Top-right: Percentage of variation explained by the first 20 eigenvectors from factor analysis. The minimum spanning tree MST has been used as a tool to characterize the distribution of species and strains in interaction trait space. The MST of a graph is a tree that connects all nodes of that graph so that the total edge weight is minimized. For a hierarchical structure, MST consists of long edges between clusters and short edges that connect the nodes inside each cluster; in this case, the SMST scales with the number of clusters. The SMST increases by divergent evolution.
The left panel of Appendix 1—figure 5 shows an example MST in the trait space of our simulation, represented in 2D for the sake of visualization. The community in this snapshot consists of strains. Here we used R-package vegan 2. The red curve is the average of all curves and the red shaded area is the corresponding standard deviation. R 1 and R 2 are approximate lengths of short branches connecting strains within clusters R 1 , and long branches R 2 connecting species.
Looking Beyond Classic Accommodation or Reduction
N approximates the number of long branches or species. From this log-log-scaled plot we see that there are two clearly different scales in the size of edges.
R 1 is a representative value for the size of clusters distance between strains within a typical species and R 2 is a representative value for the scale of the trait space typical distance between different species. N in Appendix 1—figure 5 represents approximately the number of distinct clusters species in the system. A diversity index quantifies a certain aspect of diversity in a single number. Since diversity is itself complex, no single diversity index is sufficient to describe the diversity of a community.
For example richness, i. Evenness or Shannon entropy takes into account this distribution but does not inform about the diversity of the trait of species, i. Functional diversity indexes focus on this aspect but none of them exhaustively describes properties of trait space Mouchet et al. For a comprehensive assessment of diversity and community dynamics, information about the density of species over resources, rate of extinction and emergence, and also details of community structure, for example interaction of species and topology of the network, should be considered, too.
The next plots show how SMST, an index for functional diversity, changes over time.
Note that evolutionary collapses mass extinctions occasionally occur see Appendix 1, Collapses of diversity with a probability that depends on the trade-off parameter, lifespan and mutation probability. Comparison of Appendix 1—figure 7 and Appendix 1—figure 6 confirms that diversity and dynamics are strongly associated with trade-off without a noticeable effect of lifespan except for very short lifespans, upper left panel of Appendix 1—figure 6 and bottom panel of Appendix 1—figure 7.
Univariate diversity indexes that are defined based on abundance of species, like richness and evenness, are routinely used to quantify diversity in biological communities. These indexes are most expressive if species are equal in their effect on their community and ecosystem functioning. However, in the last decades ecologists are increasingly realizing that without information on variety of functions in a community, diversity can not be correctly evaluated Mouchet et al. Inspired by the concept of Hutchinsonian niche, functional diversity FD was introduced by Rosenfeld as distribution of species in functional space Whittaker et al.
The axes of this space represent the functional features of species Mouchet et al. Thus, the distribution of interaction traits in trait space determines variety of functions in the system. Different measures of FD have been introduced, each quantifying and explaining one facet of trait distribution in trait or functional space, very similar to SMST Appendix 1, SMST and distribution of species and strains in trait space.
In the following we also use three other indexes: functional dispersion, Rao index and functional evenness Mouchet et al. Distinct, well-separated clusters in functional trait space mean that species are diversified to different functional groups. A functional group is defined in ecology either as a set of species with similar effect on their environment, or as cluster in trait space Hooper et al. In ITEEM, by using the framework of interaction-based models, these two definitions are interchangeable. The positions of functional groups in functional space define their functional niches.
The notion of functional niche was first introduced by Elton as the place of an animal in its community or its biotic environment Elton, Then Clarke Clarke, noted that the functional niche stresses the function of the species in the community, which is different from its physical niche, the latter determining its place in the habitat.
A suitable definition of functional niche is the area occupied by a species in the n -dimensional functional space Clarke, ; Whittaker et al. Following this picture and considering that there is no physical niche or habitat in our well-mixed model, we can say that in ITEEM, the position of trait vectors in functional space determines the profession function of species. For the phase diagram in Figure 4 of the main text we have synthesized a descriptive dimensionless diversity parameter by averaging over normalized values of several diversity indexes, namely richness , Shannon entropy , standard deviation of replication r , maximum distance in trait space , standard deviation of interaction terms , sum of squared lengths of minimum spanning tree of trait space , functional diversity indexes functional dispersion, Rao index and functional evenness, all three in two versions: with and without abundance , and strength of cycles.
Therefore, we report in Appendix 1—figure 8 some of the most important indexes of community state computed from our simulations for different trade-offs and lifespans. The four phases described in Figure 4 of the main text can be seen in nearly all the parameters. When disruptive selection produces clusters of localized strains in trait space this index decreases. The corresponding network is a directed network with one directed edge between each pair of nodes strains , pointing from the dominating to the dominated one, with weight between 0 and 1 according to Equation 9. Three or more directed edges in the dominance network can form cycles of strains in which each strain competes successfully against one cycle neighbor but loses against the other neighbor, a configuration corresponding to the rock-paper-scissors game.
Even in a completely connected random dominance network, a randomly selected triplet of nodes forms a cycle with a probability of 1 4. Hence, we compare number, N c y c N e t w o r k , and average strength, S c y c N e t w o r k , of cycles of the evolving network at each time step with number, N c y c R a n d o m , and average strength, S c y c R a n d o m of cycles of its equivalent shuffled random networks. For this purpose we. Then we apply the procedure described in step one on this network to measure the number of cycles and their average strengths.
The results of these three steps are plotted in Figure 3g and Figure 4a of the main text. In our analysis, a diversity collapse is defined as a sharp decrease in diversity, i. In this way we excluded small or gradual decreases in a diversity measure. Diversity collapse red line is defined as a sharp decrease in diversity. Red dots mark 5sampling time steps before the collapse.
- what happens to stock options in an ipo.
- Author Profile:.
- binary option robot review watchdog?
- Breaking the Trade-Off Between Efficiency and Service?
The red dots highlight the five sampling time steps before each observed collapse. The overall distribution of sampled values, and the distribution of the red points preceding the collapses are over plotted as two sets of contours. In order to find a qualitative explanation for diversity collapses in ITEEM, we examined the relation between diversity and average cycle strength Appendix 1—figure 9b. During simulations, these two quantities are usually correlated, but this correlation is blurred by the stochastic nature of eco-evolutionary dynamics.
Sometimes, diversity increases faster than cycle strength or vice versa.
System Analysis
If we highlight the time steps before the sudden collapses, we see that they always lie in the right part of the cycle strength distribution, which means that cycle strengths are larger than expected at these time points. We see a scaling relation with exponent of 1. In the absence of such biotic selections, speciation and extinction of different species occur randomly with a constant rate without any autocorrelation in time.
To examine if there is such a correlation we used the distribution of inter-event times, that is, the distribution of intervals between occurrence of consecutive events. For a completely random Poisson process — which is the null hypothesis for correlated speciation and extinction — this distribution follows an exponential distribution.
The dismal science
Deviation from the exponential distribution is a signature of correlation between events. Appendix 1—figure 12 compares the inter-event distribution of ITEEM data with the best fit of a geometric distribution discrete version of exponential distribution to the data. Rate of increase in diversity is calculated by fitting a line to the first generations of each simulation and averaging is over five different simulations.
Error bars show the errors estimated by the fit. Note the log-log scale of the plot. In the neutral model, reproduction probabilities should also be the same for all strains, hence we attributed the same replication in each simulation to all strains. The distribution of strains over trait space Appendix 1—figure 15 shows that genetic drift is able to spread trait vectors in trait space and to produce a small cloud of strains, but the diversity generated in this way is much smaller than that of communities evolved under biotic selection pressure mediated by competition under moderate trade-offs compare scale to that of bottom left panel of Appendix 1—figure 3.
Note the small size of the trait space in comparison to a non-neutral model bottom left panel of Appendix 1—figure 3. In order to show clearly the difference between the diversity produced in both models we also studied diversity measures and other parameters. The bottom panels illustrate the corresponding comparison for cycle formation.
The relative strengths of cycles in these simulations have large fluctuations around a value less than one, without any stable pattern over time. This means that community dynamics in the neutral model is determined by fluctuations, as expected from a model dominated by random genetic drift. A map between two spaces, for example interaction and phenotype space, can be constructed by the rules and laws that link them. The interaction of two individuals is a function of their phenotypic traits. Generally, this can be a complex relation with different functionality of different traits.
Muscle trade-offs in a power-amplified prey capture system
When this function is known, any phenotypic variation can be mapped into the interaction space. To generally investigate this map, we borrow the term competition kernel from adaptive dynamics theory as a phenotype-based model Doebeli, A competition kernel measures the competitive impact of two individuals from different strains with different traits, i. Elements of these vectors can be any relevant phenotype like size, color, expression of a gene, etc.
The dimension of the phenotype space is equal to the number of traits with each axis representing a trait, and strains are distributed according to their traits over this space Appendix 1—figure 17a. Interaction space, on the other hand, has one axis for each strain, and individuals are distributed based on their interactions with the strains that represent the axes Appendix 1—figure 17b. The dimension of interaction space is dynamic because it increases with new strains and shrinks as strains go extinct. This system consists of 3 strains with two phenotypic traits. Here, without loss of generality, we ignore intra-specific competitions.
A new, phenotypical mutant strain appears in phenotype space close to its parent Appendix 1—figure 18a and c. A relevant question about interaction based-models could be if the random Gaussian variations in the interaction traits are biologically meaningful or not.