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Register a new userSpecies assembly in model ecosystems, I: Analysis of the population model and the invasion dynamics
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Recently we have introduced a simplified model of ecosystem assembly (Capitan et al., 2009) for which we are able to map out all assembly pathways generated by external invasions in an exact manner. In this paper we provide a deeper analysis of the model, obtaining analytical results and introducing some approximations which allow us to reconstruct the results of our previous work. In particular, we show that the population dynamics equations of a very general class of trophic-level structured food-web have an unique interior equilibrium point which is globally stable. We show analytically that communities found as end states of the assembly process are pyramidal and we find that the equilibrium abundance of any species at any trophic level is approximately inversely proportional to the number of species in that level. We also find that the per capita growth rate of a top predator invading a resident community is key to understand the appearance of complex end states reported in our previous work. The sign of these rates allows us to separate regions in the space of parameters where the end state is either a single community or a complex set containing more than one community. We have also built up analytical approximations to the time evolution of species abundances that allow us to determine, with high accuracy, the sequence of extinctions that an invasion may cause. Finally we apply this analysis to obtain the communities in the end states. To test the accuracy of the transition probability matrix generated by this analytical procedure for the end states, we have compared averages over those sets with those obtained from the graph derived by numerical integration of the Lotka-Volterra equations. The agreement is excellent.
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In the companion paper of this set (Capitan and Cuesta, 2010) we have developed a full analytical treatment of the model of species assembly introduced in Capitan et al. (2009). This model is based on the construction of an assembly graph containing all viable configurations of the community, and the definition of a Markov chain whose transitions are the transformations of communities by new species invasions. In the present paper we provide an exhaustive numerical analysis of the model, describing the average time to the recurrent state, the statistics of avalanches, and the dependence of the results on the amount of available resource. Our results are based on the fact that the Markov chain provides an asymptotic probability distribution for the recurrent states, which can be used to obtain averages of observables as well as the time variation of these magnitudes during succession, in an exact manner. Since the absorption times into the recurrent set are found to be comparable to the size of the system, the end state is quickly reached (in units of the invasion time). Thus, the final ecosystem can be regarded as a fluctuating complex system where species are continually replaced by newcomers without ever leaving the set of recurrent patterns. The assembly graph is dominated by pathways in which most invasions are accepted, triggering small extinction avalanches. Through the assembly process, communities become less resilient (e.g., have a higher return time to equilibrium) but become more robust in terms of resistance against new invasions.
The competitive exclusion principle asserts that coexisting species must occupy distinct ecological niches (i.e. the number of surviving species can not exceed the number of resources). An open question is to understand if and how different resource dynamics affect this bound. Here, we analyze a generalized consumer resource model with externally supplied resources and show that -- in contrast to self-renewing resources -- species can occupy only half of all available environmental niches. This motivates us to construct a new schema for classifying ecosystems based on species packing properties.
Niche and neutral theory are two prevailing, yet much debated, ideas in ecology proposed to explain the patterns of biodiversity. Whereas niche theory emphasizes selective differences between species and interspecific interactions in shaping the community, neutral theory supposes functional equivalence between species and points to stochasticity as the primary driver of ecological dynamics. In this work, we draw a bridge between these two opposing theories. Starting from a Lotka-Volterra (LV) model with demographic noise and random symmetric interactions, we analytically derive the stationary population statistics and species abundance distribution (SAD). Using these results, we demonstrate that the model can exhibit three classes of SADs commonly found in niche and neutral theories and found conditions that allow an ecosystem to transition between these various regimes. Thus, we reconcile how neutral-like statistics may arise from a diverse community with niche differentiation.
Population dynamics of a competitive two-species system under the influence of random events are analyzed and expressions for the steady-state population mean, fluctuations, and cross-correlation of the two species are presented. It is shown that random events cause the population mean of each specie to make smooth transition from far above to far below of its growth rate threshold. At the same time, the population mean of the weaker specie never reaches the extinction point. It is also shown that, as a result of competition, the relative population fluctuations do not die out as the growth rates of both species are raised far above their respective thresholds. This behavior is most remarkable at the maximum competition point where the weaker species population statistics becomes completely chaotic regardless of how far its growth rate in raised.
In addition to their unusually long life cycle, periodical cicadas, {it Magicicada} spp., provide an exceptional example of spatially synchronized life stage phenology in nature. Within regions (broods) spanning 50,000 to 500,000 km$^2$, adults emerge synchronously every 13 or 17 years. While satiation of avian predators is believed to be a key component of the ability of these populations to reach high densities, it is not clear why populations at a single location remain entirely synchronized. We develop nonlinear Leslie matrix-type models of periodical cicadas that include predation-driven Allee effects and competition in addition to reproduction and survival. Using both analytical and numerical techniques, we demonstrate the observed presence of a single brood critically depends on the relationship between fecundity, competition, and predation. We analyze the single-brood, two-brood and all-brood equilibria in the large life-span limit using a tractable hybrid approximation to the Leslie matrix model with continuous time competition in between discrete reproduction events. Within the hybrid model we prove that the single-brood equilibrium is the only stable equilibrium. This hybrid model allows us to quantitatively predict population sizes and the range of parameters for which the stable single-brood and unstable two-brood and all-brood equilibria exist. The hybrid model yields a good approximation to the numerical results for the Leslie matrix model for the biologically relevant case of a 17-year lifespan.
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