Sharp habitat shifts, evolutionary tipping points and rescue: Quantifying the perilous path of a specialist species toward a refugium in a changing environment.

Specialist species thriving under specific environmental conditions in narrow geographic ranges are widely recognised as heavily threatened by climate deregulation. Many might rely on both their potential to adapt and to disperse toward a refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation. Here, I study the eco-evolutionary dynamics of a sexually reproducing specialist species in a two-patch quantitative genetic model with moving optima. Thanks to a separation of ecological and evolutionary time scales and the phase-line study of the selection gradient, I derive the critical environmental speed for persistence, which reflects how the existence of a refugium impacts extinction patterns and how it relates to the cost of dispersal. Moreover, the analysis provides key insights about the dynamics that arise on the path towards this refugium. I show that after an initial increase of population size, there exists a critical environmental speed above which the species crosses a tipping point, resulting into an abrupt habitat switch. In addition, when selection for local adaptation is strong, this habitat switch passes through an evolutionary "death valley", leading to a phenomenon related to evolutionary rescue, which can promote extinction for lower environmental speeds than the critical one.

Dynamics analysis of an SVEIR epidemic model in a patchy environment.

In this paper, we propose a multi-patch SVEIR epidemic model that incorporates vaccination of both newborns and susceptible populations. We determine the basic reproduction number $ R_{0} $ and prove that the disease-free equilibrium $ P_{0} $ is locally and globally asymptotically stable if $ R_{0} < 1, $ and it is unstable if $ R_{0} > 1. $ Moreover, we show that the disease is uniformly persistent in the population when $ R_{0} > 1. $ Numerical simulations indicate that vaccination strategies can effectively control disease spread in all patches while population migration can either intensify or prevent disease transmission within a patch.

Evolution and Adaptation of Anti-predation Response of Prey in a Two-Patchy Environment.

When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. We formulate a patch model to investigate the aforementioned fear effect which is indirect, in contrast to the predation as a direct effect, of the predator on the prey population. We consider not only cost but also benefit of anti-predation response of the prey, and explore their trade-offs together as well as the impact of the fear effect mediated dispersals of the prey. In the case of constant response level, if there is no dispersal and for some given response functions, the model indicates the existence of an evolutionary stable strategy which is also a convergence stable strategy for the response level; and if there is dispersal, the analysis of the model shows that it will enhance the co-persistence of the prey on both patches. Considering the trait as another variable, we continue to study the evolution of anti-predation strategy for the model with dispersal, which leads to a three-dimensional system of ordinary differential equations. We perform some numerical simulations, which demonstrate global convergence to a positive equilibrium with the response level evolving towards a positive constant level, implying the existence of an optimal anti-predation response level.

Maximizing the total population with logistic growth in a patchy environment.

This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the population of a single species with logistic growth residing in a patchy environment and study the effects of dispersal and spatial heterogeneity of patches on the total population at equilibrium. Our objective is to maximize the total population by redistributing the resources among the patches under the constraint that the total amount of resources is limited. It is shown that the global maximizer can be characterized for any number of patches when the diffusion rate is either sufficiently small or large. To show this, we compute the first variation of the total population with respect to resources in the two patches case. In the case of three or more patches, we compute the asymptotic expansion of all patches by using the Taylor expansion with respect to the diffusion rate. To characterize the shape of the global maximizer, we use a recurrence relation to determine all coefficients of all patches.

Effect of Stressors on the Carrying Capacity of Spatially Distributed Metapopulations.

Stressors such as antibiotics, herbicides, and pollutants are becoming increasingly common in the environment. The effects of stressors on populations are typically studied in homogeneous, nonspatial settings. However, most populations in nature are spatially distributed over environmentally heterogeneous landscapes with spatially restricted dispersal. Little is known about the effects of stressors in these more realistic settings. Here, we combine laboratory experiments with novel mathematical theory to rigorously investigate how a stressor's physiological effect and spatial distribution interact with dispersal to influence population dynamics. We prove mathematically that if a stressor increases the death rate and/or simultaneously decreases the population growth rate and yield, a homogeneous distribution of the stressor leads to a lower total population size than if the same amount of the stressor was heterogeneously distributed. We experimentally test this prediction on spatially distributed populations of budding yeast (Saccharomyces cerevisiae). We find that the antibiotic cycloheximide increases the yeast death rate but reduces the growth rate and yield. Consistent with our mathematical predictions, we observe that a homogeneous spatial distribution of cycloheximide minimizes the total equilibrium size of experimental metapopulations, with the magnitude of the effect depending predictably on the dispersal rate and the geographic pattern of antibiotic heterogeneity. Our study has implications for assessing the population risk posed by pollutants, antibiotics, and global change and for the rational design of strategies for employing toxins to control pathogens and pests.

Death, sex, and sugars: variations in nonstructural carbohydrate concentrations in a sexually plastic tree.

PREMISE: Environmental sex determination (ESD) is a rare sex determination system in which individuals may switch sex expression throughout their lifetimes in response to environmental factors. In sexually stable species, individuals usually bear more female flowers if the plants are larger, have greater access to limiting resources, or are in better condition. Research regarding sexually plastic species with ESD and how resources correlate with sex expression is limited. Furthermore, most research investigates resources at the population level, failing to account for resources available to individuals for growth, maintenance, or reproduction. METHODS: Acer pensylvanicum is a species that is known to switch sex. Using twig samples collected during 2014-2016 in December and May, we analyzed resource status in the form of stored nonstructural carbohydrates (NSCs) and compared this with expressed sex. RESULTS: We found that females had higher sugar concentrations than males. Furthermore, males changing expression to female had higher sugar concentrations during the prior winter than did males remaining male. We found that size was not a key predictor: neither male nor female-flowering individuals increased NSC concentrations with size. Dying female trees had high concentrations of NSCs throughout the dying process and only manifested reduced NSCs once dead. CONCLUSIONS: This is the first study showing significant correlations between NSCs and sex expression in a plant species with ESD. These findings support the hypothesis that sex switching could be a consequence of increased resource availability and that the high female mortality of A. pensylvanicum populations is likely not a direct result of carbon starvation.

Coexistence and extinction for two competing species in patchy environments.

A system of two competing species µ and ν that diffuse over a two-patch environment is investigated. When u-species has smaller birth rate in the first patch and larger birth rate in the second patch than v-species, and the average birth rate for u-species is larger than or equal to v-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas u-species drives v-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of u-species is less than that of v-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of v-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, v-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or u-species that has the greatest birth rate among both species and patches will persist and drive v-species to extinction.

Regulatory mechanisms of group distributions in a gregarious arthropod.

In a patchy environment, how social animals manage conspecific and environmental cues in their choice of habitat is a leading issue for understanding their spatial distribution and their exploitation of resources. Here, we experimentally tested the effects of environmental heterogeneities (artificial shelters) and some of their characteristics (size and fragmentation) on the aggregation process of a common species of terrestrial isopod (Crustacea). One hundred individuals were introduced into three different heterogeneous set-ups and in a homogeneous set-up. In the four set-ups, the populations split into two aggregates: one large (approx. 70 individuals) and one smaller (approx. 20 individuals). These aggregates were not randomly distributed in the arena but were formed diametrically opposite from one another. The similarity of the results among the four set-ups shows that under experimental conditions, the environmental heterogeneities have a low impact on the aggregation dynamics and spatial patterns of the isopod, merely serving to increase the probability of nucleation of the larger aggregation at these points. By contrast, the regulation of aggregate sizes and the regular distribution of groups are signatures of local amplification processes, in agreement with the short-range activator and long-range inhibitor model (scale-dependent feedbacks). In other words, we show how small-scale interactions may govern large-scale spatial patterns. This experimental illustration of spatial self-organization is an important step towards comprehension of the complex game of competition among groups in social species.

Fishing for nutrients in heterogeneous landscapes: modelling plant growth trade-offs in monocultures and mixed communities.

The problem of how best to find and exploit essential resources, the quality and locations of which are unknown, is common throughout biology. For plants, the need to grow an efficient root system so as to acquire patchily distributed soil nutrients is typically complicated by competition between plants, and by the costs of maintaining the root system. Simple mechanistic models for root growth can help elucidate these complications, and here we argue that these models can be usefully informed by models initially developed for foraging fish larvae. Both plant and fish need to efficiently search a spatio-temporally variable environment using simple algorithms involving only local information, and both must perform this task against a backdrop of intra- and inter-specific competition and background mortality. Here we develop these parallels by using simple stochastic models describing the growth and efficiency of four contrasting idealized root growth strategies. We show that plants which grow identically in isolation in homogeneous substrates will typically perform very differently when grown in monocultures, in heterogeneous nutrient landscapes and in mixed-species competition. In particular, our simulations show a consistent result that plants which trade-off rapid growth in favour of a more efficient and durable root system perform better, both on average and in terms of the best performing individuals, than more rapidly growing ephemeral root systems. Moreover, when such slower growing but more efficient plants are grown in competition, the overall community productivity can exceed that of the constituent monocultures. These findings help to disentangle many of the context-dependent behaviours seen in the experimental literature, and may form a basis for future studies at the level of complex population dynamics and life history evolution.

Patch size and distance: modelling habitat structure from the perspective of clonal growth.

BACKGROUND AND AIMS: This study considers the spatial structure of patchy habitats from the perspective of plants that forage for resources by clonal growth. Modelling is used in order to compare two basic strategies, which differ in the response of the plant to a patch boundary. The 'avoiding plant' (A) never grows out of a good (resource-rich) patch into a bad (resource-poor) region, because the parent ramet withdraws its subsidy from the offspring. The 'entering plant' (E) always crosses the boundary, as the offspring is subsidized at the expense of the parent. In addition to these two extreme scenarios, an intermediate mixed strategy (M) will also be tested. The model is used to compare the efficiency of foraging in various habitats in which the proportion of resource-rich areas (p) is varied. METHODS: A stochastic cellular automata (CA) model is developed in which habitat space is represented by a honeycomb lattice. Each cell within the lattice can accommodate a single ramet, and colonization can occur from a parent ramet's cell into six neighbouring cells. The CA consists of two layers: the population layer and the habitat. In the population layer, a cell can be empty or occupied by a ramet; in the habitat layer, a cell can be good (resource-rich) or bad (resource-poor). The habitat layer is constant; the population layer changes over time, according to the birth and death of ramets. KEY RESULTS: Strategies M and E are primarily limited by patch distance, whereas A is more sensitive to patch size. At a critical threshold of the proportion of resource-rich areas, p = 0·5, the mean patch size increases abruptly. Below the threshold, E is more efficient than A, whilst above the threshold the opposite is true. The mixed strategy (M) is more efficient than either of the pure strategies across a broad range of p values. CONCLUSIONS: The model predicts more species/genotypes with the 'entering' strategy, E, in habitats where resource-rich patches are scattered, and more plants with the 'avoiding' strategy, A, in habitats where the connectivity of resource-rich patches is high. The results suggest that the degree of physiological integration between a parent and an offspring ramet is important even across a very short distance because it can strongly influence the efficiency of foraging.