The Ideal Free Distribution with travel costs.

This article studies the effect of travel costs on population distribution in a patchy environment. The Ideal Free Distribution with travel costs is defined in the article as the distribution under which it is not profitable for individuals to move, i.e., the movement between patches ceases. It is shown that depending on the travel costs between patches, the Ideal Free Distribution may be unique, there may be infinitely many possible IFDs, or no Ideal Free Distribution exists. In the latter case, animal distribution can converge to an equilibrium of distributional dynamics at which individuals do disperse, but the net movement between patches ceases. Such distributional equilibrium corresponds to balanced dispersal.

Evolutionary emergence of plant and pollinator polymorphisms in consumer-resource mutualisms.

Mutualism is considered a major driver of biodiversity, as it enables extensive codiversification in terrestrial communities. An important case is flowering plants and their pollinators, where convergent selection on plant and pollinator traits is combined with divergent selection to minimize niche overlap within each group. In this article, we study the emergence of polymorphisms in communities structured trophically: plants are the primary producers of resources required by the primary consumers, the servicing pollinators. We model natural selection on traits affecting mutualism between plants and pollinators and competition within these two trophic levels. We show that phenotypic diversification is favored by broad plant niches, suggesting that bottom-up trophic control leads to codiversification. Mutualistic generalism, i.e., tolerance to differences in plant and pollinator traits, promotes a cascade of evolutionary branching favored by bottom-up plant competition dependent on similarity and top-down mutualistic services that broaden plant niches. Our results predict a strong positive correlation between the diversity of plant and pollinator phenotypes, which previous work has partially attributed to the trophic dependence of pollinators on plants.

The asymmetric Hawk-Dove game with costs measured as time lost.

The classic Hawk-Dove game is a symmetric game in that it does not distinguish between the winners and losers of Hawk-Hawk or Dove-Dove contests. Either of the two interacting Hawks or the two interacting Doves have the same probability to win/lose the contest. In addition, all pairwise interactions take the same time and after disbanding, the individuals pair instantaneously again. This article develops an asymmetric version of the Hawk-Dove model where all costs are measured by the time lost. These times are strategy dependent and measure the length of the conflict and, when a fight occurs between two interacting Hawks, the time an individual needs to recover and pair again. These recovery times depend on whether the Hawk won or lost the contest so that we consider an asymmetric Hawk-Dove game where we distinguish between winners and losers. However, the payoff matrix for this game does not correspond to the standard bimatrix game, because some entries are undefined. To calculate strategy payoffs we consider not only costs and benefits obtained from pairwise contests but also costs when individuals are disbanded. Depending on the interacting and recovery times, the evolutionary outcomes are: Hawk only, both Hawk and Dove, and a mixed strategy. This shows that measuring the cost in time lost leads to a new prediction since, in the classic (symmetric) Hawk-Dove model that does assume positive cost (C>0), both Hawk and Dove strategy is never an evolutionary outcome.

Reducing courtship time promotes marital bliss: The Battle of the Sexes game revisited with costs measured as time lost.

Classic bimatrix games, that are based on pair-wise interactions between two opponents belonging to different populations, do not consider the cost of time. In this article, we build on an old idea that lost opportunity costs affect individual fitness. We calculate fitnesses of each strategy for a two-strategy bimatrix game at the equilibrium distribution of the pair formation process that includes activity times. This general approach is then applied to the Battle of the Sexes game where we analyze the evolutionary outcome by finding the Nash equilibria (NE) of this time-constrained game when courtship and child rearing costs are measured by time lost. While the classic Battle of the Sexes game has either a unique strict NE (specifically, all males exhibit Philanderer behavior and either all females are Coy or all are Fast depending on model parameters), or a unique interior NE where both sexes exhibit mixed behavior, including time costs for courtship and child rearing changes this prediction. First, (Philanderer, Coy) is never a NE. Second, if the benefit of having offspring is independent of parental strategies, (Philanderer, Fast) is the unique strict NE but a second stable interior NE emerges when courtship time is sufficiently short. In fact, as courtship time becomes shorter, this mixed NE (where most males are Faithful and the Coy female population is increasing) attracts almost all initial population configurations. Third, this latter promotion of marital bliss also occurs when parents who share in child rearing receive a higher benefit from their offspring than those that don't. Finally, for courtship time of moderate duration, the same phenomenon occurs when the population size increases.

Two-strategy games with time constraints on regular graphs.

Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth-death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner's Dilemma games, but for low degree there can be marked differences. For birth-death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner's dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner's dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.

Plant coexistence mediated by adaptive foraging preferences of exploiters or mutualists.

Coexistence of plants depends on their competition for common resources and indirect interactions mediated by shared exploiters or mutualists. These interactions are driven either by changes in animal abundance (density-mediated interactions, e.g., apparent competition), or by changes in animal preferences for plants (behaviorally-mediated interactions). This article studies effects of behaviorally-mediated interactions on two plant population dynamics and animal preference dynamics when animal densities are fixed. Animals can be either adaptive exploiters or adaptive mutualists (e.g., herbivores or pollinators) that maximize their fitness. Analysis of the model shows that adaptive animal preferences for plants can lead to multiple outcomes of plant coexistence with different levels of specialization or generalism for the mediator animal species. In particular, exploiter generalism promotes plant coexistence even when inter-specific competition is too strong to make plant coexistence possible without exploiters, and mutualist specialization promotes plant coexistence at alternative stable states when plant inter-specific competition is weak. Introducing a new concept of generalized isoclines allows us to fully analyze the model with respect to the strength of competitive interactions between plants (weak or strong), and the type of interaction between plants and animals (exploitation or mutualism).

Bimatrix games that include interaction times alter the evolutionary outcome: The Owner-Intruder game.

Classic bimatrix games, that are based on pair-wise interactions between two opponents in two different roles, do not consider the effect that interaction duration has on payoffs. However, interactions between different strategies often take different amounts of time. In this article, we further develop a new approach to an old idea that opportunity costs lost while engaged in an interaction affect individual fitness. We consider two scenarios: (i) individuals pair instantaneously so that there are no searchers, and (ii) searching for a partner takes positive time and populations consist of a mixture of singles and pairs. We describe pair dynamics and calculate fitnesses of each strategy for a two-strategy bimatrix game that includes interaction times. Assuming that distribution of pairs (and singles) evolves on a faster time scale than evolutionary dynamics described by the replicator equation, we analyze the Nash equilibria (NE) of the time-constrained game. This general approach is then applied to the Owner-Intruder bimatrix game where the two strategies are Hawk and Dove in both roles. While the classic Owner-Intruder game has at most one interior NE and it is unstable with respect to replicator dynamics, differences in pair duration change this prediction in that up to four interior NE may exist with their stability depending on whether pairing is instantaneous or not. The classic game has either one (all Hawk) or two ((Hawk,Dove) and (Dove,Hawk)) stable boundary NE. When interaction times are included, other combinations of stable boundary NE are possible. For example, (Dove,Dove), (Dove,Hawk), or (Hawk,Dove) can be the unique (stable) NE if interaction time between two Doves is short compared to some other interactions involving Doves.

Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time.

Fitness is often defined as the average payoff an animal obtains when it is engaged in several activities, each taking some time. We point out that the average can be calculated with respect to either the time distribution, or to the event distribution of these activities. We show that these two averages lead to the same fitness function. We illustrate this result through two examples from foraging theory, Holling II functional response and the diet choice model, and one game-theoretic example of Hamilton's rule applied to the time-constrained Prisoner's dilemma (PD). In particular, we show that in these models, fitness defined as expected gain per unit time equals fitness defined as expected gain divided by expected time. We also show how these fitnesses predict the optimal outcome for diet choice and the prevalence of cooperation in the repeated PD game.

Competition, trait-mediated facilitation, and the structure of plant-pollinator communities.

In plant-pollinator communities many pollinators are potential generalists and their preferences for certain plants can change quickly in response to changes in plant and pollinator densities. These changes in preferences affect coexistence within pollinator guilds as well as within plant guilds. Using a mathematical model, we study how adaptations of pollinator preferences influence population dynamics of a two-plant-two-pollinator community interaction module. Adaptation leads to coexistence between generalist and specialist pollinators, and produces complex plant population dynamics, involving alternative stable states and discrete transitions in the plant community. Pollinator adaptation also leads to plant-plant apparent facilitation that is mediated by changes in pollinator preferences. We show that adaptive pollinator behavior reduces niche overlap and leads to coexistence by specialization on different plants. Thus, this article documents how adaptive pollinator preferences for plants change the structure and coexistence of plant-pollinator communities.

Beyond replicator dynamics: From frequency to density dependent models of evolutionary games.

Game theoretic models of evolution such as the Hawk-Dove game assume that individuals gain fitness (which is a proxy of the per capita population growth rate) in pair-wise contests only. These models assume that the equilibrium distribution of phenotypes involved (e.g., Hawks and Doves) in the population is given by the Hardy-Weinberg law, which is based on instantaneous, random pair formation. On the other hand, models of population dynamics do not consider pairs, newborns are produced by singles, and interactions between phenotypes or species are described by the mass action principle. This article links game theoretic and population approaches. It shows that combining distribution dynamics with population dynamics can lead to stable coexistence of Hawk and Dove population numbers in models that do not assume a priori that fitness is negative density dependent. Our analysis shows clearly that the interior Nash equilibrium of the Hawk and Dove model depends both on population size and on interaction times between different phenotypes in the population. This raises the question of the applicability of classic evolutionary game theory that requires all interactions take the same amount of time and that all single individuals have the same payoff per unit of time, to real populations. Furthermore, by separating individual fitness into birth and death effects on singles and pairs, it is shown that stable coexistence in these models depends on the time-scale of the distribution dynamics relative to the population dynamics. When explicit density-dependent fitness is included through competition over a limited resource, the combined dynamics of the Hawk-Dove model often lead to Dove extinction no matter how costly fighting is for Hawk pairs.

Interaction times change evolutionary outcomes: Two-player matrix games.

Two most influential models of evolutionary game theory are the Hawk-Dove and Prisoner's dilemma models. The Hawk-Dove model explains evolution of aggressiveness, predicting individuals should be aggressive when the cost of fighting is lower than its benefit. As the cost of aggressiveness increases and outweighs benefits, aggressiveness in the population should decrease. Similarly, the Prisoner's dilemma models evolution of cooperation. It predicts that individuals should never cooperate despite cooperation leading to a higher collective fitness than defection. The question is then what are the conditions under which cooperation evolves? These classic matrix games, which are based on pair-wise interactions between two opponents with player payoffs given in matrix form, do not consider the effect that conflict duration has on payoffs. However, interactions between different strategies often take different amounts of time. In this article, we develop a new approach to an old idea that opportunity costs lost while engaged in an interaction affect individual fitness. When applied to the Hawk-Dove and Prisoner's dilemma, our theory that incorporates general interaction times leads to qualitatively different predictions. In particular, not all individuals will behave as Hawks when fighting cost is lower than benefit, and cooperation will evolve in the Prisoner's dilemma.

A dynamical model for bark beetle outbreaks.

Tree-killing bark beetles are major disturbance agents affecting coniferous forest ecosystems. The role of environmental conditions on driving beetle outbreaks is becoming increasingly important as global climatic change alters environmental factors, such as drought stress, that, in turn, govern tree resistance. Furthermore, dynamics between beetles and trees are highly nonlinear, due to complex aggregation behaviors exhibited by beetles attacking trees. Models have a role to play in helping unravel the effects of variable tree resistance and beetle aggregation on bark beetle outbreaks. In this article we develop a new mathematical model for bark beetle outbreaks using an analogy with epidemiological models. Because the model operates on several distinct time scales, singular perturbation methods are used to simplify the model. The result is a dynamical system that tracks populations of uninfested and infested trees. A limiting case of the model is a discontinuous function of state variables, leading to solutions in the Filippov sense. The model assumes an extensive seed-bank so that tree recruitment is possible even if trees go extinct. Two scenarios are considered for immigration of new beetles. The first is a single tree stand with beetles immigrating from outside while the second considers two forest stands with beetle dispersal between them. For the seed-bank driven recruitment rate, when beetle immigration is low, the forest stand recovers to a beetle-free state. At high beetle immigration rates beetle populations approach an endemic equilibrium state. At intermediate immigration rates, the model predicts bistability as the forest can be in either of the two equilibrium states: a healthy forest, or a forest with an endemic beetle population. The model bistability leads to hysteresis. Interactions between two stands show how a less resistant stand of trees may provide an initial toe-hold for the invasion, which later leads to a regional beetle outbreak in the resistant stand.

Dispersal dynamics in food webs.

Studies of food webs suggest that limited nonrandom dispersal can play an important role in structuring food webs. It is not clear, however, whether density-dependent dispersal fits empirical patterns of food webs better than density-independent dispersal. Here, we study a spatially distributed food web, using a series of population-dispersal models that contrast density-independent and density-dependent dispersal in landscapes where sampled sites are either homogeneously or heterogeneously distributed. These models are fitted to empirical data, allowing us to infer mechanisms that are consistent with the data. Our results show that models with density-dependent dispersal fit the α, ß, and γ tritrophic richness observed in empirical data best. Our results also show that density-dependent dispersal leads to a critical distance threshold beyond which site similarity (i.e., ß tritrophic richness) starts to decrease much faster. Such a threshold can also be detected in the empirical data. In contrast, models with density-independent dispersal do not predict such a threshold. Moreover, preferential dispersal from more centrally located sites to peripheral sites does not provide a better fit to empirical data when compared with symmetric dispersal between sites. Our results suggest that nonrandom dispersal in heterogeneous landscapes is an important driver that shapes local and regional richness (i.e., α and γ tritrophic richness, respectively) as well as the distance-decay relationship (i.e., ß tritrophic richness) in food webs.

L-shaped prey isocline in the Gause predator-prey experiments with a prey refuge.

Predator and prey isoclines are estimated from data on yeast-protist population dynamics (Gause et al., 1936). Regression analysis shows that the prey isocline is best fitted by an L-shaped function that has a vertical and a horizontal part. The predator isocline is vertical. This shape of isoclines corresponds with the Lotka-Volterra and the Rosenzweig-MacArthur predator-prey models that assume a prey refuge. These results further support the idea that a prey refuge changes the prey isocline of predator-prey models from a horizontal to an L-shaped curve. Such a shape of the prey isocline effectively bounds amplitude of predator-prey oscillations, thus promotes species coexistence.

Effects of animal dispersal on harvesting with protected areas.

Effects of density dependent as well as independent dispersal modes between a harvested patch and a protected area on the maximum sustainable yield and population abundance are studied. Without dispersal, the Gordon-Schaefer harvesting model predicts that as the protected area increases, population abundance increases too but the maximum sustainable yield (MSY) decreases. This article shows that dispersal can change this prediction. For density independent balanced and fast dispersal, neither the MSY, nor population abundance depends on the protected area. For fast and unbalanced dispersal both the MSY and equilibrium population abundance are unimodal functions of the protected area size. For density dependent dispersal which is in direction of increasing fitness predictions depend on whether individuals react to mortality risk in harvested patch. When animals disregard harvesting risk, the results are similar to the case of density independent and balanced dispersal. When animals do consider harvesting risk, the results are similar to the case without dispersal. The models considered also show that dispersal reduces beneficial effect of protected areas, because population abundance is smaller when compared with no dispersal case.

Competition in di- and tri-trophic food web modules.

Competition in di- and tri-trophic food web modules with many competing species is studied. The food web modules considered are apparent competition between n species sharing a single predator and a diamond-like food web with a single resource, a single top predator and many competing middle species. The predators have either fixed preferences for their prey, or they switch between available prey in a way that maximizes their fitness. Dependence of these food web dynamics on environmental carrying capacity and food web connectance is studied. The results predict that optimal flexible foraging strongly weakens apparent competition and promotes species coexistence. Food web robustness (defined here as the proportion of surviving species) does not decrease with increased connectance in these food-webs. Moreover, it is shown that flexible prey switching leads to the same population equilibria as in corresponding food webs with highly specialized predators. The results show that flexible foraging behavior by predators can have very strong impact on species richness, as well as the response of communities to changes in resource enrichment and food-web connectance when compared to the same food-web topology with inflexible top predators. Several results on global stability using Lyapunov functions are provided.

The Allee-type ideal free distribution.

The ideal free distribution (IFD) in a two-patch environment where individual fitness is positively density dependent at low population densities is studied. The IFD is defined as an evolutionarily stable strategy of the habitat selection game. It is shown that for low and high population densities only one IFD exists, but for intermediate population densities there are up to three IFDs. Population and distributional dynamics described by the replicator dynamics are studied. It is shown that distributional stability (i.e., IFD) does not imply local stability of a population equilibrium. Thus distributional stability is not sufficient for population stability. Results of this article demonstrate that the Allee effect can strongly influence not only population dynamics, but also population distribution in space.

Behavioral refuges and predator-prey coexistence.

The effects of a behavioral refuge caused either by the predator optimal foraging or prey adaptive antipredator behavior on the Gause predator-prey model are studied. It is shown that both of these mechanisms promote predator-prey coexistence either at an equilibrium, or along a limit cycle. Adaptive prey refuge use leads to hysteresis in prey antipredator behavior which allows predator-prey coexistence along a limit cycle. Similarly, optimal predator foraging leads to sigmoidal functional responses with a potential to stabilize predator-prey population dynamics at an equilibrium, or along a limit cycle.

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds.

The population-dispersal dynamics for predator-prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator-prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.