Plant–pollinator population dynamics

Ecological theory of mutualism: Robust patterns of stability and thresholds in two-species population models

Mutualisms are ubiquitous in nature, provide important ecosystem services, and involve many species of interest for conservation. Theoretical progress on the population dynamics of mutualistic interactions, however, comparatively lagged behind that of trophic and competitive interactions, leading to the impression that ecologists still lack a generalized framework to investigate the population dynamics of mutualisms. Yet, over the last 90 years, abundant theoretical work has accumulated, ranging from abstract to detailed. Here, we review and synthesize historical models of two-species mutualisms. We find that population dynamics of mutualisms are qualitatively robust across derivations, including levels of detail, types of benefit, and inspiring systems. Specifically, mutualisms tend to exhibit stable coexistence at high density and destabilizing thresholds at low density. These dynamics emerge when benefits of mutualism saturate, whether due to intrinsic or extrinsic density dependence in intraspecific processes, interspecific processes, or both. We distinguish between thresholds resulting from Allee effects, low partner density, and high partner density, and their mathematical and conceptual causes. Our synthesis suggests that there exists a robust population dynamic theory of mutualism that can make general predictions.

What Maintains Flower Colour Variation within Populations?

Natural selection acts on phenotypic trait variation. Understanding the mechanisms that create and maintain trait variation is fundamental to understanding the breadth of diversity seen on Earth. Flower colour is among the most conspicuous and highly diverse traits in nature. Most flowering plant populations have uniform floral colours, but a minority exhibit within-population colour variation, either discrete (polymorphic) or continuous. Colour variation is commonly maintained by balancing selection through multiple pollinators, opposing selection regimes, or fluctuating selection. Variation can also be maintained by heterozygote advantage or frequency-dependent selection. Neutral processes, or a lack of selection, may maintain variation, although this remains largely untested. We suggest several prospective research directions that may provide insight into the evolutionary drivers of trait variation.

Persistence and Oscillations of Plant-Pollinator-Herbivore Systems

This paper considers plant-pollinator-herbivore systems where the plant produces food for the pollinator, the pollinator provides pollination service for the plant in return, while the herbivore consumes both the food and the plant itself without providing pollination service. Based on these resource-consumer interactions, we form a plant-pollinator-herbivore model which includes the intermediary food. Using qualitative method and Kuznetsov theorem, we show global dynamics of the subsystems, uniform persistence of the whole system and periodic oscillation by Hopf bifurcation. Rigorous analysis on the system demonstrates mechanisms by which varying parameters could make the system transition between extinction of herbivore, coexistence of the three species at steady states, coexistence in periodic oscillations and extinction of pollinator. It is shown that (i) in plant-pollinator interactions, the plant would produce food; (ii) in plant-herbivore interactions, the plant would produce toxin; (iii) in the presence of both pollinator and herbivore, the plant would produce both food and toxin, and intermediate productions are analytically given by which the plant can reach its maximal density; and (iv) an appropriate toxin production could drive the herbivore into extinction, an unappropriate one would drive the pollinator into extinction, while too much toxin production will drive the plant itself into extinction. The analysis leads to explanations for experimental observations and provides new insights.

Critical Transitions in Plant-Pollinator Systems Induced by Positive Inbreeding-Reward-Pollinator Feedbacks

In many regions of the world pollinator populations are rapidly declining, a trend that is expected to disrupt major ecosystem functions and services. These changes in pollinator abundance may be prone to critical transitions with abrupt shifts to a state strongly depleted both in pollinator and vegetation abundance. Here we develop a process-based model to investigate the effect of a positive pollinator-vegetation feedback, whereby an initial decline in plant density increases selfing thereby reducing floral resources and negatively affecting pollinators. We show that a decline in resource availability and an increase in disturbance intensity can induce an abrupt shift in vegetation and pollinator dynamics and potentially lead to the collapse of plant-pollinator systems. Thus, endogenous feedbacks can induce strong non-linearities in plant-pollinator dynamics, making them vulnerable to critical transitions to a state depleted of both plants and pollinators in response to resource deficiency and natural or anthropogenic disturbance.

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Increased selfing in plants reduces nectar quality and pollinator visitation

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Bistability can emerge from endogenous positive pollinator-vegetation feedbacks

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Plant-pollinator dynamics may exhibit critical transitions under global change

Random search with resetting as a strategy for optimal pollination

The problem of pollination is unique among a wide scope of search problems, since it requires optimization of benefits for both the searcher (pollinator) and its targets (plants). To address this challenge, we propose a pollination model which is based on a framework of first passage under stochastic restart. We derive equations for the search time and number of visited plants as functions of the distribution of nectar in the plant population and of the probability that a pollinator will leave the plant after examining a flower, thus effectively restarting the search. We demonstrate that nectar variation in plants serves as a driving force for pollination and establish conditions required for optimal pollination, which provides an efficient pollinator search strategy and the maximum number of plants visited by the pollinator.

Dynamics of a plant-nectar-pollinator model and its approximate equations

This paper studies effect of nectar on pollination-mutualisms. By rigorous analysis on a plant-nectar-pollinator model, we demonstrate mechanisms by which the decay rate of nectar, the nectar-consumption rate by pollinator and the nectar-production rate by plant could lead to persistence/extinction of pollination-mutualisms. For example, (i) when the decay rate is small, the pollinator survives in the system and pollination-mutualisms persist, in which the plant approaches an enhanced density. (ii) When the decay rate is intermediate, the pollinator can survive in the system only if the initial density of nectar is not below a threshold. (iii) When the decay rate is large, the pollinator goes to extinction even though the plant and nectar persist. Furthermore, we study the approximate equations of the model, in which nectar is assumed to rapidly approach a steady state. We exhibit complete analysis on the equations, which extends previous results. A new finding in comparing the two models is that the initial value of nectar plays a role in persistence of pollination-mutualisms. Our results coincide with experimental observations while numerical simulations confirm and extend the results.

Rejuvenating functional responses with renewal theory

Functional responses are widely used to describe interactions and resource exchange between individuals in ecology. The form given to functional responses dramatically affects the dynamics and stability of populations and communities. Despite their importance, functional responses are generally considered with a phenomenological approach, without clear mechanistic justifications from individual traits and behaviours. Here, we develop a bottom-up stochastic framework grounded in renewal theory that shows how functional responses emerge from the level of the individuals through the decomposition of interactions into different activities. Our framework has many applications for conceptual, theoretical and empirical purposes. First, we show how the mean and variance of classical functional responses are obtained with explicit ecological assumptions, for instance regarding foraging behaviours. Second, we give examples in specific ecological contexts, such as in nuptial-feeding species or size-dependent handling times. Finally, we demonstrate how to analyse data with our framework, especially highlighting that observed variability in the number of interactions can be used to infer parameters and compare functional response models.

Adaptive Foraging of Pollinators Can Promote Pollination of a Rare Plant Species

Most pollinators have the foraging flexibility to visit a wide variety of plant species. Yet few studies of pollinator-mediated processes in plants have considered the effects of variation in individual foraging patterns on plant reproductive success. In this study, we use an individual-based model of pollinator foraging economics to predict how visitation rates and pollination success of two coflowering plant species change with their frequency (relative abundance). Whereas previous studies suggested that adaptive foraging of pollinators always favors pollination of abundant plant species (positive frequency dependence), here we show that under certain conditions the per capita pollination success of a rare plant species can exceed that of a more abundant species. Specifically, when the overall flower density is sufficiently high and pollinators' perception ranges are sufficiently large, animals with limited memory of previously encountered rewards forage in a way that favors pollination of the rarer plant species. Moreover, even with perfectly informed foragers, a rare plant species benefits more from offering a higher floral reward than a more abundant species. Our results show that adaptive foraging of individual pollinators can have important implications for plant community dynamics and the persistence of rare plant species.

A plant–pollinator system: How learning versus cost-benefit can induce periodic oscillations

In this paper, we propose a model describing the interaction between two species: a plant population that gets pollinated by an insect population. We assume the plant population is divided into two groups: the first group in mutualistic relationship with the insect and the second group attracting the insects while deceiving them and not delivering any reward. In addition, we assume that the insect population reduces the number of visits to the plants after several unsuccessful visits. We are interested in the conditions for the coexistence of both species, especially in the appearance of damped or sustained oscillations. We focus the analysis on the parameters that measure the balance among deceit, the benefit that the insect gets from the plant, and the learning by the pollinators. We are especially interested in analyzing the effect of learning by the insect population due to unsuccessfully visiting the deceiving plants.

Bifurcation and temporal periodic patterns in a plant-pollinator model with diffusion and time delay effects

This paper deals with a plant-pollinator model with diffusion and time delay effects. By considering the distribution of eigenvalues of the corresponding linearized equation, we first study stability of the positive constant steady-state and existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. We then derive an explicit formula for determining the direction and stability of the Hopf bifurcation by applying the normal form theory and the centre manifold reduction for partial functional differential equations. Finally, we present an example and numerical simulations to illustrate the obtained theoretical results.

Construction of Dulac functions for mathematical models in population biology

In this paper, we present a method for constructing a Dulac function for mathematical models in population biology, in the form of systems of ordinary differential equations in the plane.

Floral advertisement and the competition for pollination services

Flowering plants are a major component of terrestrial ecosystems, and most of them depend on animal pollinators for reproduction. Thus, the mutualism between flowering plants and their pollinators is a keystone ecological relationship in both natural and agricultural ecosystems. Though plant-pollinator interactions have received considerable amount of attention, there are still many unanswered questions. In this paper, we use methods of evolutionary game theory to investigate the co-evolution of floral advertisement and pollinator preferences Our results indicate that competition for pollination services among plant species can in some cases lead to specialization of the pollinator population to a single plant species (oligolecty). However, collecting pollen from multiple plants - at least at the population level - is evolutionarily stable under a wider parameter range. Finally, we show that, in the presence of pollinators, plants that optimize their investment in attracting vs. rewarding visiting pollinators outcompete plants that do not.

Persistence of pollination mutualisms in the presence of ants

This paper considers plant-pollinator-ant systems in which the plant-pollinator interaction is mutualistic but ants have both positive and negative effects on plants. The ants also interfere with pollinators by preventing them from accessing plants. While a Beddington-DeAngelis (BD) formula can describe the plant-pollinator interaction, the formula is extended in this paper to characterize the pollination mutualism under the ant interference. Then, a plant-pollinator-ant system with the extended BD functional response is discussed, and global dynamics of the model demonstrate the mechanisms by which pollination mutualism can persist in the presence of ants. When the ant interference is strong, it can result in extinction of pollinators. Moreover, if the ants depend on pollination mutualism for survival, the strong interference could drive pollinators into extinction, which consequently lead to extinction of the ants themselves. When the ant interference is weak, a cooperation between plant-ant and plant-pollinator mutualisms could occur, which promotes survival of both ants and pollinators, especially in the case that ants (respectively, pollinators) cannot survive in the absence of pollinators (respectively, ants). Even when the level of ant interference remains invariant, varying ants' negative effect on plants can result in survival/extinction of both ants and pollinators. Therefore, our results provide an explanation for the persistence of pollination mutualism when there exist ants.

How does pollination mutualism affect the evolution of prior self-fertilization? A model

The mode of pollination is often neglected regarding the evolution of selfing. Yet the distribution of mating systems seems to depend on the mode of pollination, and pollinators are likely to interfere with selfing evolution, since they can cause strong selective pressures on floral traits. Most selfing species reduce their investment in reproduction, and display smaller flowers, with less nectar and scents (referred to as selfing syndrome). We model the evolution of prior selfing when it affects both the demography of plants and pollinators and the investment of plants in pollination. Including the selfing syndrome in the model allows to predict several outcomes: plants can evolve either toward complete outcrossing, complete selfing, or to a stable mixed-mating system, even when inbreeding depression is high. We predict that the evolution to high prior selfing could lead to evolutionary suicides, highlighting the importance of merging demography and evolution in models. The consequence of the selfing syndrome on plant-pollinator interactions could be a widespread mechanism driving the evolution of selfing in animal-pollinated taxa.

Plant-animal mutualism in biological markets: evolutionary and ecological dynamics driven by non-heritable phenotypic variance

Mutualism between plants and animals, such as in pollination and seed dispersal, is a fundamental mechanism facilitating the productivity and biodiversity of ecosystems, and it is often considered as an analog of a free-market economy. The coevolution of plant reward and animal choosiness, however, involves an apparent paradox due to incomplete information and limited mutation rates: plant rewards evolve only when animals are choosy, but choosy animals purge the heritable variations of plants, which then favors less choosy animals. Here we use a two-species mathematical model to illustrate how non-heritable phenotypic variances of plants may facilitate the coevolution of rewards and choosiness and solve the paradox with low mutation rates. We simultaneously track the ecological and evolutionary dynamics and show that the population ratio links the two processes and tunes the stable eco-evolutionary equilibrium. Numerical simulations confirm the analytic prediction with varying mutation rates (heritable variance). The efficiency of a biological market is generally suboptimal due to the information constraint and individual competition.

Invasibility of nectarless flowers in plant-pollinator systems

This paper considers plant-pollinator systems in which plants are divided into two categories: The plants that secret a substantial volume of nectar in their flowers are called secretors, while those without secreting nectar are called nonsecretors (cheaters). The interaction between pollinators and secretors is mutualistic, while the interaction between pollinators and nonsecretors is parasitic. Both interactions can be described by Beddington-DeAngelis functional responses. Using dynamical systems theory, we show global dynamics of a pollinator-secretor-cheater model and demonstrate mechanisms by which nectarless flowers/nonsecretors can invade the pollinator-secretor system and by which the three species could coexist. We define a threshold in the nonsecretors' efficiency in translating pollinator-cheater interaction into fitness, which is determined by parameters (factors) in the systems. When their efficiency is above the threshold, non-secretors can invade the pollinator-secretor system. While the nonsecretors' invasion often leads to their persistence in pollinator-secretor systems, the model demonstrates situations in which the non-secretors' invasion can drive secretors into extinction, which consequently leads to extinction of the nonsecretors themselves.

Pollinators' mating rendezvous and the evolution of floral advertisement

Successful cross-fertilization in plant species that rely on animal pollinators depends not just on the number of pollinator visits, but also on these visits' duration. Furthermore, in non-deceptive pollination, a visit's duration depends on the magnitude of the reward provided to the pollinator. Accordingly, plants that rely on biotic pollination have to partition their investment in cross-fertilization assurance between attracting pollinator visits - advertisement, and rewarding visitors to assure that the visit is of productive duration. Here we analyze these processes by a combination of optimality methods and game theoretical modeling. Our results indicate that the optimality in such allocation of resources depends on the types of reward offered to the pollinators. More precisely, we show that plants that offer both food reward and mating rendezvous to pollinators will evolve to allocate a higher proportion of their cross-fertilization assurance budget to advertisement than plants that offer only food reward. That is, our results indicate that pollinators' mating habits may play a role in floral evolution.

Uni-directional interaction and plant-pollinator-robber coexistence

A mathematical model for the plant-pollinator-robber interaction is studied to understand the factors leading to the widespread occurrence and stability of such interactions. In the interaction, a flowering plant provides resource for its pollinator and the pollinator has both positive and negative effects on the plant. A nectar robber acts as a plant predator, consuming a common resource with the pollinator, but with a different functional response. Using dynamical systems theory, mechanisms of species coexistence are investigated to show how a robber could invade the plant-pollinator system and persist stably with the pollinator. In addition, circumstances are demonstrated in which the pollinator's positive and negative effects on the plant could determine the robber's invasibility and the three-species coexistence.

Persistence of pollination mutualisms in plant-pollinator-robber systems

Interactions between pollinators, nectar robbers, defensive plants and non-defensive plants are characterized by evolutionary games, where payoffs for the four species are represented by population densities at steady states in the corresponding dynamical systems. The plant-robber system is described by a predator-prey model with the Holling II functional response, while the plant-pollinator system is described by a cooperative model with the Beddington-DeAngelis functional response. By combining dynamics of the models with properties of the evolutionary games, we show mechanisms by which pollination mutualisms could persist in the presence of nectar robbers. The analysis leads to an explanation for persistence of plant-pollinator-robber systems in real situations.