disease spread. Process algebra allows us to describe a system in terms of the stochastic behaviour of individuals, and is a technique from computer science. We review the use of process algebra in biological systems, and the variety of quantitative and qualitative analysis techniques available. The analysis illustrated here solves the changing scale problem: from the individual behaviour we can rigorously derive equations to describe the mean behaviour of the system at the level of the population. The biological problem investigated is the transmission of infection, and how this relates to individual interactions.     

Testing,tracing and isolation in compartmental models

Existing compartmental mathematical modelling methods for epidemics, such as SEIR models, cannot accurately represent effects of contact tracing. This makes them inappropriate for evaluating testing and contact tracing strategies to contain an outbreak. An alternative used in practice is the application of agent- or individual-based models (ABM). However ABMs are complex, less well-understood and much more computationally expensive. This paper presents a new method for accurately including the effects of Testing, contact-Tracing and Isolation (TTI) strategies in standard compartmental models. We derive our method using a careful probabilistic argument to show how contact tracing at the individual level is reflected in aggregate on the population level. We show that the resultant SEIR-TTI model accurately approximates the behaviour of a mechanistic agent-based model at far less computational cost. The computational efficiency is such that it can be easily and cheaply used for exploratory modelling to quantify the required levels of testing and tracing, alone and with other interventions, to assist adaptive planning for managing disease outbreaks.     

Population ecology, nonlinear dynamics, and social evolution. I. Associations among nonrelatives

Using an individual-based and genetically explicit simulation model, we explore the evolution of sociality within a population-ecology and nonlinear-dynamics framework. Assuming that individual fitness is a unimodal function of group size and that cooperation may carry a relative fitness cost, we consider the evolution of one-generation breeding associations among nonrelatives. We explore how parameters such as the intrinsic rate of growth and group and global carrying capacities may influence social evolution and how social evolution may, in turn, influence and be influenced by emerging group-level and population-wide dynamics. We find that group living and cooperation evolve under a wide range of parameter values, even when cooperation is costly and the interactions can be defined as altruistic. Greater levels of cooperation, however, did evolve when cooperation carried a low or no relative fitness cost. Larger group carrying capacities allowed the evolution of larger groups but also resulted in lower cooperative tendencies. When the intrinsic rate of growth was not too small and control of the global population size was density dependent, the evolution of large cooperative tendencies resulted in dynamically unstable groups and populations. These results are consistent with the existence and typical group sizes of organisms ranging from the pleometrotic ants to the colonial birds and the global population outbreaks and crashes characteristic of organisms such as the migratory locusts and the tree-killing bark beetles.     

The role of mobility for the emergence of diversity in victim-exploiter systems

Theoretical and empirical studies indicate that exploitation is a possible driver of exploiter and victim diversification. However, there are many factors which could promote and limit this diversification process. Using a spatially explicit individual-based model, where an exploiter's success depends on matching between its own and a victim's continuous trait, we simulate local communities of victims and exploiters. We investigate how exploiter mobility (searching ability and movement strategies) can influence diversification of victims. We find that if victim traits are under intermediate intensity of stabilizing selection, disruptive selection exerted by exploiters can indeed lead to diversification in victim population and the victim trait distribution can split into two or more groups. Searching ability and movement strategy of exploiters (local vs. global movement) play a role in determining the number of victim trait groups emerging. Moreover, they affect the proportion of infected victims and the formation of spatial patterns in the victim trait distribution. In addition, with a high searching ability, exploiters with global movement drive victims to be more diverse than exploiters with local movement.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Origin and number of founders in an introduced insular primate: estimation from nuclear genetic data

Cynomolgus macaques (Macaca fascicularis) were introduced on the island of Mauritius between 400 and 500 years ago and underwent a strong population expansion after a probable initial founding event. However, in practice, little is known of the geographical origin of the individuals that colonized the island, on how many individuals were introduced, and of whether the following demographic expansion erased any signal of this putative bottleneck. In this study, we asked whether the current nuclear genome of the Mauritius population retained a signature that would allow us to answer these questions. Altogether, 21 polymorphic autosomal and sex-linked microsatellites were surveyed from 81 unrelated Mauritius individuals and 173 individuals from putative geographical sources in Southeast Asia: Java, the Philippines islands and the Indochinese peninsula. We found that (i) the Mauritius population was closer to different populations depending on the markers we used, which suggests a possible mixed origin with Java playing most probably a major role; and (ii) the level of diversity was lower than the other populations but there was no clear and consistent bottleneck signal using either summary statistics or full-likelihood methods. However, summary statistics strongly suggest that Mauritius is not at mutation-drift equilibrium and favours an expansion rather than a bottleneck. This suggests that on a short time scale, population decline followed by growth can be difficult to deduce from genetic data based on mutation-drift theory. We then used a simple Bayesian rejection algorithm to estimate the number of founders under different demographic models (exponential, logistic and logistic with lag) and pure genetic drift. This new method uses current population size estimates and expected heterozygosity of Mauritius and source population(s). Our results indicate that a simple exponential growth is unlikely and that, under the logistic models, the population may have expanded from an initial effective number of individuals of 10-15. The data are also consistent with a logistic growth with different lag values, indicating that we cannot exclude past population fluctuation.     

Relationships between migration rates and landscape resistance assessed using individual-based simulations

Linking landscape effects on gene flow to processes such as dispersal and mating is essential to provide a conceptual foundation for landscape genetics. It is particularly important to determine how classical population genetic models relate to recent individual-based landscape genetic models when assessing individual movement and its influence on population genetic structure. We used classical Wright-Fisher models and spatially explicit, individual-based, landscape genetic models to simulate gene flow via dispersal and mating in a series of landscapes representing two patches of habitat separated by a barrier. We developed a mathematical formula that predicts the relationship between barrier strength (i.e., permeability) and the migration rate (m) across the barrier, thereby linking spatially explicit landscape genetics to classical population genetics theory. We then assessed the reliability of the function by obtaining population genetics parameters (m, F(ST) ) using simulations for both spatially explicit and Wright-Fisher simulation models for a range of gene flow rates. Next, we show that relaxing some of the assumptions of the Wright-Fisher model can substantially change population substructure (i.e., F(ST) ). For example, isolation by distance among individuals on each side of a barrier maintains an F(ST) of ～0.20 regardless of migration rate across the barrier, whereas panmixia on each side of the barrier results in an F(ST) that changes with m as predicted by classical population genetics theory. We suggest that individual-based, spatially explicit modelling provides a general framework to investigate how interactions between movement and landscape resistance drive population genetic patterns and connectivity across complex landscapes.     

Evolutionary predictions should be based on individual-level traits

Recent theoretical studies have analyzed the evolution of habitat specialization using either the logistic or the Ricker equation. These studies have implemented evolutionary change directly in population-level parameters such as habitat-specific intrinsic growth rates r or carrying capacities K. This approach is a shortcut to a more detailed analysis where evolutionary change is studied in underlying morphological, physiological, or behavioral traits at the level of the individual that contribute to r or K. Here we describe two pitfalls that can occur when such a shortcut is employed. First, population-level parameters that appear as independent variables in a population dynamical model might not be independent when derived from processes at the individual level. Second, patterns of covariation between individual-level traits are usually not conserved when mapped to the level of demographic parameters. Nonlinear mappings constrain the curvature of trade-offs that can sensibly be assumed at the population level. To illustrate these results, we derive a two-habitat version of the logistic and Ricker equations from individual-level processes and compare the evolutionary dynamics of habitat-specific carrying capacities with those of underlying individual-level traits contributing to the carrying capacities. Finally, we sketch how our viewpoint affects the results of earlier studies.