Theoretical Population Research Articles

Radar observability of near-Earth objects using EISCAT 3D

Abstract. Radar observations can be used to obtain accurate orbital elements for near-Earth objects (NEOs) as a result of the very accurate range and range rate measureables. These observations allow the prediction of NEO orbits further into the future and also provide more information about the properties of the NEO population. This study evaluates the observability of NEOs with the EISCAT 3D 233 MHz 5 MW high-power, large-aperture radar, which is currently under construction. Three different populations are considered, namely NEOs passing by the Earth with a size distribution extrapolated from fireball statistics, catalogued NEOs detected with ground-based optical telescopes and temporarily captured NEOs, i.e. mini-moons. Two types of observation schemes are evaluated, namely the serendipitous discovery of unknown NEOs passing the radar beam and the post-discovery tracking of NEOs using a priori orbital elements. The results indicate that 60–1200 objects per year, with diameters D>0.01 m, can be discovered. Assuming the current NEO discovery rate, approximately 20 objects per year can be tracked post-discovery near the closest approach to Earth. Only a marginally smaller number of tracking opportunities are also possible for the existing EISCAT ultra-high frequency (UHF) system. The mini-moon study, which used a theoretical population model, orbital propagation, and a model for radar scanning, indicates that approximately seven objects per year can be discovered using 8 %–16 % of the total radar time. If all mini-moons had known orbits, approximately 80–160 objects per year could be tracked using a priori orbital elements. The results of this study indicate that it is feasible to perform routine NEO post-discovery tracking observations using both the existing EISCAT UHF radar and the upcoming EISCAT 3D radar. Most detectable objects are within 1 lunar distance (LD) of the radar. Such observations would complement the capabilities of the more powerful planetary radars that typically observe objects further away from Earth. It is also plausible that EISCAT 3D could be used as a novel type of an instrument for NEO discovery, assuming that a sufficiently large amount of radar time can be used. This could be achieved, for example by time-sharing with ionospheric and space-debris-observing modes.

Prey refuge and morphological defense mechanisms as nonlinear triggers in an intraguild predation food web

Intraguild predation (IGP) is a type of interaction in which a top predator simultaneously competes and predates an intermediate prey that shares a third prey species with the top predator. While common in nature, most theoretical population dynamics models proposed in the literature predict that this three species interaction usually leads to extinction of the intermediate prey population. Predator-induced defense as well as refuge mechanisms are widely seen in these systems and should be incorporated in IGP models to promote coexistence. With this aim, we introduce a nonlinear response to the predation of IG-predator on IG-prey modelling both prey refuge and morphological defenses. The phase diagram of species coexistence is obtained as function of the attack efficiency and the degree of nonlinearity of the defense mechanisms. Further we show how the nonlinearity affects the equilibrium populations. We unveil that there is an optimal nonlinearity at which the convergence towards the stationary coexistence regime is the fastest.

Effects of triphenyl phosphate on ciliate protozoa Tetrahymena thermophila following acute exposure and sub-chronic exposure

Triphenyl phosphate (TPHP) is one of the most widely used organophosphate flame retardants (OPFRs) and is frequently detected in a variety of environmental media. Previous studies reported that TPHP had toxic effects on vertebrates, but little toxic information was available in lower trophic aquatic organisms which were more sensitive to the exposure of many toxic substances. In this study, protozoa Tetrahymena thermophila (T. thermophila) were exposed to 0, 0.01, 0.17 or 2.35mg/L TPHP for 5 days to study the effects of sub-chronic exposure on theoretical population, cell viability, cell size and number of cilia. Additionally, the effects of TPHP on gene transcription were assessed by transcriptome sequencing technique (RNA-Seq). Cell viability and number of cilia were significantly reduced in all TPHP exposure groups compared with the control. In addition, exposure to 0.17 or 2.35mg/L TPHP significantly reduced the theoretical population, circumference and body width, and there was a significant decrease in body length in the 2.35mg/L exposure group. Comparative transcriptome sequencing identified a total of 4105 up- and 4487 down-regulated genes after exposure to 2.35mg/L TPHP for 5 days compared with the control. KEGG analysis showed that dysfunction of pathways associated with ribosome, spliceosome, phagosome, proteasome and protein processing in endoplasmic reticulum in this study might be responsible for the toxicity of T. thermophila caused by TPHP. In general, the results indicated that TPHP had an adverse effect on the protozoa T. thermophila.

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.

Population estimate and rate of increase of southern right whales Eubalaena australis in southeastern Australia

In Australian waters, southern right whales Eubalaena australis form 2 genetically distinct populations that have shown contrasting patterns of recovery since whaling ceased: a western population in South Australia and Western Australia and an eastern population in southeastern Australia (Tasmania, Victoria and New South Wales). Here, we provide an abundance estimate derived from a breeding female superpopulation mark-recapture model for the southeastern southern right whale population. The population comprises 268 individuals (68 breeding females) and has increased at a rate of 4.7% per annum between 1996 and 2017. There has been no significant change in the annual abundance of mother-calf pairs sighted at the only calving ground (Logans Beach in Victoria) over the last 3 decades. The total number of southern right whales (i.e. all adults and calves) using the southeastern Australian coastline has increased by 7% since 1985. Unlike the population estimate (which was restricted to breeding females sighted prior to the post-breeding southward migration), this estimate is likely to include transiting whales from the southwestern population. The theoretical population model predicts 19 breeding females at Logans Beach in 2018 and 28 in 2028; the actual number of breeding females, as of 2018, is 14. This study provides the first complete estimate of population size and rate of increase of southern right whales along the southeastern Australian coastline. This knowledge is critical for assessing population status and recovery of southern right whales in Australia. It provides a basis for monitoring persistence and responses of the population to environmental stressors.

Evidence from high-mass X-ray binaries that Galactic WR components of WR+O binaries end their life with a supernova explosion

Context. Theoretical population number studies of binaries with at least one black hole (BH) component obviously depend on whether or not BHs receive a (natal) kick during their formation. Aims. Several observational facts seem to indicate that BHs do indeed receive a kick during their formation. In the present paper, we discuss additional evidence of this. Methods. The progenitors of wind-fed high-mass X-ray binaries (HMXB) with a BH component (BH HMXB) are WR+OB binaries where the Wolf–Rayet (WR) star will finally collapse and form the BH. Starting from the observed population of WR+OB binaries in the solar neighborhood, we predict the population of wind-fed BH HMXBs as a function of the BH-natal kick. Results. The simulations reveal that when WR stars collapse into a BH with a zero or low kick, we should expect 100 or more wind-fed BH HMXBs in the solar neighborhood, whereas only one is observed (Cyg X-1). We consider this as evidence that either WR components in binaries end their life as a neutron star or that they collapse into BHs, both accompanied by a supernova explosion imparting significant (natal) kicks.

Effects of tris (2-chloroethyl) phosphate (TCEP) on growth, reproduction and gene transcription in the protozoan Tetrahymena thermophila

As a typical organophosphorus flame retardant, tris (2-chloroethyl) phosphate (TCEP) has been widely detected in various environmental media. Toxicity of TCEP to vertebrates have been investigated, but potential effects on lower trophic level species were unknown to date. In this study, toxic effects and molecular mechanisms of toxic actions of TCEP on the aquatic protozoan Tetrahymena thermophila were evaluated by use of phenotypic observations, transcriptome sequencing analysis and real-time quantitative PCR detection. Exposure to 0.044, 0.411 or 4.26 mg/L TCEP for 5 days decreased the theoretical population, cell viability, number of cilia and cell size of Tetrahymena thermophila in a time- and dose-dependent manner. Meanwhile, RNA-Seq analysis indicated that exposure to 4.26 mg/L TCEP significantly changed expression of 2932 genes (up-regulation: 1228; down-regulation: 1704). Of these, expressions of 9, 10 and 17 genes that were enriched in soluble N-ethylmaleimide-sensitive factor attachment protein receptor (SNARE) interaction in vesicular transport, proteasome and endocytosis pathway respectively were down-regulated. Data collected during this study suggested that exposure to high concentrations of TCEP might affect growth and reproduction of Tetrahymena thermophila through down-regulating transcriptional levels of genes encoding proteins associated with vesicle trafficking, proteasome and endocytosis.

Evolution of self-incompatibility in the Brassicaceae: Lessons from a textbook example of natural selection.

Self‐incompatibility (SI) is a self‐recognition genetic system enforcing outcrossing in hermaphroditic flowering plants and results in one of the arguably best understood forms of natural (balancing) selection maintaining genetic variation over long evolutionary times. A rich theoretical and empirical population genetics literature has considerably clarified how the distribution of SI phenotypes translates into fitness differences among individuals by a combination of inbreeding avoidance and rare‐allele advantage. At the same time, the molecular mechanisms by which self‐pollen is specifically recognized and rejected have been described in exquisite details in several model organisms, such that the genotype‐to‐phenotype map is also pretty well understood, notably in the Brassicaceae. Here, we review recent advances in these two fronts and illustrate how the joint availability of detailed characterization of genotype‐to‐phenotype and phenotype‐to‐fitness maps on a single genetic system (plant self‐incompatibility) provides the opportunity to understand the evolutionary process in a unique perspective, bringing novel insight on general questions about the emergence, maintenance, and diversification of a complex genetic system.

A Joint Mass–Radius–Period Distribution of Exoplanets

Abstract The radius–period distribution of exoplanets has been characterized by the Kepler survey, and the empirical mass–radius relation by the subset of Kepler planets with mass measurements. We combine the two in order to constrain the joint mass–radius–period distribution of Kepler transiting planets. We employ hierarchical Bayesian modeling and mixture models to formulate four models with varying complexity and fit these models to the data. We find that the most complex models that treat planets with significant gaseous envelopes, evaporated core planets, and intrinsically rocky planets as three separate populations are preferred by the data and provide the best fit to the observed distribution of Kepler planets. We use these models to calculate occurrence rates of planets in different regimes and to predict masses of Kepler planets, revealing the model-dependent nature of both. When using models with envelope mass loss to calculate η ⊕, we find nearly an order of magnitude drop, indicating that many Earth-like planets discovered with Kepler may be evaporated cores which do not extrapolate out to higher orbital periods. This work provides a framework for higher-dimensional studies of planet occurrence and for using mixture models to incorporate different theoretical populations of planets.

The interplay between fishery yield and top predator culling in a multispecies fishery context.

In food webs, fishery can play the role of top predator, competing thus with other top predators for valuable food resources. In this view, it has been claimed in fisheries management that culling of top predators can be a means to improve fishery yield. To investigate this hypothesis, we use theoretical population models to assess in a multispecies context how fishery yield from target species harvest responds to top predator cull. Defying crisp summary, the four analysed food web models show that this response may be either positive or negative or both, indicating that in terms of multispecies fishery management the harvest yield may not accrue as a consequence of predator removal. In addition, this multitude of behaviours points also to the fact that the response of fishery yield to top predator cull may be difficult to assess.

A Population Genomics Lexicon.

Population genomics is a growing field stemming from soon a 100 years of developments in population genetics. Here, we summarize the main concepts and terminology underlying both theoretical and empirical statistical population genomics studies. We provide the reader with pointers toward the original literature as well as methodological and historical reviews.

On competition models under allee effect: Asymptotic behavior and traveling waves

<p style='text-indent:20px;'>In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species (<inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula>). Under one-side Allee effect on <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>-species, the model demonstrates complexity on its coexistence and <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula>-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant <inline-formula><tex-math id="M5">\begin{document}$ K $\end{document}</tex-math></inline-formula> is large relative to other biological parameters, the asymptotic stability of the <inline-formula><tex-math id="M6">\begin{document}$ v $\end{document}</tex-math></inline-formula>-dominance state <inline-formula><tex-math id="M7">\begin{document}$ (0,\:1) $\end{document}</tex-math></inline-formula> indicates the competitive exclusion of the <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula>-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the <inline-formula><tex-math id="M9">\begin{document}$ u $\end{document}</tex-math></inline-formula>-dominance states), there exist traveling wave solutions flowing from the <inline-formula><tex-math id="M10">\begin{document}$ u $\end{document}</tex-math></inline-formula>-dominance states to the <inline-formula><tex-math id="M11">\begin{document}$ v $\end{document}</tex-math></inline-formula>-dominance state. The asymptotic rates of the traveling waves at <inline-formula><tex-math id="M12">\begin{document}$ \xi \rightarrow \mp \infty $\end{document}</tex-math></inline-formula> are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.

Bayesian compartmental models and associated reproductive numbers for an infection with multiple transmission modes.

Zoonotic visceral leishmaniasis (ZVL) is a serious neglected tropical disease that is endemic in 98 countries. ZVL is primarily transmitted via a sand fly vector. In the United States, it is enzootic in some canine populations; it is transmitted from infectious mother to pup transplacentally, and vector-borne transmission is absent. This absence affords a unique opportunity to study (1) vertical transmission dynamics in dogs and (2) the importance of vertical transmission in maintaining an infectious reservoir in the presence of a vector. In this paper, we present Bayesian compartmental models and reproductive number formulations to examine (1) and (2), providing a mechanism to plan and evaluate interventions in regions where both transmission modes are present. First, we propose an individual-level susceptible, infectious, removed (SIR) model to study the effect of maternal infection status during pregnancy on pup infection progression. We provide evidence that pups born to diagnostically positive mothers during pregnancy are more likely to become diagnostically positive both earlier in life, and at some point during their lifetime, than those born to diagnostically negative mothers. Second, we propose a population-level SIR model to study the impact of a vertically maintained reservoir on propagating infection in a naive canine population through emergent vector transmission using simulation studies. We also present reproductive numbers to quantify contributions of vertically infected and vector-infected dogs to maintaining infection in the population. We show that a vertically maintained canine reservoir can propagate infection in a theoretical naive population in the presence of avector.

Generating normal networks via leaf insertion and nearest neighbor interchange

BackgroundGalled trees are studied as a recombination model in theoretical population genetics. This class of phylogenetic networks has been generalized to tree-child networks and other network classes by relaxing a structural condition imposed on galled trees. Although these networks are simple, their topological structures have yet to be fully understood.ResultsIt is well-known that all phylogenetic trees on n taxa can be generated by the insertion of the n-th taxa to each edge of all the phylogenetic trees on n−1 taxa. We prove that all tree-child (resp. normal) networks with k reticulate nodes on n taxa can be uniquely generated via three operations from all the tree-child (resp. normal) networks with k−1 or k reticulate nodes on n−1 taxa. Applying this result to counting rooted phylogenetic networks, we show that there are exactly frac {(2n)!}{2^{n} (n-1)!}-2^{n-1} n! binary phylogenetic networks with one reticulate node on n taxa.ConclusionsThe work makes two contributions to understand normal networks. One is a generalization of an enumeration procedure for phylogenetic trees into one for normal networks. Another is simple formulas for counting normal networks and phylogenetic networks that have only one reticulate node.

Artificial Intelligence for Ecological and Evolutionary Synthesis

The grand ambition of theorists studying ecology and evolution is to discover the logical and mathematical rules driving the world's biodiversity at every level from genetic diversity within species to differences between populations, communities, and ecosystems. This ambition has been difficult to realize in great part because of the complexity of biodiversity. Theoretical work has led to a complex web of theories, each having non-obvious consequences for other theories. Case in point, the recent realization that genetic diversity involves a great deal of temporal and spatial stochasticity forces theoretical population genetics to consider abiotic and biotic factors generally reserved to ecosystem ecology. This interconnectedness may require theoretical scientists to adopt new techniques adapted to reason about large sets of theories. Mathematicians have solved this problem by using formal languages based on logic to manage theorems. However, theories ecology and evolution are not mathematical theorems, they involve uncertainty. Recent work in Artificial Intelligence in bridging logic and probability theory offers the opportunity to build rich knowledge bases that combine logic's ability to represent complex mathematical ideas with probability theory's ability to model uncertainty. We describe these hybrid languages and explore how they could be used to build a unified knowledge base of theories for ecology and evolution.

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.

Role of base strength, cluster structure and charge in sulfuric-acid-driven particle formation

Abstract. In atmospheric sulfuric-acid-driven particle formation, bases are able to stabilize the initial molecular clusters and thus enhance particle formation. The enhancing potential of a stabilizing base is affected by different factors, such as the basicity and abundance. Here we use weak (ammonia), medium strong (dimethylamine) and very strong (guanidine) bases as representative atmospheric base compounds, and we systematically investigate their ability to stabilize sulfuric acid clusters. Using quantum chemistry, we study proton transfer as well as intermolecular interactions and symmetry in clusters, of which the former is directly related to the base strength and the latter to the structural effects. Based on the theoretical cluster stabilities and cluster population kinetics modeling, we provide molecular-level mechanisms of cluster growth and show that in electrically neutral particle formation, guanidine can dominate formation events even at relatively low concentrations. However, when ions are involved, charge effects can also stabilize small clusters for weaker bases. In this case the atmospheric abundance of the bases becomes more important, and thus ammonia is likely to play a key role. The theoretical findings are validated by cluster distribution experiments, as well as comparisons to previously reported particle formation rates, showing a good agreement.

Cost-Effectiveness of Continuous Support From a Layperson During a Woman’s First Two Births

ObjectiveTo evaluate the cost-effectiveness and health outcomes related to continuous support from a layperson during a woman’s first two births in a theoretical population. DesignCost-effectiveness analysis. ParticipantsA theoretical cohort of 1.2 million women based on an approximation of annual low-risk, nulliparous, term, singleton births in the United States with the assumption that these women have second births. This reflects the average number of births per woman in the United States. MethodsWe designed a cost-effectiveness model to compare outcomes in women with continuous support from relatives, friends, or community members with minimal to no training (excluding trained doulas) during labor and birth compared with outcomes for women with no continuous support. Outcomes included mode of birth, uterine rupture, hysterectomy, maternal death, cost, and quality-adjusted life years (QALYs). We derived probabilities from the literature and set a cost-effectiveness threshold at $100,000/QALY. ResultsIn this theoretical model, continuous support by a layperson during the first birth resulted in fewer cesarean births, decreased costs, and increased QALYs for the first and subsequent births. Women with support from laypersons had 71,090 fewer cesarean births, 35 fewer uterine ruptures, 9 fewer hysterectomies, and 16 fewer maternal deaths, which saved $364 million with 2,673 increased QALYs. Sensitivity analyses showed that continuous support in the first birth was cost-effective even when varying the estimate of lost wages of the support person up to $708. ConclusionContinuous labor support from a layperson leads to fewer cesarean births, improved outcomes, decreased costs, and increased QALYs. This highlights the need to increase women’s access to continuous layperson support during labor and birth uninhibited by financial and institutional barriers.

Occurrence of the fungus mycotoxin, ustiloxin A, in surface waters of paddy fields in Enshi, Hubei, China, and toxicity in Tetrahymena thermophila

There has been an increasing incidence rate of rice false smut in global rice cultivation areas. However, there is a dearth of studies on the environmental concentrations and hazards of ustiloxin A (UA), which is the major mycotoxin produced by a pathogenic fungus of the rice false smut. Here, the concentrations of UA in the surface waters of two paddy fields located in Enshi city, Hubei province, China, were measured, and its toxicity in T. Thermophila was evaluated. This is the first study to detect UA in the surface waters of the two paddy fields, and the measured mean concentrations were 2.82 and 0.26 μg/L, respectively. Exposure to 2.19, 19.01 or 187.13 μg/L UA for 5 days significantly reduced the theoretical population and cell size of T. thermophila. Furthermore, treatment with 187.13 μg/L UA changed the percentages of T. thermophila cells in different cell-cycle stages, and with an increased malformation rate compared with the control, suggesting the disruption of the cell cycle. The expressions of 30 genes involved in the enriched proteasome pathway, 7 cyclin genes (cyc9, cyc10, cyc16, cyc22, cyc23, cyc26, cyc33) and 2 histone genes (mlh1 and hho1) were significantly down-regulated, which might be the modes of action responsible for the disruption of cell cycling due to UA exposure.

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.