Theoretical Population Biology Research Articles

Using gridCoal to assess whether standard population genetic theory holds in the presence of spatio-temporal heterogeneity in population size.

Spatially explicit population genetic models have long been developed, yet have rarely been used to test hypotheses about the spatial distribution of genetic diversity or the genetic divergence between populations. Here, we use spatially explicit coalescence simulations to explore the properties of the island and the two-dimensional stepping stone models under a wide range of scenarios with spatio-temporal variation in deme size. We avoid the simulation of genetic data, using the fact that under the studied models, summary statistics of genetic diversity and divergence can be approximated from coalescence times. We perform the simulations using gridCoal, a flexible spatial wrapper for the software msprime (Kelleher et al., 2016, Theoretical Population Biology, 95, 13) developed herein. In gridCoal, deme sizes can change arbitrarily across space and time, as well as migration rates between individual demes. We identify different factors that can cause a deviation from theoretical expectations, such as the simulation time in comparison to the effective deme size and the spatio-temporal autocorrelation across the grid. Our results highlight that FST , a measure of the strength of population structure, principally depends on recent demography, which makes it robust to temporal variation in deme size. In contrast, the amount of genetic diversity is dependent on the distant past when Ne is large, therefore longer run times are needed to estimate Ne than FST . Finally, we illustrate the use of gridCoal on a real-world example, the range expansion of silver fir (Abies alba Mill.) since the last glacial maximum, using different degrees of spatio-temporal variation in deme size.

Learning enables adaptation in cooperation for multi-player stochastic games.

Interactions among individuals in natural populations often occur in a dynamically changing environment. Understanding the role of environmental variation in population dynamics has long been a central topic in theoretical ecology and population biology. However, the key question of how individuals, in the middle of challenging social dilemmas (e.g. the 'tragedy of the commons'), modulate their behaviours to adapt to the fluctuation of the environment has not yet been addressed satisfactorily. Using evolutionary game theory, we develop a framework of stochastic games that incorporates the adaptive mechanism of reinforcement learning to investigate whether cooperative behaviours can evolve in the ever-changing group interaction environment. When the action choices of players are just slightly influenced by past reinforcements, we construct an analytical condition to determine whether cooperation can be favoured over defection. Intuitively, this condition reveals why and how the environment can mediate cooperative dilemmas. Under our model architecture, we also compare this learning mechanism with two non-learning decision rules, and we find that learning significantly improves the propensity for cooperation in weak social dilemmas, and, in sharp contrast, hinders cooperation in strong social dilemmas. Our results suggest that in complex social-ecological dilemmas, learning enables the adaptation of individuals to varying environments.

Rich Bifurcation Structure of Prey–Predator Model Induced by the Allee Effect in the Growth of Generalist Predator

Prey–predator models are building blocks for many food-chain and food-web models in theoretical population biology. These models can be divided into two groups depending on the nature of predators, namely, specialist predator and generalist predator. Generalist predators can survive in the absence of prey but specialist predators go to extinction. Prey–predator models with specialist predator and Allee effect in prey growth have been investigated by several researchers and various types of interesting dynamics have been reported. In this paper, we consider a prey–predator model with generalist predator subject to Allee effect in predator’s growth rate. In general, a prey–predator system with saturating functional response can be destabilized due to the increase of the carrying capacity of prey which is known as paradox of enrichment. In our model with Allee effect in predator growth, we have shown that increase in carrying capacity of prey helps the populations to survive in a coexistence steady state. The considered model is capable of producing bistable dynamics for a reasonable range of parameter values. The complete dynamics of the system are quite rich and all possible local and global bifurcations are studied to understand the dynamics of the model. Analytical results are verified with numerical examples and successive bifurcations are identified with the help of bifurcation diagrams.

When do factors promoting genetic diversity also promote population persistence? A demographic perspective on Gillespie’s SAS-CFF model

Classical stochastic demography predicts that environmental stochasticity reduces population growth rates and, thereby, can increase extinction risk. In contrast, in a 1978 Theoretical Population Biology paper, Gillespie demonstrated with his stochastic additive scale and concave fitness function (SAS-CFF) model that environmental stochasticity can promote genetic diversity. Extending the SAS-CFF to account for demography, I examine the simultaneous effects of environmental stochasticity on genetic diversity and population persistence. Explicit expressions for the per-capita growth rates of rare alleles and the population at low-density are derived. Consistent with Gillespie’s analysis, if the log-fitness function is concave and allelic responses to the environment are not perfectly correlated, then per-capita growth rates of rare alleles are positive and genetic diversity is maintained in the sense of stochastic persistence i.e. allelic frequencies tend to stay away from zero almost-surely and in probability. Alternatively, if the log-fitness function is convex, then per-capita growth rates of rare alleles are negative and an allele asymptotically fixates with probability one. If the population’s low-density, per-capita growth rate is positive, then the population persists in the sense of stochastic persistence, else it goes asymptotically extinct with probability one. In contrast to per-capita growth rates of rare alleles, the population’s per-capita growth rate is a decreasing function of the concavity of the log-fitness function. Moreover, when the log-fitness function is concave, allelic diversity increases the population’s per-capita growth rate while decreasing the per-capita growth rate of rare alleles; when the log-fitness function is convex, environmental stochasticity decreases the per-capita growth rate of rare alleles, but increases the population’s per-capita growth rate. Collectively, these results (i) highlight how mechanisms promoting population persistence may be at odds with mechanisms promoting genetic diversity, and (ii) provide conditions under which population persistence relies on existing standing genetic variation.

Foraging behavior of pigeons (Columba livia) in situations of diminishing returns using a reinforcement schedule that controls the energy expenditure of responses

Animal foraging behavior has received much attention in the fields of biology and psychology. Many studies have examined this behavior using optimal foraging theory as the theoretical framework, which explains that animals behave in a way that optimizes net energy intake (calculated by dividing the difference in energy obtained from prey and energy expended for foraging by the total time engaging in these behaviors). This study simulated the foraging situation assumed in Charnov’s (Theoretical Population Biology, 9, 129–136, 1976) marginal value theorem—namely, where food is located in patches separated by areas without food. To quantitatively analyze the behavior of homing pigeons (Columba livia) in terms of the marginal value theorem, we conducted a series of experiments using a reinforcement schedule that controls the energy of responses (Kono, Learning & Behavior, 41,425–432, 2013). Experiment 1 revealed that pigeons’ behavior was qualitatively consistent with the prediction of the marginal value theorem, but the energy schedule weakly controlled the energy expenditure of behavior. In Experiment 2, the procedure was modified to more strictly control energy expenditure. The results showed that the pigeons’ behavior was relatively close to that of the predicted value of the marginal value theorem. Overall, then, pigeons behaved optimally in foraging as predicted by the marginal value theorem. However, individual differences were observed, particularly when schedule requirements were high. Future studies should examine the underlying cause of this result.

Some topics in theoretical population genetics: Editorial commentaries on a selection of Marc Feldman’s TPB papers

This article consists of commentaries on a selected group of papers of Marc Feldman published in Theoretical Population Biology from 1970 to the present. The papers describe a diverse set of population-genetic models, covering topics such as cultural evolution, social evolution, and the evolution of recombination. The commentaries highlight Marc Feldman’s role in providing mathematically rigorous formulations to explore qualitative hypotheses, in many cases generating surprising conclusions.

Illuminating Women's Hidden Contribution to Historical Theoretical Population Genetics.

While productivity in academia is measured through authorship, not all scientific contributors have been recognized as authors. We consider nonauthor "acknowledged programmers" (APs), who developed, ran, and sometimes analyzed the results of computer programs. We identified APs in Theoretical Population Biology articles published between 1970 and 1990, finding that APs were disproportionately women (P = 4.0 × 10-10). We note recurrent APs who contributed to several highly-cited manuscripts. The occurrence of APs decreased over time, corresponding to the masculinization of computer programming and the shift of programming responsibilities to individuals credited as authors. We conclude that, while previously overlooked, historically, women have made substantial contributions to computational biology. For a video of this abstract, see: https://vimeo.com/313424402.

An application of the Caputo-Fabrizio operator to replicator-mutator dynamics: Bifurcation, chaotic limit cycles and control

The physical behaviors of replicator-mutator processes found in theoretical biophysics, physical chemistry, biochemistry and population biology remain complex with unlimited expressibility. People languages, for instance, have impressively and unpredictably changed over the time in human history. This is mainly due to the collection of small changes and collaboration with other languages. In this paper, the Caputo-Fabrizio operator is applied to a replicator-mutator dynamic taking place in midsts with movement. The model is fully analyzed and solved numerically via the Crank-Nicolson scheme. Stability and convergence results are provided. A concrete application to replicator-mutator dynamics for a population with three strategies is performed with numerical simulations provided for some fixed values of the physical position of the population symbolized by r and the grid points. Physically, it happens that limit cycles appear, not only in function of the mutation parameter μ but also in function of the values given to r . The amplitudes of limit cycles also appear to be proportional to r but the stability of the system remains unaffected. However, those limit cycles instead of disappearing as expected, are immediately followed by chaotic and unpredictable behaviors certainly due to the non-singular kernel used in the model and suitable to non-linear dynamics. Hence, the appearance and disappearance of limit cycles might be controlled by the position variable r which can also apprehend chaos.

StagePop: modelling stage‐structured populations in r

Summary We present an r‐package, stagePop, which is a tool for predicting the deterministic dynamics and interactions of stage‐structured populations (i.e. where the life cycle consists of distinct stages, for example eggs, juveniles and reproductive adults). The continuous‐time formulation enables stagePop to easily simulate time‐varying stage durations, overlapping generations and density‐dependent vital rates. The package can be used to predict predator–prey interactions, host–parasitoid interactions, resource competition, intra‐specific competition and the effects of environmental change on stage‐structured (and non‐stage structured) species. It can be used for predicting the effects of bio‐control and climate change, investigating mechanisms for maintaining diversity, predicting the dynamics of complex food webs following perturbation and so on. The code is based on the formulation by Nisbet and Gurney (Theoretical Population Biology, 23, 1983, 114) using delay‐differential equations, which are solved using the r‐packages deSolve or PBSddesolve.

Variance, Invariance and Statistical Explanation

The most compelling extant accounts of explanation casts all explanations as causal. Yet there are sciences, theoretical population biology in particular, that explain their phenomena by appeal to statistical, non-causal properties of ensembles. I develop a generalised account of explanation. An explanation serves two functions: metaphysical and cognitive. The metaphysical function is discharged by identifying a counterfactually robust invariance relation between explanans event and explanandum. The cognitive function is discharged by providing an appropriate description of this relation. I offer examples of explanations from portfolio theory and population genetics that meet this characterisation. In each case the invariance relation holds between a statistical property of an ensemble and a change in structure of the ensemble. In neither case, however, does the statistical property cause the outcome it explains. There are genuine statistical, non-causal scientific explanations.

Tail probabilities of extinction time in a large number of experimental populations.

Determining the distribution of population extinction times is a fundamental problem in theoretical population biology. In particular, the tail properties, patterns in the probability of long-term persistence, have not been studied. Further, until now there have been no experimental or observational data sets with which to empirically test the "rare event" predictions of the standard stochastic theory of extinction, which holds that extinction times should be exponentially distributed. I performed an experimental study of extinction in a large number of replicate (n = 1076) laboratory populations of the waterflea Daphnia pulicaria. Observed extinction time ranged from 1 to 1239 days. Statistical models supported the hypothesis of a power-law distribution over the exponential distribution and other alternatives. This pattern contradicts the notion that population extinction time has an exponential tail, questioning its ubiquitous use in theoretical ecology. It is also a rare instance of a data set that exhibits power-law scaling under appropriate statistical criteria.

Frederick Edward Smith: 1920–2012

Frederick Edward Smith: 1920–2012

Spatio-Temporal Measurements and Modeling of Genetic Expression

Single-molecule, single-cell studies of genetic expression have provided key insights into how cells respond to external stimuli [Munsky, B., et al., Science (2012)]. By directly measuring copy numbers of individual bio-molecules in cells, it is now possible to obtain statistical measures of the spatio-temporal distributions of key signaling and regulatory networks. Such comprehensive datasets can be used to infer system-level models that yield quantitative insight into cellular regulation, predict cellular responses in new experimental conditions, and suggest more revealing experiments to uncover regulatory dynamics. The integration of single-molecule spectroscopy, biochemistry, and numerical modeling is a powerful multi-disciplinary approach to investigating cellular response at the genetic level. A key issue we seek to address is what types of fluctuations are most informative about the underlying gene regulatory process. In other words, how much experimental resources should be spent to measure (i) temporal, (ii) spatial, or (iii) cell-to-cell fluctuations? As an example, we studied Interluekin 1-alpha (IL1α) mRNA expression within human THP-1 cells during stimulus response to lipopolysaccharide (LPS). By spatially resolving individual mRNA using multiplexed single molecule FISH [Femino A.M., et al., Science (1998), Raj A., et al., Nat Meth (2008)] in large populations of single cells at multiple times points, we quantified all three fluctuation types. We expanded the common bursting gene expression model [Peccoud, J., Theoretical Population Biology (1995)] and derived a set of linear ODEs to describe the mean, variance, and co-variance of nuclear and cytoplasmic IL1α mRNA. We fit this model to multiple single-cell datasets. Comparing models inferred from each data set, we are able to draw conclusions on which fluctuation types are most revealing about the underlying system's mechanisms and parameters, providing feedback for new experiments. The approach developed here is applicable to any eukaryotic gene expression pathway.

Integrating mechanistic organism–environment interactions into the basic theory of community and evolutionary ecology

This paper presents an overview of how mechanistic knowledge of organism-environment interactions, including biomechanical interactions of heat, mass and momentum transfer, can be integrated into basic theoretical population biology through mechanistic functional responses that quantitatively describe how organisms respond to their physical environment. Integrating such functional responses into simple community and microevolutionary models allows scaling up of the organism-level understanding from biomechanics both ecologically and temporally. For community models, Holling-type functional responses for predator-prey interactions provide a classic example of the functional response affecting qualitative model dynamics, and recent efforts are expanding analogous models to incorporate environmental influences such as temperature. For evolutionary models, mechanistic functional responses dependent on the environment can serve as fitness functions in both quantitative genetic and game theoretic frameworks, especially those concerning function-valued traits. I present a novel comparison of a mechanistic fitness function based on thermal performance curves to a commonly used generic fitness function, which quantitatively differ in their predictions for response to environmental change. A variety of examples illustrate how mechanistic functional responses enhance model connections to biologically relevant traits and processes as well as environmental conditions and therefore have the potential to link theoretical and empirical studies. Sensitivity analysis of such models can provide biologically relevant insight into which parameters and processes are important to community and evolutionary responses to environmental change such as climate change, which can inform conservation management aimed at protecting response capacity. Overall, the distillation of detailed knowledge or organism-environment interactions into mechanistic functional responses in simple population biology models provides a framework for integrating biomechanics and ecology that allows both tractability and generality.

On chaos, transient chaos and ghosts in single population models with Allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)∞ occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle–node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects.

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

Toward a unified view of diversity partitioning

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On beta diversity decomposition: Trouble shared is not trouble halved

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Comments on “Improving estimates of the basic reproductive ratio: Using both the mean and the dispersal of transition times”

Comments on Heffernan, J. M., Wahl, L. M. (2006). Improving estimates of the basic reproductive ratio: Using both the mean and the dispersal of transition times. Theoretical Population Biology, 70, 135–145.

The strategy of “The strategy of model building in population biology”

In this essay, I argue for four related claims. First, Richard Levins’ classic “The Strategy of Model Building in Population Biology” was a statement and defense of theoretical population biology growing out of collaborations between Robert MacArthur, Richard Lewontin, E. O. Wilson, and others. Second, I argue that the essay served as a response to the rise of systems ecology especially as pioneered by Kenneth Watt. Third, the arguments offered by Levins against systems ecology and in favor of his own methodological program are best construed as “pragmatic”. Fourth, I consider limitations of Levins’ arguments given contemporary population biology.