Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates

International audienceWe study the competition of two species for a single resource in a chemostat. In the simplest space-homogeneous situation, it is known that only one species survives, namely the best competitor. In order to exhibit coexistence phenomena, where the two competitors are able to survive, we consider a space dependent situation: we assume that the two species and the resource follow a di usion process in space, on top of the competition process. Besides, and in order to consider the most general case, we assume each population is associated with a distinct di usion constant. This is a key di culty in our analysis: the speci c (and classical) case where all di usion constants are equal, leads to a particular conservation law, which in turn allows to eliminate the resource in the equations, a fact that considerably simpli fies the analysis and the qualitative phenomena. Using the global bifurcation theory, we prove that the underlying 2-species, stationary, di usive, chemostat-like model, does possess coexistence solutions, where both species survive. On top of that, we identify the domain, in the space of the relevant bifurcation parameters, for which the system does have coexistence solutions

A slow-fast dynamic decomposition links neutral and non-neutral coexistence in interacting multi-strain pathogens

Understanding the dynamics of multi-type microbial ecosystems remains a challenge, despite advancing molecular technologies for diversity resolution within and between hosts. Analytical progress becomes difficult when modelling realistic levels of community richness, relying on computationally-intensive simulations and detailed parametrisation. Simplification of dynamics in polymorphic pathogen systems is possible usingaggregation methods and the slow-fast dynamics approach. Here we develop one new such framework, tailored to the epidemiology of an endemic multi-strain pathogen. We apply Goldstone’s idea of slow dynamics resulting from spontaneously broken symmetries, to study direct interactions in co-colonization, ranging from competition to facilitation between strains. The slow-fast dynamics approach interpolates between a neutral and non-neutral model for multi-strain coexistence, and quantifies the exact asymmetries that are important for the maintenance and stabilisation of diversity

Bistability induced by generalist natural enemies can reverse pest invasions

Reaction-diffusion analytical modeling of predator-prey systems has shown that specialist natural enemies can slow, stop and even reverse pest invasions, assuming that the prey population displays a strong Allee effect in its growth. Few additional analytical results have been obtained for other spatially distributed predator-prey systems, as traveling waves of non-monotonous systems are notoriously difficult to obtain. Traveling waves have indeed recently been shown to exist in predator-prey systems, but the direction of the wave, an essential item of information in the context of the control of biological invasions, is generally unknown. Preliminary numerical explorations have hinted that control by generalist predators might be possible for prey populations displaying logistic growth. We aimed to formalize the conditions in which spatial biological control can be achieved by generalists, through an analytical approach based on reaction-diffusion equations. The population of the focal prey - the invader - is assumed to grow according to a logistic function. The predator has a type II functional response and is present everywhere in the domain, at its carrying capacity, on alternative hosts. Control, defined as the invader becoming extinct in the domain, may result from spatially independent demographic dynamics or from a spatial extinction wave. Using comparison principles, we obtain sufficient conditions for control and for invasion, based on scalar bistable partial differential equations (PDEs). The searching efficiency and functional response plateau of the predator are identified as the main parameters defining the parameter space for prey extinction and invasion. Numerical explorations are carried out in the region of those control parameters space between the super-and subso-lutions, in which no conclusion about controllability can be drawn on the basis of analytical solutions. The ability of generalist predators to control prey populations with logistic growth lies in the bis-table dynamics of the coupled system, rather than in the bistability of prey-only dynamics as observed for specialist predators attacking prey populations displaying Allee effects. The consideration of space in predator-prey systems involving generalist predators with a parabolic functional response is crucial. Analysis of the ordinary differential equations (ODEs) system identifies parameter regions with monostable (extinction) and bistable (extinction or invasion) dynamics. By contrast, analysis of the associated PDE system distinguishes different and additional regions of invasion and extinction. Depending on the relative positions of these different zones, four patterns of spatial dynamics can be identified : traveling waves of extinction and invasion, pulse waves of extinction and heterogeneous stationary positive solutions of the Turing type. As a consequence, prey control is predicted to be possible when space is considered in additional situations other than those identified without considering space. The reverse situation is also possible. None of these considerations apply to spatial predator-prey systems with specialist natural enemies

Global behavior of N competing species with strong diffusion: diffusion leads to exclusion

International audienceIt is known that the competitive exclusion principle holds for a large kind of models involving several species competing for a single resource in an homogeneous environment. Various works indicate that the coexistence is possible in an heterogeneous environment. We propose a spatially heterogeneous system modeling the competition of several species for a single resource. If spatial movements are fast enough, we show that our system can be well approximated by a spatially homogeneous system, called aggregated model, which can be explicitly computed. Moreover, we show that if the competitive exclusion principle holds for the aggregated model, it holds for the spatially heterogeneous model too

Auxiliary files for invasion resistance in multispecies systems based on the replicator equation

<p>In this study we propose a replicator equation framework to model multi-species dynamics. In this replicator equation, the coefficients describe pairwise invasion fitnesses between constituent members, and an explicit quadratic term represents the systemic invasion resistance. This invasion resistance (<em>system trait</em>) is dependent on species frequencies and can be linked with specific structures of their pairwise invasion fitness matrix. Within this replicator framework, mean invasion fitness arises, evolves dynamically, and may undergo critical shifts with global environmental changes (e.g. mean growth rate, mean propensity for co-colonization). In the paper, extending an analogy with an SIS epidemiological model, we derive the conceptual mechanistic link between such replicator equation and <em>N </em>microbial species' growth and interaction traits, stemming from micro-scale environmental modification. We also study several specific invasion matrix structures in detail, their role for the quality of species dynamics and also for systemic invasion resistance. In this Dryad repository, we provide some auxiliary files and links to simulation codes, used and presented in our paper, to aid a mathematical understanding of invasion resistance using the replicator equation. We propose the framework can be applied to study colonization resistance in a wide range of microbial ecosystems. </p><p>Funding provided by: Fundação para a Ciência e Tecnologia<br>Crossref Funder Registry ID: https://ror.org/00snfqn58<br>Award Number: 5666/44637YE</p><p>Funding provided by: Fundação para a Ciência e Tecnologia<br>Crossref Funder Registry ID: https://ror.org/00snfqn58<br>Award Number: CEECIND/03051/2018</p><p>Funding provided by: Le Studium<br>Crossref Funder Registry ID: https://ror.org/03qgg4v31<br>Award Number: 2020-2001-230- Y20V7</p><p>Funding provided by: Fundação para a Ciência e Tecnologia<br>Crossref Funder Registry ID: https://ror.org/00snfqn58<br>Award Number: 2022.03060.PTDC</p&gt

Comparison of the global dynamics for two chemostat-like models: random temporal variation versus spatial heterogeneity

This article is dedicated to the study and comparison of two chemostat-like competition models in a heterogeneous environment. The first model is a probabilistic model where we build a PDMP simulating the effect of the temporal heterogeneity of an environment over the species in competition. Its study uses classical tools in this field. The second model is a gradostat-like model simulating the effect of the spatial heterogeneity of an environment over the same species. Despite the fact that the nature of the two models is very different, we will see that their long time behavior is globally very similar. We define for both model quantities called invasion rates which model the growth rate of a species when it is near to extinction. We show that the signs of these invasion rates essentially determine the long time behavior for both systems. In particular, we exhibit a new example of bistability between a coexistence steady state and a semi-trivial steady state

The key to complexity in interacting systems with multiple strains

Ecological community structure, persistence and stability are shaped by multiple forces, acting on multiple scales. These include patterns of resource use and limitation, spatial heterogeneities, drift and migration. Pathogen strains co-circulating in a host population are a special type of an ecological community. They compete for colonization of susceptible hosts, and sometimes interact via altered susceptibilities to co-colonization. Diversity in such pairwise interaction traits enables the multiple strains to create dynamically their niches for growth and persistence, and 'engineer' their common environment. How such a network of interactions with others mediates collective coexistence remains puzzling analytically and computationally difficult to simulate. Furthermore, the gradients modulating stability-complexity regimes in such multi-player systems remain poorly understood. In a recent study, we presented an analytic framework for N-type coexistence in an SIS epidemiological system with co-colonization interactions. The multi-strain complexity was reduced from O(N 2) dimensions of population structure to only N equations for strain frequency evolution on a long timescale. Here, we examine the key drivers of coexistence regimes in such a system. We find the ratio of single to co-colonization µ critically determines the type of equilibrium for multi-strain dynamics. This key quantity in the model encodes a trade-off between overall transmission intensity R 0 and mean interaction coefficient in strain space k. Preserving a given coexistence regime, under fixed trait variation, can only be achieved from a balance between higher competition in favourable environments, and higher cooperation in harsher environments, consistent with the stress gradient hypothesis in ecology. Multi-strain coexistence regimes are more stable when µ is small, whereas as µ increases, dynamics tends to increase in complexity. There is an intermediate ratio that maximizes the existence and stability of a unique coexistence equilibrium between strains. This framework provides a foundation for linking invariant principles in collective coexistence across biological systems, and for understanding critical shifts in community dynamics, driven by simple and random pairwise interactions but potentiated by mean-field and environmental gradients