Food Quality in Producer-Grazer Models: A Generalized Analysis

Stoichiometric constraints play a role in the dynamics of natural populations, but are not explicitly considered in most mathematical models. Recent theoretical works suggest that these constraints can have a significant impact and should not be neglected. However, it is not yet resolved how stoichiometry should be integrated in population dynamical models, as different modeling approaches are found to yield qualitatively different results. Here we investigate a unifying framework that reveals the differences and commonalities between previously proposed models for producer-grazer systems. Our analysis reveals that stoichiometric constraints affect the dynamics mainly by increasing the intraspecific competition between producers and by introducing a variable biomass conversion efficiency. The intraspecific competition has a strongly stabilizing effect on the system, whereas the variable conversion efficiency resulting from a variable food quality is the main determinant for the nature of the instability once destabilization occurs. Only if the food quality is high an oscillatory instability, as in the classical paradox of enrichment, can occur. While the generalized model reveals that the generic insights remain valid in a large class of models, we show that other details such as the specific sequence of bifurcations encountered in enrichment scenarios can depend sensitively on assumptions made in modeling stoichiometric constraints.Comment: Online appendixes include

Resilience through adaptation

<div><p>Adaptation of agents through learning or evolution is an important component of the resilience of Complex Adaptive Systems (CAS). Without adaptation, the flexibility of such systems to cope with outside pressures would be much lower. To study the capabilities of CAS to adapt, social simulations with agent-based models (ABMs) provide a helpful tool. However, the value of ABMs for studying adaptation depends on the availability of methodologies for sensitivity analysis that can quantify resilience and adaptation in ABMs. In this paper we propose a sensitivity analysis methodology that is based on comparing time-dependent probability density functions of output of ABMs with and without agent adaptation. The differences between the probability density functions are quantified by the so-called earth-mover’s distance. We use this sensitivity analysis methodology to quantify the probability of occurrence of critical transitions and other long-term effects of agent adaptation. To test the potential of this new approach, it is used to analyse the resilience of an ABM of adaptive agents competing for a common-pool resource. Adaptation is shown to contribute positively to the resilience of this ABM. If adaptation proceeds sufficiently fast, it may delay or avert the collapse of this system.</p></div

A New Statistical Method to Determine the Degree of Validity of Health Economic Model Outcomes against Empirical Data.

The validation of health economic (HE) model outcomes against empirical data is of key importance. Although statistical testing seems applicable, guidelines for the validation of HE models lack guidance on statistical validation, and actual validation efforts often present subjective judgment of graphs and point estimates

Time-dependent histograms at <i>E</i>_<i>h</i> = 0.07.

<p>(a): Histogram of the ABM without adaptation. (b): Histogram of the ABM with adaptation. (c): The value of <i>Q</i> as a function of time. The value of <i>Q</i> increases as without adaptation an increasing number of runs goes to extinction.</p

Time-averaged histograms of the number of agents (<i>n</i>).

<p>The blue histograms are measured over 10,000 replicate runs at <i>t</i> = 1000, and the green histograms over the time-steps of a single model run between <i>t</i> = 1000 and <i>t</i> = 100,000. All parameters are at their default values. Fig 6a indicates that the ABM without adaptation is ergodic, and Fig 6b indicates that the ABM with adaptation is not ergodic.</p

Effects of adaptation.

<p>(a): Means of the pdfs <i>P</i>_<i>a</i>(<i>green</i>) and <i>P</i>_<i>b</i> (blue) at the default parameter settings. The thinner lines show the mean plus or minus one standard deviation. Both the mean and standard deviation were estimated based on 1000 replicate runs. (b): Plot of <i>Q</i> between the pdfs <i>P</i>_<i>a</i> and <i>P</i>_<i>b</i>, as functions of time. The plot shows that on long time-scales the effects of adaptation increase. (c): Comparison between the earth-mover’s distance <i>d</i>_<i>e</i> (black), the Jensen-Shannon divergence <i>d</i>_<i>J</i> (red), and the Euclidian distance <i>d</i>_<i>E</i> (orange). To fit in the same graph, we plot <i>d</i>_<i>e</i> divided by its maximimum value. For large values of <i>t</i>, <i>d</i>_<i>J</i> and <i>d</i>_<i>u</i> become insensitive to changes in the pdfs when there is no longer overlap between the pdfs. In contrast, <i>d</i>_<i>e</i> continues to increase.</p

Flow chart of the proposed method to measure adaptation.

<p>The method is based on a comparison between versions of an ABM with and without adaptation. See the main text for further explanation.</p

Snapshot of a typical simulation run of the ABM test-case.

<p>The ABM is composed of a square grid of sites. Dark colours indicate sites with high resource densities, while light colours indicate low densities. The red arrows show the current locations of agents.</p

Effect of adaptation on resilience.

<p>(a): Plot of the percentage of model runs with a positive population <i>N</i>_<i>pos</i>(<i>t</i>) as a function of time. We used 100 replicate runs at <i>E</i>_<i>h</i> = 0.07, for both versions of the ABM. The dashed lines correspond to exponential fits (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0171833#pone.0171833.e010" target="_blank">Eq 9</a>). Without adaptation, an increasing number of runs go to extinction, whereas with adaptation there is extinction only on short time-scales. (b): Mean value of <i>w</i>_{<i>harvest</i>} over the agents of all the 100 replicate runs of the ABMs with adaptation and without adaptation. The upper and lower lines show the mean plus or minus one standard deviation. (c): The mean value of <i>w</i>_<i>move</i> over the agents of all the 100 replicate runs of the ABMs with and without adaptation. The upper and lower lines show the mean plus or minus one standard deviation.</p