The evolution of partner specificity in mutualisms

Mutualistic species vary in their level of partner specificity, which has important evolutionary, ecological, and management implications. Yet, the evolutionary mechanisms which underpin partner specificity are not fully understood. Most work on specialization focuses on the trade-off between generalism and specialism, where specialists receive more benefits from preferred partners at the expense of benefits from non-preferred partners, while generalists receive similar benefits from all partners. Because all mutualisms involve some degree of both cooperation and conflict between partners, we highlight that specialization to a mutualistic partner can be cooperative, increasing benefit to a focal species and a partner, or antagonistic, increasing resource extraction by a focal species from a partner. We devise an evolutionary game theoretic model to assess the evolutionary dynamics of cooperative specialization, antagonistic specialization, and generalism. Our model shows that cooperative specialization leads to bistability: stable equilibria with a specialist host and its preferred partner excluding all others. We also show that under cooperative specialization with spatial effects, generalists can thrive at the boundaries between differing specialist patches. Under antagonistic specialization, generalism is evolutionarily stable. We provide predictions for how a cooperation-antagonism continuum may determine the patterns of partner specificity that develop within mutualistic relationships.

Variability of complex oscillatory regimes in the stochastic model of cold-flame combustion of a hydrocarbon mixture

Processes of the cold-flame combustion of a mixture of two hydrocarbons are studied on the base of a three-dimensional nonlinear dynamical model. Bifurcation analysis of the deterministic model reveals mono- and bistability parameter zones with equilibrium and oscillatory attractors. For this model, effects of random disturbances in the bistability parameter zone are studied. We show that random forcing causes transitions between coexisting stable equilibria and limit cycles with the formation of complex stochastic mixed-mode oscillations. Properties of these oscillatory regimes are studied by means of statistics of interspike intervals. A phenomenon of anti-coherence resonance is discussed. This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.

Stability criteria for the consumption and exchange of essential resources

Models of pairwise interactions have informed our understanding of when ecological communities will have stable equilibria. However, these models do not explicitly include the effect of the resource environment, which has the potential to refine or modify our understanding of when a group of interacting species will coexist. Recent consumer-resource models incorporating the exchange of resources alongside competition exemplify this: such models can lead to either stable or unstable equilibria, depending on the resource supply. On the other hand, these recent models focus on a simplified version of microbial metabolism where the depletion of resources always leads to consumer growth. Here, we model an arbitrarily large system of consumers governed by Liebig's law, where species require and deplete multiple resources, but each consumer's growth rate is only limited by a single one of these multiple resources. Consumed resources that do not lead to growth are leaked back into the environment, thereby tying the mismatch between depletion and growth to cross-feeding. For this set of dynamics, we show that feasible equilibria can be either stable or unstable, once again depending on the resource environment. We identify special consumption and production networks which protect the community from instability when resources are scarce. Using simulations, we demonstrate that the qualitative stability patterns we derive analytically apply to a broader class of network structures and resource inflow profiles, including cases in which species coexist on only one externally supplied resource. Our stability criteria bear some resemblance to classic stability results for pairwise interactions, but also demonstrate how environmental context can shape coexistence patterns when ecological mechanism is modeled directly.

Mathematical modelling of SigE regulatory network reveals new insights into bistability of mycobacterial stress response

Background The ability to rapidly adapt to adverse environmental conditions represents the key of success of many pathogens and, in particular, of Mycobacterium tuberculosis. Upon exposition to heat shock, antibiotics or other sources of stress, appropriate responses in terms of genes transcription and proteins activity are activated leading part of a genetically identical bacterial population to express a different phenotype, namely to develop persistence. When the stress response network is mathematically described by an ordinary differential equations model, development of persistence in the bacterial population is associated with bistability of the model, since different emerging phenotypes are represented by different stable steady states. Results In this work, we develop a mathematical model of SigE stress response network that incorporates interactions not considered in mathematical models currently available in the literature. We provide, through involved analytical computations, accurate approximations of the system’s nullclines, and exploit the obtained expressions to determine, in a reliable though computationally efficient way, the number of equilibrium points of the system. Conclusions Theoretical analysis and perturbation experiments point out the crucial role played by the degradation pathway involving RseA, the anti-sigma factor of SigE, for coexistence of two stable equilibria and the emergence of bistability. Our results also indicate that a fine control on RseA concentration is a necessary requirement in order for the system to exhibit bistability.

Delayed Feedback Control of Hidden Chaos in the Unified Chaotic System between the Sprott C System and Yang System

In this paper, the influence of delayed feedback on the unified chaotic system from the Sprott C system and Yang system is studied. The Hopf bifurcation and dynamic behavior of the system are fully studied by using the central manifold theorem and bifurcation theory. The explicit formula, bifurcation direction, and stability of the periodic solution of bifurcation are given correspondingly. The Hopf bifurcation diagram and chaotic phenomenon are also analyzed by numerical simulation to prove the correctness of the theory. It shows that this delay control can only be applied to the hidden chaos with two stable equilibria.

Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure

A new algorithm for the estimation of the robustness of a dynamical system’s equilibrium is presented. Unlike standard approaches, the algorithm does not aim to identify the entire basin of attraction of the solution. Instead, it iteratively estimates the so-called local integrity measure, that is, the radius of the largest hypersphere entirely included in the basin of attraction of a solution and centred in the solution. The procedure completely overlooks intermingled and fractal regions of the basin of attraction, enabling it to provide a significant engineering quantity in a very short time. The algorithm is tested on four different mechanical systems of increasing dimension, from 2 to 8. For each system, the variation of the integrity measure with respect to a system parameter is evaluated, proving the engineering relevance of the results provided. Despite some limitations, the algorithm proved to be a viable alternative to more complex and computationally demanding methods, making it a potentially appealing tool for industrial applications.

Competition and Predation Models

The chapter “Competition and Predation Models” considers models with two or more interacting species. What needs to happen for there to be “stable equilibria” that contain all possible members of a system? This is where simple models can be useful: these interactions can be represented by mathematical equations, and then solved for conditions that allow species to coexist. This chapter shows three techniques that make it possible to take a model system and determine whether the system has a stable equilibrium with all members present. The basic principles of model stability are presented, as well as how mathematical models can be used to address basic ecological questions in competition and predator-prey systems. Isocline analysis and analytical stability analysis are explained as ways to predict model behavior and are then used to draw inferences about the processes acting in the real world.

The dynamics of cooperation, power, and inequality in a group-structured society

Most human societies are characterized by the presence of different identity groups which cooperate but also compete for resources and power. To deepen our understanding of the underlying social dynamics, we model a society subdivided into groups with constant sizes and dynamically changing powers. Both individuals within groups and groups themselves participate in collective actions. The groups are also engaged in political contests over power which determines how jointly produced resources are divided. Using analytical approximations and agent-based simulations, we show that the model exhibits rich behavior characterized by multiple stable equilibria and, under some conditions, non-equilibrium dynamics. We demonstrate that societies in which individuals act independently are more stable than those in which actions of individuals are completely synchronized. We show that mechanisms preventing politically powerful groups from bending the rules of competition in their favor play a key role in promoting between-group cooperation and reducing inequality between groups. We also show that small groups can be more successful in competition than large groups if the jointly-produced goods are rivalrous and the potential benefit of cooperation is relatively small. Otherwise large groups dominate. Overall our model contributes towards a better understanding of the causes of variation between societies in terms of the economic and political inequality within them.

Critical Configurations and Tube of Typical Trajectories for the Potts and Ising Models with Zero External Field

We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.

Counting equilibria of large complex systems by instability index

We consider a nonlinear autonomous system of N≫1 degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.