The stability of ecosystems: a brief overview of the paradox of enrichment

The effects of inducible defenses on population stability in Paramecium aurelia

Predator-prey dynamics have been studied across many different systems over the past 80 years. The outcomes of this past research have yielded useful theoretical and empirical models of predator-prey systems. However, what stabilizes predator-prey dynamics is often debated and not well understood. One proposed stabilizing mechanism is that the inducible defenses of prey decrease predation risk by creating a portion of the prey population that is invulnerable to predation, leading to a reduction in trophic interaction strength. We investigated the potential stabilizing effects of inducible morphological defenses in the protozoan, Paramecium aurelia, across a range of nutrient concentrations to better understand a potential stabilizing mechanism of systems under nutrient enrichment (Paradox of Enrichment). Using P. aurelia clones that differ in their ability to induce defenses, we found that the most susceptible clone that does not express any known inducible defense showed reduced survival along a gradient of increasing nutrient concentrations. Clones expressing either inducible or permanent morphological defenses (increasing body width in response to predation threat) were not significantly affected by increasing nutrients demonstrating a potential benefit of these defenses. However, when evaluating population stability (coefficient of variation) rather than survival, we found a stabilizing effect of increasing nutrients on all P. aurelia populations. Our results demonstrate varied effects of increasing nutrients on population stability depending on the level of defense expression and stability metric used. Our results reinforce that choice of stability metric can alter conclusions about population stability and persistence, highlighting the need to adopt multiple metrics and approaches.

Assembly Graph as the Rosetta Stone of Ecological Assembly

Ecological assembly-the process of ecological community formation through species introductions-has recently seen exciting theoretical advancements across dynamical, informational, and probabilistic approaches. However, these theories often remain inaccessible to non-theoreticians, and they lack a unifying lens. Here, I introduce the assembly graph as an integrative tool to connect these emerging theories. The assembly graph visually represents assembly dynamics, where nodes symbolise species combinations and edges represent transitions driven by species introductions. Through the lens of assembly graphs, I review how ecological processes reduce uncertainty in random species arrivals (informational approach), identify graphical properties that guarantee species coexistence and examine how the class of dynamical models constrain the topology of assembly graphs (dynamical approach), and quantify transition probabilities with incomplete information (probabilistic approach). To facilitate empirical testing, I also review methods to decompose complex assembly graphs into smaller, measurable components, as well as computational tools for deriving empirical assembly graphs. In sum, this math-light review of theoretical progress aims to catalyse empirical research towards a predictive understanding of ecological assembly.

How Can Materials Chemists Contribute to Food Supply Security in the Age of AI and Robotics?

Global factors are compromising the security of the food supply. Many of us in our community might be unsure about how to help with our seemingly inapplicable expertise. In this piece, I describe how materials chemistry and materials chemists will be essential in addressing this problem, but not only in the way we might think. There is a gigantic and largely unexplored opportunity for real societal impact if we accept a broader view of what academic materials chemistry can be.

On formal limitations of causal ecological networks

Causal multivariate time-series analysis, combined with network theory, provide a powerful tool for studying complex ecological interactions. However, these methods have limitations often underestimated when used in graphical modelling of ecological systems. In this opinion article, I examine the relationship between formal logic methods used to describe causal networks and their inherent statistical and epistemological limitations. I argue that while these methods offer valuable insights, they are restricted by axiomatic assumptions, statistical constraints and the incompleteness of our knowledge. To prove that, I first consider causal networks as formal systems, define causality and formalize their axioms in terms of modal logic and use ecological counterexamples to question the axioms. I also highlight the statistical limitations when using multivariate time-series analysis and Granger causality to develop ecological networks, including the potential for spurious correlations among other data characteristics. Finally, I draw upon Gödel's incompleteness theorems to highlight the inherent limits of fully understanding complex networks as formal systems and conclude that causal ecological networks are subject to initial rules and data characteristics and, as any formal system, will never fully capture the intricate complexities of the systems they represent. This article is part of the theme issue 'Connected interactions: enriching food web research by spatial and social interactions'.

How do productivity gradient and diffusion shape patterns in a plant-herbivore grazing system?

Natural systems show heterogeneous patchy distributions of vegetation over large landscapes. Reaction-diffusion systems can demonstrate such heterogeneity of species distributions. Here, we analyse a reaction-diffusion model of plant-herbivore interactions in two-dimensional space to illustrate non-homogeneous distributions of plants and herbivores. The non-spatial system shows bottom-up control, where herbivore density is low under low and high primary productivity but increased at intermediate productivity. In addition, the non-spatial system provides bistability between a dense vegetation state devoid of herbivores and a coexisting state of plants and herbivores. In the spatiotemporal model, we give analytical conditions of occurring diffusion-driven (Turing) instability, where a novel point in our model is the relative dispersal of herbivores, which represents the movement of herbivores from a higher to a lower vegetation state in addition to the self-diffusion of both species. It is shown that heterogeneity in the population distribution does not occur if the relative dispersal of herbivores is low, but it appears in the opposite case. Due to bistability in the underlying non-spatial system, the spatiotemporal model produces initial value-dependent patterns. The two initial values make different patterns despite having the same primary productivity and relative dispersal rate. As productivity increases with a given relative herbivore dispersal, pattern transition occurs from a blend of stripes and spots of low vegetation state to a predominantly low-density vegetation state with smaller patches of densely vegetated states with one initial value. On the contrary, a discernible change in vegetation patterns from cold spots in the dense vegetation to hot stripes in the primarily low-vegetated state is noticed under the other initial population value. Furthermore, the population distributions of plants and herbivores in the entire domain after a long period are heterogeneous for both initial values, provided the relative herbivore dispersal is substantial. We estimated mean population densities to observe species fitness in the whole domain under variable productivity. When productivity is high, the mean population density of plants may go up or down, depending on the herbivore's relative dispersal rate. In contrast to the bottom-up control dynamics of the non-spatial system, the system exhibits a top-down control under high relative dispersal, where the herbivore regulates vegetation growth under high productivity. On the other hand, herbivores are extinct under high productivity if the relative dispersal is low.

Modelling dynamics between free-living amoebae and bacteria

Free-living amoebae (FLA) serve as hosts for a variety of endosymbionts, which are microorganisms that reside and multiply within the FLA. Some of these endosymbionts pose a pathogenic threat to humans, animals, or both. The symbiotic relationship with FLA not only offers these microorganisms protection but also enhances their survival outside their hosts and assists in their dispersal across diverse habitats, thereby escalating disease transmission. This review is intended to offer an exhaustive overview of the existing mathematical models that have been applied to understand the dynamics of FLA, especially concerning their interactions with bacteria. An extensive literature review was conducted across Google Scholar, PubMed, and Scopus databases to identify mathematical models that describe the dynamics of interactions between FLA and bacteria, as published in peer-reviewed scientific journals. The literature search revealed several FLA-bacteria model systems, including Pseudomonas aeruginosa, Pasteurella multocida, and Legionella spp. Although the published mathematical models account for significant system dynamics such as predator-prey relationships and non-linear growth rates, they generally overlook spatial and temporal heterogeneity in environmental conditions, such as temperature, and population diversity. Future mathematical models will need to incorporate these factors to enhance our understanding of FLA-bacteria dynamics and to provide valuable insights for future risk assessment and disease control measures.

The Demographic-Wealth model for cliodynamics

Cliodynamics is a still a relatively new research area with the purpose of investigating and modelling historical processes. One of its first important mathematical models was proposed by Turchin and called "Demographic-Fiscal Model" (DFM). This DFM was one of the first and is one of a few models that link population with state dynamics. In this work, we propose a possible alternative to the classical Turchin DFM, which contributes to further model development and comparison essential for the field of cliodynamics. Our "Demographic-Wealth Model" (DWM) aims to also model link between population and state dynamics but makes different modelling assumptions, particularly about the type of possible taxation. As an important contribution, we employ tools from nonlinear dynamics, e.g., existence theory for periodic orbits as well as analytical and numerical bifurcation analysis, to analyze the DWM. We believe that these tools can also be helpful for many other current and future models in cliodynamics. One particular focus of our analysis is the occurrence of Hopf bifurcations. Therefore, a detailed analysis is developed regarding equilibria and their possible bifurcations. Especially noticeable is the behavior of the so-called coexistence point. While changing different parameters, a variety of Hopf bifurcations occur. In addition, it is indicated, what role Hopf bifurcations may play in the interplay between population and state dynamics. There are critical values of different parameters that yield periodic behavior and limit cycles when exceeded, similar to the "paradox of enrichment" known in ecology. This means that the DWM provides one possible avenue setup to explain in a simple format the existence of secular cycles, which have been observed in historical data. In summary, our model aims to balance simplicity, linking to the underlying processes and the goal to represent secular cycles.

How realistic features affect the stability of an Arctic marine food web model

Rapid sea-ice decline and warmer waters are threatening the stability of Arctic ecosystems and potentially forcing their restructuring. Mathematical models that support observational evidence are becoming increasingly important. We develop a food web model for the Southern Beaufort Sea based on species with high ecological significance. Generalized modeling is applied to study the impact of realistic characteristics on food web stability; a powerful method that provides a linear stability analysis for systems with uncertainty in data and underlying physical processes. We find that including predator-specific foraging traits, weighted predator-prey interactions, and habitat constraints increase food-web stability. The absence of a fierce top predator (killer whale, polar bear, etc.) also significantly increases the portion of stable webs. Adding ecosystem background noise in terms of a collective impact of latent, minor ecosystem members shows a peak in stability at an optimum, relatively small background pressure. These results indicate that refining models with more realistic detail to account for the complexity of the ecological system may be key to bridge the gap between empirical observations and model predictions in ecosystem stability.

Cyanosphere Dynamic During Dolichospermum Bloom: Potential Roles in Cyanobacterial Proliferation

Under the effect of global change, management of cyanobacterial proliferation becomes increasingly pressing. Given the importance of interactions within microbial communities in aquatic ecosystems, a handful of studies explored the potential relations between cyanobacteria and their associated bacterial community (i.e., cyanosphere). Yet, most of them specifically focused on the ubiquitous cyanobacteria Microcystis, overlooking other genera. Here, based on 16s rDNA metabarcoding analysis, we confirmed the presence of cyanosphere representing up to 30% of the total bacterial community diversity, during bloom episode of another preponderant cyanobacterial genus, Dolichospermum. Moreover, we highlighted a temporal dynamic of this cyanosphere. A sPLS-DA model permits to discriminate three important dates and 220 OTUs. With their affiliations, we were able to show how these variations potentially imply a turnover in ecological functions depending on bloom phases. Although more studies are necessary to quantify the impacts of these variations, we argue that cyanosphere can have an important, yet underestimated, role in the modulation of cyanobacterial blooms.

Siderophore-mediated iron partition promotes dynamical coexistence between cooperators and cheaters

Microbes shape their habitats by consuming resources and producing a diverse array of chemicals that can serve as public goods. Despite the risk of exploitation by cheaters, genes encoding sharable molecules like siderophores are widely found in nature, prompting investigations into the mechanisms that allow producers to resist invasion by cheaters. In this work, we presented the chemostat-typed "resource partition model" to demonstrate that dividing the iron resource between private and public siderophores can promote stable or dynamic coexistence between producers and cheaters in a well-mixed environment. Moreover, our analysis shows that when microbes not only consume but also produce resources, chemical innovation leads to stability criteria that differ from those of classical consumer resource models, resulting in more complex dynamics. Our work sheds light on the role of chemical innovations in microbial communities and the potential for resource partition to facilitate dynamical coexistence between cooperative and cheating organisms.

Combining generalized modeling and specific modeling in the analysis of ecological networks

The complexity of real food webs involves uncertainty in data and in underlying ecological processes, and modeling approaches deal with these challenges differently. Generalized modeling provides a linear stability analysis without narrow specification of all processes, and conventional dynamical systems models approximate functional forms to discuss trajectories in phase space. This study compares results and ecological interpretations from both methods in four-species ecological networks at steady state. We find that a specific (dynamical systems) model only provides a subset of stability data from the generalized model, which spans many plausible dynamic scenarios, allowing for conflicting results. Nevertheless, both approaches reveal that fixed points become stable when nutrient flows to predators are fettered and even more when the basal growth rate approaches a maximum. The specific model identifies a distinct ecosystem response to bottom-up forcing, the enrichment of lower trophic levels. Enrichment stabilizes a fixed point when basal species are in a resource-deprived environment but destabilizes it if resources become more abundant. The generalized model provides less specific information since infinitely many paths of enrichment are hypothetical. Nevertheless, generalized modeling of ecological systems is a powerful technique that enables a meta analysis of these uncertain complex systems.

Evolution of prudent predation in complex food webs

Prudent predators catch sufficient prey to sustain their populations but not as much as to undermine their populations' survival. The idea that predators evolve to be prudent has been dismissed in the 1970s, but the arguments invoked then are untenable in the light of modern evolution theory. The evolution of prudent predation has repeatedly been demonstrated in two-species predator-prey metacommunity models. However, the vigorous population fluctuations that these models predict are not widely observed. Here we show that in complex model food webs prudent predation evolves as a result of consumer-mediated ('apparent') competitive exclusion of resources, which disadvantages aggressive consumers and does not generate such fluctuations. We make testable predictions for empirical signatures of this mechanism and its outcomes. Then we discuss how these predictions are borne out across freshwater, marine and terrestrial ecosystems. Demonstrating explanatory power of evolved prudent predation well beyond the question of predator-prey coexistence, the predicted signatures explain unexpected declines of invasive alien species, the shape of stock-recruitment relations of fish, and the clearance rates of pelagic consumers across the latitudinal gradient and 15 orders of magnitude in body mass. Specific research to further test this theory is proposed.

A phytoplankton-zooplankton-fish model with chaos control: In the presence of fear effect and an additional food

The interplay of phytoplankton, zooplankton, and fish is one of the most important aspects of the aquatic environment. In this paper, we propose to explore the dynamics of a phytoplankton-zooplankton-fish system, with fear-induced birth rate reduction in the middle predator by the top predator and an additional food source for the top predator fish. Phytoplankton-zooplankton and zooplankton-fish interactions are handled using Holling type IV and II responses, respectively. First, we prove the well-posedness of the system, followed by results related to the existence of possible equilibrium points. Conditions under which a different number of interior equilibria exist are also derived here. We also show this existence numerically by varying the intrinsic growth rate of phytoplankton species, which demonstrates the model's vibrant nature from a mathematical point of view. Furthermore, we performed the local and global stability analysis around the above equilibrium points, and the transversality conditions for the occurrence of Hopf bifurcations and transcritical bifurcations are established. We observe numerically that for low levels of fear, the system behaves chaotically, and as we increase the fear parameter, the solution approaches a stable equilibrium by the route of period-halving. The chaotic behavior of the system at low levels of fear can also be controlled by increasing the quality of additional food. To corroborate our findings, we constructed several phase portraits, time-series graphs, and one- and two-parametric bifurcation diagrams. The computation of the largest Lyapunov exponent and a sketch of Poincaré maps verify the chaotic character of the proposed system. On varying the parametric values, the system exhibits phenomena like multistability and the enrichment paradox, which are the basic qualities of non-linear models. Thus, the current study can also help ecologists to estimate the parameters to study and obtain such important findings related to non-linear PZF systems. Therefore, from a biological and mathematical perspective, the analysis and the corresponding results of this article appear to be rich and interesting.

Effective Resource Competition Model for Species Coexistence

Local coexistence of species in large ecosystems is traditionally explained within the broad framework of niche theory. However, its rationale hardly justifies rich biodiversity observed in nearly homogeneous environments. Here we consider a consumer-resource model in which a coarse-graining procedure accounts for a variety of ecological mechanisms and leads to effective spatial effects which favor species coexistence. Herein, we provide conditions for several species to live in an environment with very few resources. In fact, the model displays two different phases depending on whether the number of surviving species is larger or smaller than the number of resources. We obtain conditions whereby a species can successfully colonize a pool of coexisting species. Finally, we analytically compute the distribution of the population sizes of coexisting species. Numerical simulations as well as empirical distributions of population sizes support our analytical findings.

Phase tipping: how cyclic ecosystems respond to contemporary climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

Phenotypic plasticity as a cause and consequence of population dynamics

Predicting complex species-environment interactions is crucial for guiding conservation and mitigation strategies in a dynamically changing world. Phenotypic plasticity is a mechanism of trait variation that determines how individuals and populations adapt to changing and novel environments. For individuals, the effects of phenotypic plasticity can be quantified by measuring environment-trait relationships, but it is often difficult to predict how phenotypic plasticity affects populations. The assumption that environment-trait relationships validated for individuals indicate how populations respond to environmental change is commonly made without sufficient justification. Here we derive a novel general mathematical framework linking trait variation due to phenotypic plasticity to population dynamics. Applying the framework to the classical example of Nicholson's blowflies, we show how seemingly sensible predictions made from environment-trait relationships do not generalise to population responses. As a consequence, trait-based analyses that do not incorporate population feedbacks risk mischaracterising the effect of environmental change on populations.

Modelling the spatiotemporal complexity of interactions between pathogenic bacteria and a phage with a temperature-dependent life cycle switch

We apply mathematical modelling to explore bacteria-phage interaction mediated by condition-dependent lysogeny, where the type of the phage infection cycle (lytic or lysogenic) is determined by the ambient temperature. In a natural environment, daily and seasonal variations of the temperature cause a frequent switch between the two infection scenarios, making the bacteria-phage interaction with condition-dependent lysogeny highly complex. As a case study, we explore the natural control of the pathogenic bacteria Burkholderia pseudomallei by its dominant phage. B. pseudomallei is the causative agent of melioidosis, which is among the most fatal diseases in Southeast Asia and across the world. We assess the spatial aspect of B. pseudomallei-phage interactions in soil, which has been so far overlooked in the literature, using the reaction-diffusion PDE-based framework with external forcing through daily and seasonal parameter variation. Through extensive computer simulations for realistic biological parameters, we obtain results suggesting that phages may regulate B. pseudomallei numbers across seasons in endemic areas, and that the abundance of highly pathogenic phage-free bacteria shows a clear annual cycle. The model predicts particularly dangerous soil layers characterised by high pathogen densities. Our findings can potentially help refine melioidosis prevention and monitoring practices.

Stability of generalized ecological-network models

The stability of ecological networks of varying topologies and predator-prey relationships is explored by applying the concept of generalized modeling. The effects of omnivory, complexity, enrichment, number of top predators, and predatory response are discussed. The degree of omnivory plays a large role in governing web stability at steady state. Complexity as measured from connectance and network size is not a perfect indicator of stability; large, highly connected webs can be just as stable as smaller, less connected ones. Learning behavior as expressed in Holling's type III predatory response is stabilizing for food webs and provides exceptions to the paradox of enrichment for some topologies.

Exploring the role of spatial and stoichiometric heterogeneity in the top-down control in eutrophic planktonic ecosystems

Understanding the impact of eutrophication on the dynamics of aquatic food webs, remains a long-term challenge in ecology. Mathematical models generally predict the destabilisation of such webs, under increasing eutrophication levels, with large oscillations of species densities that can result in their extinction. This is at odds with a number of ecological observations that show stable dynamics even for high nutrient loads. The apparent discrepancy between theory and observations is known as the Rosenzweig's 'paradox of enrichment' and various solutions to the problem have been proposed over the years. In this study, we explore the stabilisation of dynamics of a tri-trophic plankton model in a eutrophic environment which occurs as a result of interplay of space heterogeneity, ecological stoichiometry, and food taxis of predators. We build a variety of models of increasing complexity, to explore various scenarios of phytoplankton growth, zooplankton food-dependent vertical movement, and different stoichiometric limitations of zooplankton. We show that the synergy among the vertical gradient in phytoplankton growth, phytoplankton structuring in terms of their stoichiometric ratio, and food-dependent vertical movement of zooplankton, would result in a postponing of destabilisation of eutrophic systems as compared to a well-mixed system. Our approach reveals a high complexity of the bifurcation structure of the system when key model parameters, such as the degree of eutrophication and light shading, are varied. We find coexistence of limit cycles and stable equilibria and that the possibility of multiple attractors in the system can result in hysteresis phenomena when the nutrient load is manipulated. These results are relevant and should be taken into account in lake restoration programs.

Revealing Evolutionarily Optimal Strategies in Self-Reproducing Systems via a New Computational Approach

Modelling the evolution of complex life history traits and behavioural patterns observed in the natural world is a challenging task. Here, we develop a novel computational method to obtain evolutionarily optimal life history traits/behavioural patterns in population models with a strong inheritance. The new method is based on the reconstruction of evolutionary fitness using underlying equations for population dynamics and it can be applied to self-reproducing systems (including complicated age-structured models), where fitness does not depend on initial conditions, however, it can be extended to some frequency-dependent cases. The technique provides us with a tool to efficiently explore both scalar-valued and function-valued traits with any required accuracy. Moreover, the method can be implemented even in the case where we ignore the underlying model equations and only have population dynamics time series. As a meaningful ecological case study, we explore optimal strategies of diel vertical migration (DVM) of herbivorous zooplankton in the vertical water column which is a widespread phenomenon in both oceans and lakes, generally considered to be the largest synchronised movement of biomass on Earth. We reveal optimal trajectories of daily vertical motion of zooplankton grazers in the water column depending on the presence of food and predators. Unlike previous studies, we explore both scenarios of DVM with static and dynamic predators. We find that the optimal pattern of DVM drastically changes in the presence of dynamic predation. Namely, with an increase in the amount of food available for zooplankton grazers, the amplitude of DVM progressively increases, whereas for static predators DVM would abruptly cease.

Multi-year dynamics of fine-scale marine cyanobacterial populations are more strongly explained by phage interactions than abiotic, bottom-up factors

Currently defined ecotypes in marine cyanobacteria Prochlorococcus and Synechococcus likely contain subpopulations that themselves are ecologically distinct. We developed and applied high-throughput sequencing for the 16S-23S rRNA internally transcribed spacer (ITS) to examine ecotype and fine-scale genotypic community dynamics for monthly surface water samples spanning 5 years at the San Pedro Ocean Time-series site. Ecotype-level structure displayed regular seasonal patterns including succession, consistent with strong forcing by seasonally varying abiotic parameters (e.g. temperature, nutrients, light). We identified tens to thousands of amplicon sequence variants (ASVs) within ecotypes, many of which exhibited distinct patterns over time, suggesting ecologically distinct populations within ecotypes. Community structure within some ecotypes exhibited regular, seasonal patterns, but not for others, indicating other more irregular processes such as phage interactions are important. Network analysis including T4-like phage genotypic data revealed distinct viral variants correlated with different groups of cyanobacterial ASVs including time-lagged predator-prey relationships. Variation partitioning analysis indicated that phage community structure more strongly explains cyanobacterial community structure at the ASV level than the abiotic environmental factors. These results support a hierarchical model whereby abiotic environmental factors more strongly shape niche partitioning at the broader ecotype level while phage interactions are more important in shaping community structure of fine-scale variants within ecotypes.

Hybrid approach to modeling spatial dynamics of systems with generalist predators

We consider hybrid spatial modeling approaches for ecological systems with a generalist predator utilizing a prey and either a second prey or an allochthonous resource. While spatial dispersion of populations is often modeled via stepping-stone (discrete spatial patches) or continuum (one connected spatial domain) formulations, we shall be interested in hybrid approaches which we use to reduce the dimension of certain components of the spatial domain, obtaining either a continuum model of varying spatial dimensions, or a mixed stepping-stone-continuum model. This approach results in models consisting of partial differential equations for some of the species which are coupled via reactive boundary conditions to lower dimensional partial differential equations or ordinary differential equations for the other species. In order to demonstrate the use of this approach, we consider two case studies. In the first case study, we consider a one-predator two-prey interaction between beavers, wolves and white-tailed deer in Voyageurs National Park. In the second case study, we consider predator-prey-allochthonous resource interactions between bears, berries and salmon on Kodiak Island. For each case study, we compare the results from the hybrid modeling approach with corresponding stepping-stone and continuum model results, highlighting benefits and limitations of the method. In some cases, we find that the hybrid modeling approach allows for solutions which are easier to simulate (akin to stepping-stone models) while maintaining seemingly more realistic spatial dynamics (akin to full continuum models).

Historical contingency and productivity effects on food-chain length

Food-chain length (FCL) is a fundamental ecosystem attribute, integrating information on both food web composition and ecosystem processes. It remains untested whether FCL also reflects the history of community assembly known to affect community composition and ecosystem functioning. Here, we performed microcosm experiments with a copepod (top predator), two ciliate species (intermediate consumers), and bacteria (producers), and modified the sequence of species introduction into the microcosm at four productivity levels to jointly test the effects of historical contingency and productivity on FCL. FCL increased when the top predator was introduced last; thus, the trophic position of the copepod reflected assembly history. A shorter FCL occurred at the highest productivity level, probably because the predator switched to feeding at the lower trophic levels because of the abundant basal resource. Thus, we present empirical evidence that FCL was determined by historical contingency, likely caused by priority effects, and by productivity.

Zooplankton selectivity and nutritional value of phytoplankton influences a rich variety of dynamics in a plankton population model

Mathematical modeling may be an excellent tool to analyze and explain complex biological phenomena. In this paper, we use a mathematical model to reveal various interesting dynamical features of phytoplankton-zooplankton interaction and attempt to explain the reason for contrasting dynamics shown by different laboratory and field experiments. Our study shows that the phytoplankton-zooplankton interaction in a pelagic system is very complex and the plankton dynamics, including the bloom phenomenon, strongly depends on the selective predation of zooplankton and the nutritional value of phytoplankton. The study supports the existing hypothesis that decoupling at the plant-animal interface may occur due to strong fish predation on zooplankton. In addition, we argue that decoupling of the food chain may also occur under low to intermediate nutrient inflow if zooplankton feeds on phytoplankton having lower nutritional value. It is also shown that nutrient enrichment can destabilize an otherwise stable system if zooplankton feeds on highly nutritious prey, but unable to destabilize the system if zooplankton feeds on low-nutritious prey. This may be one possible explanation to the longstanding question: Why do some experiments show the paradox of enrichment and others do not?

EXPLORING THE ROLE OF VERTICAL HETEROGENEITY IN THE STABILIZATION OF PLANKTONIC ECOSYSTEMS UNDER EUTROPHICATION

Understanding plankton dynamics in marine and lake ecosystems under eutrophication is currently a hot topic in the literature. Simple theoretical models predict appearance of large amplitude oscillations of species densities in nutrient-rich waters; however, such predictions do not always correspond to field observations. Recent models taking into account heterogeneity of the growth rate of phytoplankton and active food-searching behavior of zooplankton demonstrated that grazers can efficiently control phytoplankton densities at low values even for a high nutrient stock. In this paper, we extend the previous modeling findings on the role of fast-moving plankton grazers by exploring a more realistic case where the limiting nutrient is a dynamical variable. Thus, the growth of phytoplankton across the water column depends on both light attenuation and dynamical depletion of nutrients. We also consider a more realistic scenario of a depth-dependent vertical turbulent diffusion. Most of the previous results on stabilization of planktonic ecosystems still hold; however, some alternative mechanisms of bloom suppression can also be possible. In particular, we demonstrate that the foraging of zooplankton according to the ideal free distribution (IFD) of food (which was previously considered to be a crucial condition for stabilization) may be less stabilizing than random foraging of zooplankton. We also show that stable top-down control in the ecosystem would be highly dependent on values of vertical diffusion and on the nutrient concentration in deep layers.

Optimal harvesting of a prey–predator model with variable carrying capacity

This work deals with a prey–predator model in an environment where the carrying capacities are assumed to be variable with time and one species feeds upon the other. Independent harvesting efforts are applied in either species and asymmetrical intraguild predation occurs. A common resource is consumed by two competing species and at the same time predator also consumes the prey. At first we discuss the model under constant carrying capacity and make the conclusion that no limit cycle exists in this case. Then we discuss the model without intraspecific competition. Our main concern is to cover the above mentioned two cases together, i.e. the model with variable carrying capacity and intraspecific competition. We determine the steady states and examine the dynamical behavior. We also analyze the local and global stability of the interior equilibrium by Routh–Hurwitz criterion and a suitable Lyapunov function respectively. A Hopf bifurcation occurs with respect to a parameter which is the ratio of predator’s and prey’s intrinsic growth rate. The possibility of bionomic equilibrium has been considered. The optimal harvest policy is formulated and solved with Pontryagin’s maximum principle. Some numerical simulations are given to explain most of the analytical results.

Predator-prey-subsidy population dynamics on stepping-stone domains

Predator-prey-subsidy dynamics on stepping-stone domains are examined using a variety of network configurations. Our problem is motivated by the interactions between arctic foxes (predator) and lemmings (prey) in the presence of seal carrion (subsidy) provided by polar bears. We use the n-Patch Model, which considers space explicitly as a "Stepping Stone" system. We consider the role that the carrying capacity, predator migration rate, input subsidy rate, predator mortality rate, and proportion of predators surviving migration play in the predator-prey-subsidy population dynamics. We find that for certain types of networks, added mobility will help predator populations, allowing them to survive or coexist when they would otherwise go extinct if confined to one location, while in other situations (such as when sparsely distributed nodes in the network have few resources available) the added mobility will hurt the predator population. We also find that a combination of favourable conditions for the prey and subsidy can lead to the formation of limit cycles (boom and bust dynamic) from stable equilibrium states. These modifications to the dynamics vary depending on the specific network structure employed, highlighting the fact that network structure can strongly influence the predator-prey-subsidy dynamics in stepping-stone domains.

Scrounging by foragers can resolve the paradox of enrichment

Theoretical models of predator-prey systems predict that sufficient enrichment of prey can generate large amplitude limit cycles, paradoxically causing a high risk of extinction (the paradox of enrichment). Although real ecological communities contain many gregarious species, whose foraging behaviour should be influenced by socially transmitted information, few theoretical studies have examined the possibility that social foraging might resolve this paradox. I considered a predator population in which individuals play the producer-scrounger foraging game in one-prey-one-predator and two-prey-one-predator systems. I analysed the stability of a coexisting equilibrium point in the one-prey system and that of non-equilibrium dynamics in the two-prey system. The results revealed that social foraging could stabilize both systems, and thereby resolve the paradox of enrichment when scrounging behaviour (i.e. kleptoparasitism) is prevalent in predators. This suggests a previously neglected mechanism underlying a powerful effect of group-living animals on the sustainability of ecological communities.

Resource-dependent antagonistic coevolution leads to a new paradox of enrichment

The classical, ecological, paradox of enrichment describes a phenomenon that resource enrichment destabilizes predator-prey systems by exacerbating population oscillations. Here we suggest a new, evolutionary, paradox of enrichment. Resource enrichment can lead to more asymmetrical predator-prey coevolution (i.e., extremely high levels of prey defenses against predators) that decreases predator abundances and increases predator extinction risk. A major reason for this is that high resource availability can reduce fitness costs associated with prey defenses. In our experiments with a bacterium and its lytic phage, nutrient-balanced resource enrichment led to patterns in population demography and coevolutionary dynamics consistent with this coevolution-based paradox of enrichment; in particular, phage population extinction events were observed under nutrient-rich, not nutrient-poor, conditions. Consistent with ecological studies, carbon-biased resource enrichment (with carbon availability disproportionately increased relative to other nutrients) did not destabilize dynamics, and the asymmetry of coevolution was not altered in this context. Our work highlights the importance of integrating ecological and evolutionary thinking for studies of the consequences of nutrient pollution and other types of environmental changes.

Enhanced stability in host-parasitoid interactions with autoparasitism and parasitoid migration

Previous studies based on simple non-spatial model have suggested that autoparasitism, in which females develop as primary endoparasitoids of hosts while males develop at the expense of primary parasitoids, stabilizes host-parasitoid steady state. To date, however, how the stabilizing role of autoparasitism would be affected by more complex spatial factors has not been adequately investigated. To address the issue, here we analyzed a spatially extended two-patch host-parasitoid model and compared it with the corresponding non-spatial model. Results showed that in the non-spatial model and the case of autoparasitoid, the host-parasitoid steady states can be unstable if the host׳s intrinsic rate of growth and/or carrying capacity is sufficiently large. However, in the spatially extended two-patch model with parasitoid migration, the unstable host-parasitoid steady states in each local patch may become stable, provided there is certain spatial unevenness in host growth and/or carrying capacity. Therefore, the migration of parasitoid together with spatial unevenness in host growth and/or carrying capacity stabilizes the host-parasitoid interactions. The stabilizing effects are stronger with the host density-dependent migration of parasitoid than with the random migration of parasitoid. In the case of primary parasitoid, the model demonstrated similar stabilizing effects associated with the migration of parasitoid. However, the parameter conditions for stability are much more stringent than in the case of autoparasitoid. We concluded that the stabilizing effects of parasitoid migration and autoparasitism can add to each other, leading to more stable host-parasitoid interactions.

Revisiting the Stability of Spatially Heterogeneous Predator-Prey Systems Under Eutrophication

We employ partial integro-differential equations to model trophic interaction in a spatially extended heterogeneous environment. Compared to classical reaction-diffusion models, this framework allows us to more realistically describe the situation where movement of individuals occurs on a faster time scale than on the demographic (population) time scale, and we cannot determine population growth based on local density. However, most of the results reported so far for such systems have only been verified numerically and for a particular choice of model functions, which obviously casts doubts about these findings. In this paper, we analyse a class of integro-differential predator-prey models with a highly mobile predator in a heterogeneous environment, and we reveal the main factors stabilizing such systems. In particular, we explore an ecologically relevant case of interactions in a highly eutrophic environment, where the prey carrying capacity can be formally set to 'infinity'. We investigate two main scenarios: (1) the spatial gradient of the growth rate is due to abiotic factors only, and (2) the local growth rate depends on the global density distribution across the environment (e.g. due to non-local self-shading). For an arbitrary spatial gradient of the prey growth rate, we analytically investigate the possibility of the predator-prey equilibrium in such systems and we explore the conditions of stability of this equilibrium. In particular, we demonstrate that for a Holling type I (linear) functional response, the predator can stabilize the system at low prey density even for an 'unlimited' carrying capacity. We conclude that the interplay between spatial heterogeneity in the prey growth and fast displacement of the predator across the habitat works as an efficient stabilizing mechanism. These results highlight the generality of the stabilization mechanisms we find in spatially structured predator-prey ecological systems in a heterogeneous environment.

Dynamic Responses in a Plant-Insect System to Fertilization by Cormorant Feces

Theoretical arguments suggest that increased plant productivity may not only increase consumer densities but also their fluctuations. While increased consumer densities are commonly observed in fertilization experiments, experiments are seldom performed at a spatial and temporal scale where effects on population fluctuations may be observed. In this study we used a natural gradient in soil fertility caused by cormorant nesting. Cormorants feed on fish but defecate on their nesting islands. On these islands we studied soil nutrient availability, plant nutrient content and the density of Galerucella beetles, main herbivores feeding on Lythrum salicaria. In a common garden experiment, we followed larval development on fertilized plants and estimated larval stoichiometry. Soil nutrient availability varied among islands, and several cormorant islands had very high N and P soil content. Plant nutrient content, however, did not vary among islands, and there was no correlation between soil and plant nutrient contents. Beetle densities increased with plant nutrient content in the field study. However, there was either no effect on temporal fluctuations in beetle density or that temporal fluctuations decreased (at high P). In the common garden experiment, we found limited responses in either larval survival or pupal weights to fertilization. A possible mechanism for the limited effect of fertilization on density fluctuations may be that the distribution of L. salicaria on nesting islands was restricted to sites with a lower N and P content, presumably because high N loads are toxic.

Confronting the paradox of enrichment to the metacommunity perspective

Resource enrichment can potentially destabilize predator-prey dynamics. This phenomenon historically referred as the "paradox of enrichment" has mostly been explored in spatially homogenous environments. However, many predator-prey communities exchange organisms within spatially heterogeneous networks called metacommunities. This heterogeneity can result from uneven distribution of resources among communities and thus can lead to the spreading of local enrichment within metacommunities. Here, we adapted the original Rosenzweig-MacArthur predator-prey model, built to study the paradox of enrichment, to investigate the effect of regional enrichment and of its spatial distribution on predator-prey dynamics in metacommunities. We found that the potential for destabilization was depending on the connectivity among communities and the spatial distribution of enrichment. In one hand, we found that at low dispersal regional enrichment led to the destabilization of predator-prey dynamics. This destabilizing effect was more pronounced when the enrichment was uneven among communities. In the other hand, we found that high dispersal could stabilize the predator-prey dynamics when the enrichment was spatially heterogeneous. Our results illustrate that the destabilizing effect of enrichment can be dampened when the spatial scale of resource enrichment is lower than that of organismss movements (heterogeneous enrichment). From a conservation perspective, our results illustrate that spatial heterogeneity could decrease the regional extinction risk of species involved in specialized trophic interactions. From the perspective of biological control, our results show that the heterogeneous distribution of pest resource could favor or dampen outbreaks of pests and of their natural enemies, depending on the spatial scale of heterogeneity.

The dynamics of food chains under climate change and nutrient enrichment

Warming has profound effects on biological rates such as metabolism, growth, feeding and death of organisms, eventually affecting their ability to survive. Using a nonlinear bioenergetic population-dynamic model that accounts for temperature and body-mass dependencies of biological rates, we analysed the individual and interactive effects of increasing temperature and nutrient enrichment on the dynamics of a three-species food chain. At low temperatures, warming counteracts the destabilizing effects of enrichment by both bottom-up (via the carrying capacity) and top-down (via biological rates) mechanisms. Together with increasing consumer body masses, warming increases the system tolerance to fertilization. Simultaneously, warming increases the risk of starvation for large species in low-fertility systems. This effect can be counteracted by increased fertilization. In combination, therefore, two main drivers of global change and biodiversity loss can have positive and negative effects on food chain stability. Our model incorporates the most recent empirical data and may thus be used as the basis for more complex forecasting models incorporating food-web structure.

Does predator interference cause alternative stable states in multispecies communities?

Whereas it is well known that simple ecological mechanisms may promote stability in simple species models, their consequences for stability and resilience in multispecies communities are largely unexplored. Here, we studied the effect of predator interference on the occurrence of alternative attractors and complex dynamics in randomly constructed multispecies predator-prey communities. We studied three types of interference: random interference ("asymmetric"), random interference but symmetrical between pairs of predators ("symmetric"), and interference among only the same species ("conspecific"). In all cases predator interference increased the average number of alternative attractors, whereas at the same time it reduced the emergence of oscillatory or chaotic dynamics. Our findings demonstrate a contrasting effect of predator interference on the stability of a community: on the one hand it reduces cycles and chaos in the dynamics, on the other hand predator interference increases the likelihood that communities may undergo critical transitions between multiple stable states.

Diversity-dependent stability under mowing and nutrient addition: evidence from a 7-year grassland experiment

Anthropogenic perturbations may affect biodiversity and ecological stability as well as their relationships. However, diversity-stability patterns and associated mechanisms under human disturbances have rarely been explored. We conducted a 7-year field experiment examining the effects of mowing and nutrient addition on the diversity and temporal stability of herbaceous plant communities in a temperate steppe in northern China. Mowing increased population and community stability, whereas nutrient addition had the opposite effects. Stability exhibited positive relationships with species richness at population, functional group and community levels. Treatments did not alter these positive diversity-stability relationships, which were associated with the stabilising effect of species richness on component populations, species asynchrony and portfolio effects. Despite the difficulty of pinpointing causal mechanisms of diversity-stability patterns observed in nature, our results suggest that diversity may still be a useful predictor of the stability of ecosystems confronted with anthropogenic disturbances.