Shaken, Not Stirred: On Permanence in Ecological Communities

All models of ecological communities are approximations: it would be pointless to burden them with too many contingencies and details. On the other hand, they would be of little help if they were not robust against the kind of perturbations and shocks to which a real ecosystem is ceaselessly exposed. They have to exhibit some kind of stability. But which kind? For a considerable time, it used to be understood that stability meant local stability of an equilibrium, i.e., asymptotic stability in the sense of Lyapunov: any sufficiently small perturbation is promptly offset by a small countermove back to the stationary state. But this notion, which was developed by physicists and engineers and serves perfectly well to describe the stability of a control mechanism or a mixture of chemicals, is not reall

Scale-free memory model for multiagent reinforcement learning. Mean field approximation and rock-paper-scissors dynamics

A continuous time model for multiagent systems governed by reinforcement learning with scale-free memory is developed. The agents are assumed to act independently of one another in optimizing their choice of possible actions via trial-and-error search. To gain awareness about the action value the agents accumulate in their memory the rewards obtained from taking a specific action at each moment of time. The contribution of the rewards in the past to the agent current perception of action value is described by an integral operator with a power-law kernel. Finally a fractional differential equation governing the system dynamics is obtained. The agents are considered to interact with one another implicitly via the reward of one agent depending on the choice of the other agents. The pairwise interaction model is adopted to describe this effect. As a specific example of systems with non-transitive interactions, a two agent and three agent systems of the rock-paper-scissors type are analyzed in detail, including the stability analysis and numerical simulation. Scale-free memory is demonstrated to cause complex dynamics of the systems at hand. In particular, it is shown that there can be simultaneously two modes of the system instability undergoing subcritical and supercritical bifurcation, with the latter one exhibiting anomalous oscillations with the amplitude and period growing with time. Besides, the instability onset via this supercritical mode may be regarded as "altruism self-organization". For the three agent system the instability dynamics is found to be rather irregular and can be composed of alternate fragments of oscillations different in their properties.Comment: 17 pages, 7 figur

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

In theory, enrichment of resource in a predator-prey model leads to destabilization of the system, thereby collapsing the trophic interaction, a phenomenon referred to as "the paradox of enrichment". After it was first proposed by Rosenzweig (1971), a number of subsequent studies were carried out on this dilemma over many decades. In this article, we review these theoretical and experimental works and give a brief overview of the proposed solutions to the paradox. The mechanisms that have been discussed are modifications of simple predator-prey models in the presence of prey that is inedible, invulnerable, unpalatable and toxic. Another class of mechanisms includes an incorporation of a ratio-dependent functional form, inducible defence of prey and density-dependent mortality of the predator. Moreover, we find a third set of explanations based on complex population dynamics including chaos in space and time. We conclude that, although any one of the various mechanisms proposed so far might potentially prevent destabilization of the predator-prey dynamics following enrichment, in nature different mechanisms may combine to cause stability, even when a system is enriched. The exact mechanisms, which may differ among systems, need to be disentangled through extensive field studies and laboratory experiments coupled with realistic theoretical models

Prey dispersal and predator persistence

To understand how patchiness influences population dynamics of a tri-trophic interaction, a tractable model is formulated in terms of differential equations. Motivated by the structure of systems such as plants, phytophagous mites and predatory mites, the model takes dispersal into account at the middle trophic level. The effect of dispersal for the middle level in a tri-trophic system could be either stabilising or destabilising since the middle level acts both as prey and as predator. First a simple model with logistic growth for the lowest level is formulated. A model with logistic growth for the lowest level and instantaneous dispersal has a globally stable three-species equilibrium, if this equilibrium exists. Addition of a middle level dispersal phase of non-negligible duration influences equilibrium stability. In the absence of the top trophic level a limit cycle can occur, caused by the delay that exists in the reaction of the middle level to the changes in the lowest level. With low predator efficiency, it is also possible to have an unstable three-species equilibrium. So addition of a middle level dispersal phase of non-negligible duration can work destabilising. Next the persistence of the third trophic level is studied. Even when the three-species equilibrium exists, the third trophic level need not be persistent. A two-species limit cycle can keep its stability when a three-species equilibrium exists; the system is then bistable. It is argued that such a bistability can offer an alternative explanation for pesticide-induced outbreaks of spider mites and failure of predator introduction

Density-dependent dispersal may explain the mid-season crash in some aphid populations

Copyright 2008 Elsevier B.V., All rights reserved.Aphid population dynamics during the season show a characteristic pattern with rapid increase in numbers at the beginning followed by a sudden drop in the middle of the season. This pattern is usually associated with predation and/or change in food quality during the summer. By developing a mechanistic model of aphid population dynamics we show that this pattern can arise from density-dependent dispersal behaviour of aphids. The dynamics produced by the model were similar to those observed in real populations of the alder aphid (Pterocallis alni). The two mechanisms required for these oscillations to arise were the perception of density through the number of contacts with other individuals and the inter-generational transfer of information (the maternal effect). Both mechanisms are examples of delayed density-dependence and, therefore, this study adds to the evidence that delayed density-dependence might cause complex population dynamics. To reproduce the seasonal dynamics of the alder aphid with the model, the maternal effect was essential, indicating that this could be an important factor in alder aphid dynamics. According to our model, external regulations (e.g., predation and/or change in food quality) were not required to explain the highly oscillatory population dynamics of aphids during a season.Peer reviewe