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The most important fact in the field of theoretical ecology and evolutionary biology is the strategy of predation for predators and the avoidance of prey from predator attack. A lot of experimental works suggest that the reduction of prey depends on both direct predation and fear of predation. We explore the impact of fear effect and mutual interference into a three-species food chain model. In this manuscript, we have considered a tri-topic food web model with Beddington-DeAngelis functional response between interacting species, incorporating the reduction of prey and intermediate predator growth because of the fear of intermediate and top predator respectively. We have provided parametric conditions on the existence of biologically feasible equilibria as well as their local and global stability also. We have established conditions of transcritical, saddle-node and Hopf bifurcation in vicinity of different equilibria. Finally, we performed some numerical investigations to justify analytical findings.Mathematics Subject Classification : 39A30, 92D25, 92D50.

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with [Formula: see text] other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n. We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d, namely [Formula: see text] or [Formula: see text] with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution [Formula: see text] (component-wise) to the equilibrium linear equation: [Formula: see text]where [Formula: see text] is the [Formula: see text] vector with components 1 and [Formula: see text] is a large sparse random matrix, accounting for the interactions between species. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Bizeul and Najim (2021).

The concept of fitness is often reduced to a single component, such as the replication rate in a given habitat. For species with multi-step life cycles, this can be an unjustified oversimplification, as every step of the life cycle can contribute to the overall reproductive success in a specific way. In particular, this applies to microbes that spend part of their life cycles associated to a host. In this case, there is a selection pressure not only on the replication rates, but also on the phenotypic traits associated to migrating from the external environment to the host and vice-versa (i.e., the migration rates). Here, we investigate a simple model of a microbial lineage living, replicating, migrating and competing in and between two compartments: a host and an environment. We perform a sensitivity analysis on the overall growth rate to determine the selection gradient experienced by the microbial lineage. We focus on the direction of selection at each point of the phenotypic space, defining an optimal way for the microbial lineage to increase its fitness. We show that microbes can adapt to the two-compartment life cycle through either changes in replication or migration rates, depending on the initial values of the traits, the initial distribution across the two compartments, the intensity of competition, and the time scales involved in the life cycle versus the time scale of adaptation (which determines the adequate probing time to measure fitness). Overall, our model provides a conceptual framework to study the selection on microbes experiencing a host-associated life cycle.

A central goal of ecological research has been to understand the limits on the maximum number of species that can coexist under given constraints. However, we know little about the assembly and disassembly processes under which a community can reach such a maximum number, or whether this number is in fact attainable in practice. This limitation is partly due to the challenge of performing experimental work and partly due to the lack of a formalism under which one can systematically study such processes. Here, we introduce a formalism based on algebraic topology and homology theory to study the space of species coexistence formed by a given pool of species. We show that this space is characterized by ubiquitous discontinuities that we call coexistence holes (that is, empty spaces surrounded by filled space). Using theoretical and experimental systems, we provide direct evidence showing that these coexistence holes do not occur arbitrarily—their diversity is constrained by the internal structure of species interactions and their frequency can be explained by the external factors acting on these systems. Our work suggests that the assembly and disassembly of ecological systems is a discontinuous process that tends to obey regularities. Drawing on theory from the field of mathematics known as topology, the authors characterize sets of species that can or cannot coexist in hypergraphs.

Ecology cannot yet fully explain why so many tree species coexist in natural communities such as tropical forests. A major difficulty is linking individual-level processes to community dynamics. We propose a combination of tree spatial data, spatial statistics and dynamical theory to reveal the relationship between spatial patterns and population-level interaction coefficients and their consequences for multispecies dynamics and coexistence. Here we show that the emerging population-level interaction coefficients have, for a broad range of circumstances, a simpler structure than their individual-level counterparts, which allows for an analytical treatment of equilibrium and stability conditions. Mechanisms such as animal seed dispersal, which result in clustering of recruits that is decoupled from parent locations, lead to a rare-species advantage and coexistence of otherwise neutral competitors. Linking spatial statistics with theories of community dynamics offers new avenues for explaining species coexistence and calls for rethinking community ecology through a spatial lens., Tree spatial data, spatial statistics and dynamical theory reveal the relationship between spatial patterns and population-level interaction coefficients and their consequences for multispecies dynamics and coexistence.

Plasmids are autonomous genetic elements that can be exchanged between microorganisms via horizontal gene transfer (HGT). Despite the central role they play in antibiotic resistance and modern biotechnology, our understanding of plasmids’ natural ecology is limited. Recent experiments have shown that plasmids can spread even when they are a burden to the cell, suggesting that natural plasmids may exist as parasites. Here, we use mathematical modeling to explore the ecology of such parasitic plasmids. We first develop models of single plasmids and find that a plasmid’s population dynamics and optimal infection strategy are strongly determined by the plasmid’s HGT mechanism. We then analyze models of co-infecting plasmids and show that parasitic plasmids are prone to a “tragedy of the commons” in which runaway plasmid invasion severely reduces host fitness. We propose that this tragedy of the commons is averted by selection between competing populations and demonstrate this effect in a metapopulation model. We derive predicted distributions of unique plasmid types in genomes—comparison to the distribution of plasmids in a collection of 17,725 genomes supports a model of parasitic plasmids with positive plasmid–plasmid interactions that ameliorate plasmid fitness costs or promote the invasion of new plasmids.

The enhanced or reduced growth of juvenile masu salmon (Oncorhynchus masou masou) may result from climate changes to their environment and thus impact on the eco-evolutionary dynamics of their life-history choices. Male juveniles with status, i.e., if their body size is larger than a threshold, stay in the stream and become resident males reproducing for multiple years, while those with smaller status, i.e., their body size is below the threshold, migrate to the ocean and return to the stream one year later to reproduce only once. Since juvenile growth is suppressed by the density of resident males, the fraction of resident males may stay in equilibrium or fluctuate wildly over a 2-year period. When the threshold value evolves, the convergence stable strategy may generate either an equilibrium or large fluctuations of male residents. If environmental changes occur faster than the rate of evolutionary adaptation, the eco-evolutionary dynamics exhibit a qualitative shift in the population dynamics. We also investigated the relative assessment models, in which individual life-history choices are made based on the individual’s relative status within the juvenile population. The eco-evolutionary dynamics are very different from the absolute assessment model, demonstrating the importance of understanding the mechanisms of life history choices when predicting the impacts of climate change.

Critical transitions, sudden responses to slow changes in environmental drivers, are inherent in many dynamic processes, prompting a search for early warning signals. We apply this framework to understanding the coffee rust disease, which experienced an unprecedented outbreak in Mesoamerica in 2012–2013, likely a critical transition. Based on monthly infection data from 128 study quadrats in a 45-ha plot in southern Mexico from 2014 to 2020, we find that the persistent seasonal epidemic following the initial outbreak collapses in an evident subsequent critical transition. Characteristic signals of “critical slowing down” precede this collapse and are correlated with reduced rainfall, as expected from climate change, and planting of rust-resistant varieties, an ongoing management intervention. Recoveries from catastrophes may themselves be experienced as a critical transition and managers should consider the larger dynamical landscape for the possibility of subsequent transitions. Early warning signals could therefore be useful when evaluating mitigation effectiveness.

Nonlinear dynamics, where a change in the input is not proportional to a change in the output, are often found throughout nature, for example in biochemical kinetics. Because of the complex suite of interacting abiotic and biotic variables present in ecosystems, animal population dynamics are often thought to be driven in a nonlinear, state-dependent fashion. However, so far these have only been identified in model organisms and some natural systems. Here we show that nonlinear population dynamics are ubiquitous in nature. We use nonlinear forecasting to analyse 747 datasets of 228 species to find that insect population trends were highly nonlinear (74%), followed by mammals (58%), bony fish (49%) and birds (35%). This indicates that linear, equilibrium-based model assumptions may fail at predicting population dynamics across a wide range of animal taxa. We show that faster-reproducing animals are more likely to have nonlinear and high-dimensional dynamics, supporting past ecological theory. Lastly, only a third of time series were predictable beyond two years; therefore, the ability to predict animal population trends using these methods may be limited. Our results suggest that the complex dynamics necessary to cause regime shifts and other transitions may be inherent in a wide variety of animals. Analysing the population dynamics of 228 terrestrial and aquatic species, the authors show that nonlinearity is a ubiquitous feature of animal populations and is more prevalent among fast-reproducing species.

Across the tree of life, populations have evolved the capacity to contend with suboptimal conditions by engaging in dormancy, whereby individuals enter a reversible state of reduced metabolic activity. The resulting seed banks are complex, storing information and imparting memory that gives rise to multi-scale structures and networks spanning collections of cells to entire ecosystems. We outline the fundamental attributes and emergent phenomena associated with dormancy and seed banks, with the vision for a unifying and mathematically based framework that can address problems in the life sciences, ranging from global change to cancer biology., Seed banks are generated when individuals enter a dormant state, a phenomenon that has evolved among diverse taxa, but that is also found in stem cells, brains, and tumors. Here, Lennon et al. synthesize the fundamentals of seed-bank theory and the emergence of complex patterns and dynamics in mathematics and the life sciences.

The successful invasion of a multi-species resident system by mutants has received a great deal of attention in theoretical ecology but less is known about what happens after the successful invasion. Here, in the framework of Lotka–Volterra (LV) systems, we consider the general question where there is one resident phenotype in each species and the evolutionary outcome after invasion remains one phenotype in each species, but these include all the mutant phenotypes. In the first case, called evolutionary substitution, a mutant appears in only one species, the resident phenotype in this species dies out, and the mutant coexists with the original phenotypes of the other species. In the second case, called evolutionary replacement, a mutant appears in each species, all resident phenotypes die out, and the evolutionary outcome is coexistence among all the mutant phenotypes. For general LV systems, we show that dominance of the resident phenotype by the mutant (i.e. the mutant is always more fit) in each species where the mutant appears leads to evolutionary substitution/replacement. However, it is shown by example that, when dominance is weakened to only assuming the average fitness of the mutants is greater than the average for the resident phenotype, the residents may not die out. We also show evolutionary substitution occurs in two-species competitive LV systems when the initial invasion of the resident system (respectively, of the new coexistence system) is successful (respectively, unsuccessful). Moreover, if sequential evolutionary substitution occurs for either order that the two mutant phenotypes appear (called historically independent replacement), then it is shown evolutionary replacement occurs using a generalization of the dominance argument.

Correlation analyses are often included in bioinformatic pipelines as methods for inferring taxon–taxon interactions. In this perspective, we highlight the pitfalls of inferring interactions from covariance and suggest methods, study design considerations, and additional data types for improving high-throughput interaction inferences. We conclude that correlation, even when augmented by other data types, almost never provides reliable information on direct biotic interactions in real-world ecosystems. These bioinformatically inferred associations are useful for reducing the number of potential hypotheses that we might test, but will never preclude the necessity for experimental validation.

An essential topic in theoretical ecology is the extinction of populations subject to demographic stochasticity. Mechanistic models of demographic stochasticity, such as birth-death processes, can be analytically intractable, so are frequently approximated with stochastic differential equations (SDEs). Here, we consider two pitfalls in this type of approximation. First, familiar deterministic models are not always appropriate for use in an SDE. Second, the common practice of starting directly from an SDE without explicitly constructing a mechanistic model leaves the noise term up to the modeler’s discretion. Since the stability of stochastic models depends on the global properties of both the noise and the deterministic model, overly phenomenological deterministic models, or heuristic choices of noise, can lead to models that are unrealistically stable. The goal of this article is to provide an example of how both of these effects can undermine seemingly reasonable models. Following Dennis et al. (Theor Ecol 9:323–335 2016) and Levine and Meerson (PRE, 87:032127 2013), we compare the persistence of stochastic extensions of standard logistic and Allee models. We show that, for common choices of noise, stochastic logistic models become exponentially less extinction prone when a strong Allee effect is introduced. This apparent paradox can be resolved by recognizing that common models of an Allee effect introduce overcompensation that dominates the extinction dynamics, even when the deterministic model is rescaled to account for overcompensation. These problems can be resolved by mechanistic treatment of the deterministic model and the noise.

Maternal risk-management, an extension of r/K selection, is an indispensable tool for understanding the natural selection pressures that shape the evolution of reproduction. Central to the construct of maternal risk-management is its definition of reproductive success as replacement fitness (w = 2), the survival of one breeding daughter to replace the female and one outbreeding son to replace her mate. Here, I apply maternal risk-management as a theoretical framework to explain multiple reproductive adaptations by loggerhead sea turtles nesting on a barrier island off the southern coast of Florida, US, from 1988 to 2004. Extrapolated over a 30-year reproductive span, nesting females averaged 4000–4500 eggs. I show that, rather than “putting all their eggs in one basket,” females divided eggs into 40 clutches of variable size (50–165 eggs). To deposit clutches, females migrated to the barrier island 10–12 times at unpredictable intervals of 2–8 years. Each nesting season, females deposited 1–7 clutches over diversified time intervals at diversified locations on the beach. Despite devastating clutch losses caused by ten catastrophic hurricanes, hundreds of erratic thunderstorms and dozens of predation events during this study, 72% of clutches produced by nesting females on this barrier island were undisturbed—median hatching success for these clutches was an astonishing 92%. I conclude that diversified maternal investments over time and space by nesting females are reproductive adaptations that have successfully offset clutch losses, thus enabling populations of loggerhead females to meet or exceed their reproductive goal of replacement fitness.

Body size is a key life-history trait that influences community assembly by affecting how ecological processes operate at the organism level. However, the extent to which the relative influences of ecological processes mediate the assembly of differentially sized soil organisms is still unknown. Here, we investigate the community assembly of differentially sized soil microorganisms and microfauna using a continental-scale sampling effort combined with a global-scale meta-analysis. Our results reveal a general relationship between organism body size and the stochastic-deterministic balance operating on community assembly. The smallest microorganisms (bacteria) are relatively more influenced by dispersal-based stochastic processes, while larger ones (fungi, protists and nematodes) are more structured by selection-based deterministic processes. This study elucidates a significant and consistent relationship between an organism life-history trait and how distinct ecological processes operate in mediating their respective community assemblages, thus providing a better understanding of the mechanisms supporting soil biodiversity., It is unclear whether body size affects community assembly mechanisms of soil biota. Here, the authors analyse soil microbial and nematode communities sampled along a 4000-km transect in China and global soil microbiome data to show that bacterial assembly is governed by high dispersal, whereas larger taxa are more influenced by deterministic processes.

Empirical studies show that plant-soil feedbacks (PSF) can generate negative density dependent (NDD) recruitment capable of maintaining plant community diversity at landscape scales. However, the observation that common plants often exhibit relatively weaker NDD than rare plants at local scales is difficult to reconcile with the maintenance of overall plant diversity. We develop a spatially explicit simulation model that tracks the community dynamics of microbial mutualists, pathogens, and their plant hosts. We find that net PSF effects vary as a function of both host abundance and key microbial traits (e.g., host affinity) in ways that are compatible with both common plants exhibiting relatively weaker local NDD, while promoting overall species diversity. The model generates a series of testable predictions linking key microbial traits and the relative abundance of host species, to the strength and scale of PSF and overall plant community diversity., Microbial plant-soil feedbacks (PSF) are fundamentally important for plant diversity. The authors present a spatially explicit dynamic model that separates the effects of microbial mutualists and pathogens, thereby presenting a testable mechanistic framework to reconcile previously puzzling observations of the strength and direction of PSF with diversity maintenance.

Ecosystems are composed of complex networks of many species interacting in different ways. While ecologists have long studied food webs of feeding interactions, recent studies increasingly focus on mutualistic networks including plants that exchange food for reproductive services provided by animals such as pollinators. Here, we synthesize both types of consumer-resource interactions to better understand the controversial effects of mutualism on ecosystems at the species, guild, and whole-community levels. We find that consumer-resource mechanisms underlying plant-pollinator mutualisms can increase persistence, productivity, abundance, and temporal stability of both mutualists and non-mutualists in food webs. These effects strongly increase with floral reward productivity and are qualitatively robust to variation in the prevalence of mutualism and pollinators feeding upon resources in addition to rewards. This work advances the ability of mechanistic network theory to synthesize different types of interactions and illustrates how mutualism can enhance the diversity, stability, and function of complex ecosystems., Aside from their pollination function, pollinators consume and are consumed by other members of ecological communities; these relationships could explain the controversial effects of pollinators on ecological networks. Here the authors show that when mutualists such as pollinators are introduced into food webs, they increase ecosystem biodiversity, stability, and function.

Shifting ecosystem disturbance patterns due to climate change (for example, storms, droughts and wildfires) or direct human interference (for example, harvests and nutrient loading) highlight the importance of quantifying and strengthening the resilience of desired ecological regimes. Although existing metrics capture resilience to isolated shocks, gradual parameter changes, and continual noise, quantifying resilience to repeated, discrete disturbance events requires different analytical tools. Here, we introduce a mathematical flow–kick framework that uses dynamical systems tools to quantify resilience to disturbances explicitly in terms of their magnitude and frequency. We identify a boundary between disturbance regimes that cause either escape from, or stabilization within, a basin of attraction. We use the boundary to define resilience metrics tailored to repeated, discrete perturbations. The flow–kick model suggests that the distance-to-threshold resilience metric overestimates resilience in the context of repeated perturbations. It also reveals counterintuitive triggers for regime shifts. These include increasing the periods between disturbance events in proportion to increases to disturbance magnitude, and—in systems with multiple dynamic variables—increasing time periods between disturbances of constant magnitude. Sustainability depends on the resilience of natural, social and engineered systems. This theoretical study quantifies resilience to repeated disturbances, synthesizing understanding of how the sizes of shocks, or ‘kicks’, and recovery, or ‘flows’, contribute to maintaining systems in desirable states.

Climate change research often focuses on trends in the mean and variance. However, analyses of palaeoclimatic and contemporary dynamics reveal that climate memory - as measured for instance by temporal autocorrelation - may also change substantially over time. Here, we show that elevated temporal autocorrelation in climatic variables should be expected to increase the chance of critical transitions in climate-sensitive systems with tipping points. We demonstrate that this prediction is consistent with evidence from forests, coral reefs, poverty traps, violent conflict and ice sheet instability. In each example, the duration of anomalous dry or warm events elevates chances of invoking a critical transition. Understanding the effects of climate variability thus requires research not only on variance, but also on climate memory.

Reservoir-based chemical defense (= reducible defense) is a widespread mechanism to repel predators in many invertebrates. We investigated the influence of macronutrients on the availability and regeneration of defensive secretions and parametrize a theoretical functional response model for reducible defense to predict nutritional effects on predator–prey dynamics. Our modeling approach showed that initially high amounts of defensive secretions provided an effective short-time defense, while higher regeneration rates were favorable under permanent predation pressure. Regeneration rates were correlated to the amount of dietary fat and carbohydrates, indicating an adaptive connection of macronutrients on chemical defense and predatory success—an effect covering two trophic levels. Our results underpin the urgent need to integrate dynamical aspects of chemical defense into the modeling of predator–prey interactions in food webs.

Resilience in ecosystems and resistance to regime shifts has been a major focus in ecological research. How migration and general network dynamics affect the resilience of populations or induce regime shift cascades is a particularly challenging open question in theoretical ecology. We focus on regime shifts in populations with variable-strength Allee effects to demonstrate the effect of migration on resilience in two-population systems with critical transitions. The result is a mathematical model that justifies the assumption that resilience can be averaged across connected populations and suggests several management strategies to either avoid or induce regime shift cascades.

International audience; Although many environments like soils are constantly subjected to invasion by alien microbes, invaders usually fail to succeed, succumbing to the robust diversity often found in nature. So far, only successful invasions have been explored, and it remains unknown to what extent an unsuccessful invasion can impact resident communities. Here we hypothesized that unsuccessful invasions can cause impacts to soil functioning by decreasing the diversity and niche breadth of resident bacterial communities, which could cause shifts to community composition and niche structure\textemdashan effect that is likely exacerbated when diversity is compromised. To examine this question, diversity gradients of soil microbial communities were subjected to invasion by the frequent, yet oft-unsuccessful soil invader, Escherichia coli, and evaluated for changes to diversity, bacterial community composition, niche breadth, and niche structure. Contrary to expectations, diversity and niche breadth increased across treatments upon invasion. Community composition and niche structure were also altered, with shifts of niche structure revealing an escape by the resident community away from the invader's resources. Importantly, the extent of the escape varied in response to the community's diversity, where less diverse communities experienced larger shifts. Thus, although transient and unsuccessful, the invader competed for resources with resident species and caused tangible impacts that modified both the diversity and functioning of resident communities, which can likely generate a legacy effect that influences future invasion attempts.

The mathematics of counting holes reveals hidden paths to species coexistence and the obstacles that block the way.

Predacious syrphid fly species, also known as flower flies or hover flies, are cosmopolitan diptera that play two important ecological roles: predator and pollinator. In decades past, syrphid flies were studied by agricultural researchers due to their larvae’s ability to function as a biological control agent. In recent years, the global decline in both honey bees and various important wild bee species has led ecological researchers to investigate the role of syrphid fly pollination in both natural systems and agriculture. While these two roles have often been considered separately, they are rarely considered together in single studies. Syrphid fly population fluctuations in natural and agricultural systems are understudied, prompting calls for further study into the fundamental drivers of population dynamics of syrphid communities. In order to develop a deeper understanding of the fundamental dynamics of syrphid ecology, the present study offers a community model where both syrphid predation and pollination are incorporated into a single dynamic model. Using populations of predacious syrphid flies, herbivorous insects, and a shared resource flowering plant, the model is used to investigate community dynamics and persistence across different levels of plant reproductive dependence on syrphid pollination. Results indicate distinct levels of community viability across different pollination relationships as well as a tendency toward chaotic dynamics inherent to the trophic interactions of the community.

A large-scale, cross-taxa analysis reveals high nonlinearity and limited long-term predictability in the dynamics of animal populations.

Nature’s complexity is intriguing, but the circumstances determining whether or how order emerges from such complexity remains a matter of extensive research. Using the geographical distributions and food preferences of all terrestrial mammal species with masses >3 kg, we show that large mammals group into feeding guilds (species exploiting similar resources) and that these guilds form trophic structures that vary across biomes globally. We identify five trophic structures closely matching climate variability and named them boreal, temperate, semiarid, seasonal tropical and humid tropical owing to their relative overlap with the distribution of biomes. We also find that human activities simplify trophic structures, generally transitioning them to species-poorer states. Detected transitions include boreal and temperate structures becoming depauperate or seasonal- and humid-tropical becoming semiarid. Whether the observed generalities among trophic structures of large mammals are indicative of patterns across whole food webs is matter for further investigation. The results help refine projections of the effects of environmental change on the trophic structure of large mammals., M.M. was originally funded through 1/SAESCTN/ALENT-07–0224-FEDER-001755 led by MBA. MBA also thanks a Portuguese Foudation for Science and Technology project PTDC/ AAG-MAA/3764/2014. Both are now funded through the Spanish Ministry of Science, Innovation, and Universities (PGC2018–099363-B-I00).

The use of spatially explicit models (SEMs) in ecology has grown enormously in the past two decades. One major advancement has been that fine-scale details of landscapes, and of spatially dependent biological processes, such as dispersal and invasion, can now be simulated with great precision, due to improvements in computer technology. Many areas of modeling have shifted toward a focus on capturing these fine-scale details, to improve mechanistic understanding of ecosystems. However, spatially implicit models (SIMs) have played a dominant role in ecology, and arguments have been made that SIMs, which account for the effects of space without specifying spatial positions, have an advantage of being simpler and more broadly applicable, perhaps contributing more to understanding. We address this debate by comparing SEMs and SIMs in examples from the past few decades of modeling research. We argue that, although SIMs have been the dominant approach in the incorporation of space in theoretical ecology, SEMs have unique advantages for addressing pragmatic questions concerning species populations or communities in specific places, because local conditions, such as spatial heterogeneities, organism behaviors, and other contingencies, produce dynamics and patterns that usually cannot be incorporated into simpler SIMs. SEMs are also able to describe mechanisms at the local scale that can create amplifying positive feedbacks at that scale, creating emergent patterns at larger scales, and therefore are important to basic ecological theory. We review the use of SEMs at the level of populations, interacting populations, food webs, and ecosystems and argue that SEMs are not only essential in pragmatic issues, but must play a role in the understanding of causal relationships on landscapes.

Ecosystem and community ecology have evolved along different pathways, with little overlap. However, to meet societal demands for predicting changes in ecosystem services, the functional and structural view dominating these two branches of ecology, respectively, must be integrated. Biodiversity–ecosystem function research has addressed this integration for two decades, but full integration that makes predictions relevant to practical problems is still lacking. We argue that full integration requires going, in both branches, deeper by taking into account individual organisms and the evolutionary and physico-chemical principles that drive their behavior. Individual-based models are a major tool for this integration. They have matured by using individual-level mechanism to replace the demographic thinking which dominates classical theoretical ecology. Existing individual-based ecosystem models already have proven useful both for theory and application. Still, next-generation individual-based models will increasingly use standardized and re-usable submodels to represent behaviors and mechanisms such as growth, uptake of nutrients, foraging, and home range behavior. The strategy of pattern-oriented modeling then helps make such ecosystem models structurally realistic by developing theory for individual behaviors just detailed enough to reproduce and explain patterns observed at the system level. Next-generation ecosystem scientists should include the individual-based approach in their toolkit and focus on addressing real systems because theory development and solving applied problems go hand-in-hand in individual-based ecology.

The study of spatial pattern formation through diffusion-driven instability of reaction–diffusion models of interacting species has long been one of the fundamental problems in mathematical ecology. The present article is concerned with interacting predator–prey reaction–diffusion model with Beddington–DeAngelis type functional response. The essential conditions for Hopf and Turing bifurcations are derived on the spatial domain. The parameter space for Turing spatial structure is established. Based on the bifurcation analysis, the spatial pattern formation in Turing space through numerical simulations is carried out in order to study the evolution procedure of the proposed model system in the vicinity of coexistence equilibrium point. The consequences of the results obtained reveal that the effects of self- and cross-diffusion play significant role on the steady state spatiotemporal pattern formation of the reaction–diffusion predator–prey model system which concerns the influence of intra-species competition among predators. Finally, ecological implications of the present results obtained are discussed at length towards the end in order to validate the applicability of the model under consideration.

Counter-intuitive responses of population density to changes in parameter values were used by Nilsson and McCann (Theor Ecol 9:59–71, 2016, Theoretical Ecology) to argue for the superiority of a recently proposed measure of interaction strength. They argued that one of these responses (decreasing consumer density in response to increasing per capita resource attack rate) is rarely or never discussed and is distinct from responses to consumer mortality. In fact, there is a long history of work on responses to altered attack rates, and they are linked to responses to mortality because the latter very often produce coupled changes in attack rate. This earlier literature does not support a qualitative difference between the impacts of these two types of parameter change and does not clearly support the desirability of any particular measure of interaction strength.

Classical Lotka-Volterra (LV) competition equation has shown that coexistence of competitive species is only possible when intraspecific competition is stronger than interspecific competition, i.e., the species inhibit their own growth more than the growth of the other species. Note that density effect is assumed to be linear in a classical LV equation. In contrast, in wild populations we can observed that mortality rate often increases when population density is very high, known as crowding effects. Under this perspective, the aggregation models of competitive species have been developed, adding the additional reduction in growth rates at high population densities. This study shows that the coexistence of a few species is promoted. However, an unsolved question is the coexistence of many competitive species often observed in natural communities. Here, we build an LV competition equation with a nonlinear crowding effect. Our results show that under a weak crowding effect, stable coexistence of many species becomes plausible, unlike the previous aggregation model. An analysis indicates that increased mortality rate under high density works as elevated intraspecific competition leading to the coexistence. This may be another mechanism for the coexistence of many competitive species leading high species diversity in nature.

Multiple hypotheses propose an ostensibly disparate array of drivers of food chain length (FCL), with contradictory support from natural settings. Here we posit that the magnitude of vertical energy flux in food webs underlies several drivers of FCL. We show that rising energy flux fuels top-heavy biomass pyramids, promoting omnivory, thereby reducing FCL. We link this theory to commonly evaluated hypotheses for environmental drivers of FCL (productivity, ecosystem size) and demonstrate that effects of these drivers should be context-dependent. We evaluate support for this theory in lake and marine ecosystems and demonstrate that ecosystem size is the most important driver of FCL in low-productivity ecosystems (positive relationship) while productivity is most important in large and high-productivity ecosystems (negative relationship). This work stands in contrast to classical hypotheses, which predict a positive effect of productivity on FCL, and may help reconcile the contradictory nature of published results for drivers of FCL., Multiple environmental drivers of food chain length (FCL) have been proposed, but empirical support has been contradictory. Here the authors argue that the magnitude of vertical energy flux in ecological communities underlies two commonly evaluated drivers of FCL and show that the effects of these two drivers are context-dependent.

The Trivers–Willard theory proposes that the sex ratio of offspring should vary with maternal condition when it has sex‐specific influences on offspring fitness. In particular, mothers in good condition in polygynous and dimorphic species are predicted to produce an excess of sons, whereas mothers in poor condition should do the opposite. Despite the elegance of the theory, support for it has been limited. Here we extend and generalize the Trivers–Willard theory to explain the disparity between predictions and observations of offspring sex ratio. In polygynous species, males typically have higher mortality rates, different age‐specific reproductive schedules and more risk‐prone life history tactics than females; however, these differences are not currently incorporated into the Trivers–Willard theory. Using two‐sex models parameterized with data from free‐living mammal populations with contrasting levels of sex differences in demography, we demonstrate how sex differences in life history traits over the entire lifespan can lead to a wide range of sex allocation tactics, and show that correlations between maternal condition and offspring sex ratio alone are insufficient to conclude that mothers adaptively adjust offspring sex ratio.

A recent result for a reaction-diffusion equation is that a population diffusing at any rate in an environment in which resources vary spatially will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This has so far been proven by Lou for the case in which the reaction term has only one parameter, $$m(x)$$ , varying with spatial location $$x$$ , which serves as both the intrinsic growth rate coefficient and carrying capacity of the population. However, this striking result seems rather limited when applies to real populations. In order to make the model more relevant for ecologists, we consider a logistic reaction term, with two parameters, $$r(x)$$ for intrinsic growth rate, and $$K(x)$$ for carrying capacity. When $$r(x)$$ and $$K(x)$$ are proportional, the logistic equation takes a particularly simple form, and the earlier result still holds. In this paper we have established the result for the more general case of a positive correlation between $$r(x)$$ and $$K(x)$$ when dispersal rate is small. We review natural and laboratory systems to which these results are relevant and discuss the implications of the results to population theory and conservation ecology.

Resilience is an important aspect of the non-linear restoration of disturbed ecosystems. Fenced grassland patches on the northern Tibetan Plateau can be used to examine the resistance and resilience of degraded alpine grasslands to grazing and to a changing climate. To examine the non-linearity of restoration, we used moderate resolution imaging spectroradiometer (MODIS) normalized difference vegetation index (NDVI) as a proxy for productivity during a ten-year restoration by fencing. Degraded alpine grasslands exhibited three restoration trajectories: an equilibrium in meadows, a non-linear increase across steppes, and an abrupt impulse in desert-steppes following a slight increase in productivity. Combined with weather conditions, the ten-year grazing exclusion has successfully enhanced the NDVI on the most degraded steppes, but did not do so efficiently on either meadows or desert-steppes. Warming favors the NDVI enhancement of degraded meadows, but higher temperatures limited the restoration of degraded steppes and desert-steppes. Precipitation is necessary to restore degraded alpine grasslands, but more precipitation might be useless for meadows due to lower temperatures and for desert-steppes due to limitations caused by the small species pool. We suggest that detailed field observations of community compositional changes are necessary to better understand the mechanisms behind such non-linear ecological restorations.

Natural communities are well known to be maintained by many complex processes. Despite this, the practical aspects of studying them often require some simplification, such as the widespread assumption that direct, additive competition captures the important details about how interactions between species impact community diversity. More complex non-additive 'higher-order' interactions are assumed to be negligible or absent. Notably, these assumptions are poorly supported and have major consequences for the accuracy with which patterns of natural diversity are modelled and explained. We present a mathematically simple framework for incorporating biologically meaningful complexity into models of diversity by including non-additive higher-order interactions. We further provide empirical evidence that such higher-order interactions strongly influence species' performance in natural plant communities, with variation in seed production (as a proxy for per capita fitness) explained dramatically better when at least some higher-order interactions are considered. Our study lays the groundwork for a long-overdue shift in how species interactions are used to study the diversity of natural communities.

Evolution of cooperation is still a puzzle in evolutionary and socio biology. Based on the asymmetric interactions in a dominance hierarchy system, a simple theoretical framework is developed to reveal the effect of “induced cooperation” (i.e., the cooperative behavior of subordinate individuals to dominant individuals) on the evolution of cooperation. Extending the classic Prisoner’s Dilemma (PD) game, we define the concept of the defection cost of subordinate individuals to measure the effect of induced cooperation, i.e., a subordinate defector will incur a cost when it plays against a dominant defector. The analysis of the repeated PD game with linear dominance hierarchy and with cyclic dominance hierarchy shows clearly that induced cooperation of subordinate individuals may lead to a population’s full cooperation, and that the coexistence of cooperation and defection in a population with hierarchy dominance is possible. Our results are the first step to develop a new theoretical approach for understanding the emergence of cooperation; namely, that induced cooperation is one of the most important forces driving the evolution of cooperation as pointed out by May (in May R, McLean A (eds) Theoretical ecology 3rd edn., (2007)).

The chaos and its controllability is one of the main objectives of mathematical ecology today. In this paper, we propose a new chaos control technique for a three species predator prey system by supplying additional food to predators. A thorough mathematical analysis in this direction revealed the vital role of quality and quantity of the additional food in the controllability of the predator–prey system. The local stability analysis of boundary and interior equilibrium points of the system are done. We have determined the persistence conditions of the system. Bifurcation analysis of the proposed model is done with respect to quality and quantity of additional food. A Hopf point bifurcation phenomena is introduced to detect periodic behaviour of the system. By varying quality and quantity of additional food one can control the chaotic dynamics of a food chain. We establish that chaotic population dynamics of predator–prey system can be controlled by supplying additional food to predators. This study introduce a new chaos control mechanism in a predator–prey system which has applications in fishery management and biological conservation of prey predator species. This model has a special importance in the systems, such as caves, some small marine islands etc.

Outcomes of interspecific competition, and especially the possibility of coexistence, have been extensively studied in theoretical ecology because of their implications in community assemblages. During the last decades, the influence of different time scales through the local/regional dynamics of animal communities has received an increasing attention. Nevertheless, different time scales involved in interspecific competition can result form other processes than spatial dynamics. Here, we envision and analyze a new theoretical framework that couples a game theory approach for competition with a demographic model. We take advantage of these two time scales to derive a reduced model governing the total densities of the two populations and we study how these two time scales interfere and influence outcomes of species competition. We find that a competition process occurring on a faster time scale than demography yields a ''priority effect'' where the first species introduced outcompetes the other one. We then confirm previous findings stipu- lating that species coexistence is favored by large difference in time scales because the extinction/recolonization process. Our results then highlight that an integration of demographic and competition time scales at both local and regional levels is mandatory to explain communities assemblages and should become a research priority.

Behind pressing scientific questions of sustainability, unexplored areas of theoretical and mathematical knowledge await discovery. A fresh take on the notion of resilience provides a glimpse of what to expect.

When ecological and evolutionary dynamics occur on similar timescales, otherwise inaccessible stable and diverse communities can spontaneously assemble.

Several recent theoretical studies develop tools to predict species diversity in large model ecosystems, setting a new benchmark for understanding the mechanism of species coexistence in natural ecosystems.

Classical theory shows that large communities are destabilized by random interactions among species pairs, creating an upper bound on ecosystem diversity. However, species interactions often occur in high-order combinations, whereby the interaction between two species is modulated by one or more other species. Here, by simulating the dynamics of communities with random interactions, we find that the classical relationship between diversity and stability is inverted for high-order interactions. More specifically, while a community becomes more sensitive to pairwise interactions as its number of species increases, its sensitivity to three-way interactions remains unchanged, and its sensitivity to four-way interactions actually decreases. Therefore, while pairwise interactions lead to sensitivity to the addition of species, four-way interactions lead to sensitivity to species removal, and their combination creates both a lower and an upper bound on the number of species. These findings highlight the importance of high-order species interactions in determining the diversity of natural ecosystems., A major challenge of theoretical ecology is explaining the stable coexistence of diverse multi-species communities. Here, the authors show that when the interactions among two species depend on the presence of other species, the diversity of the community becomes a necessity rather than an obstacle for its stability.

The question of long term survival of species in a model governed by a system of four autonomous differential equations is considered. The criteria used is the ecologically realistic one of permanence that is populations with all initial values positive must remain positive, greater than some fixed positive numbers. We propose and analyse a four species mathematical model in which mutualism occurs with prey. The present work is in continuation of the work of Rai and Singh (J Int Acad Phys Sci 12:151–172, 2008), in which we have discussed stability analysis along with boundedness of solutions of initial value problems. Here we will discuss conditions for permanence of all the species with special reference to reversal of outcome due to presence of mutualist in a competitive ecosystem.

Forty years ago, Robert May questioned a central belief in ecology by proving that sufficiently large or complex ecological networks have probability of persisting close to zero. To prove this point, he analyzed large networks in which species interact at random. However, in natural systems pairs of species have well-defined interactions (e.g., predator-prey, mutualistic or competitive). Here we extend May's results to these relationships and find remarkable differences between predator-prey interactions, which increase stability, and mutualistic and competitive, which are destabilizing. We provide analytic stability criteria for all cases. These results have broad applicability in ecology. For example, we show that, surprisingly, the probability of stability for predator-prey networks is decreased when we impose realistic food web structure or we introduce a large preponderance of weak interactions. Similarly, stability is negatively impacted by nestedness in bipartite mutualistic networks., Comment: 12 pages 7 figures 1 table

Everyone knows the logistic model of population growth. Populations cannot grow indefinitely; hence the per capita growth rate must decrease as the population density N increases; the simplest decreasing function is linear; and so we have d N dt = [r(1 − N/K )]N , where r is the intrinsic growth rate and K is called the carrying capacity of the environment. Conscientious textbooks will add in parentheses, “(assuming r > 0)”. Indeed, with r < 0, the equation predicts unbounded growth, a biological nonsense, for populations whose initial size exceeds K . But an experimentalist is free to choose any positive initial density, and r < 0 is certainly not nonsense: r < 0 holds when the death rate exceeds the birth rate even at low population densities, as for an unfit mutant or a species outside its region of viability. This shows that we cannot meaningfully vary r and K independently. But how they should be connected is unclear: The information is simply not in the equation. A population is an ensemble of individuals, and its behaviour is ultimately a consequence of the behaviour of the individuals. In order to predict population behaviour, we should first describe the behaviour of the individuals and then use it to derive a model of the population. When the logistic model is derived in this way, it is clear that K is not an externally fixed parameter (the “carrying capacity of the environment”).