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We introduced a more general predator-prey model to analyze the paradox of enrichment. We hope the results obtained for the model can guide us on identifying real field paradox of enrichment.
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We present a new non-Archimedean model of evolutionary dynamics, in which the genomes are represented by p-adic numbers. In this model the genomes have a variable length, not necessarily bounded, in contrast with the classical models where the length is fixed. The time evolution of the concentration of a given genome is controlled by a p-adic evolution equation. This equation depends on a fitness function f and on mutation measure Q. By choosing a mutation measure of Gibbs type, and by using a p-adic version of the Maynard Smith Ansatz, we show the existence of threshold function M_{c}(f,Q), such that the long term survival of a genome requires that its length grows faster than M_{c}(f,Q). This implies that Eigens paradox does not occur if the complexity of genomes grows at the right pace. About twenty years ago, Scheuring and Poole, Jeffares, Penny proposed a hypothesis to explain Eigens paradox. Our mathematical model shows that this biological hypothesis is feasible, but it requires p-adic analysis instead of real analysis. More exactly, the Darwin-Eigen cycle proposed by Poole et al. takes place if the length of the genomes exceeds M_{c}(f,Q).
The stabilizing effects of local enrichment are revisited. Diffusively coupled host-parasitoid and predator-prey metapopulations are shown to admit a stable fixed point, limit cycle or stable torus with a rich bifurcation structure. A linear toy model that yields many of the basic qualitative features of this system is presented. The further nonlinear complications are analyzed in the framework of the marginally stable Lotka-Volterra model, and the continuous time analog of the unstable, host-parasitoid Nicholson-Bailey model. The dependence of the results on the migration rate and level of spatial variations is examined, and the possibility of nonlocal effect of enrichment, where local enrichment induces stable oscillations at a distance, is studied. A simple method for basic estimation of the relative importance of this effect in experimental systems is presented and exemplified.
Marine species reproduce and compete while being advected by turbulent flows. It is largely unknown, both theoretically and experimentally, how population dynamics and genetics are changed by the presence of fluid flows. Discrete agent-based simulations in continuous space allow for accurate treatment of advection and number fluctuations, but can be computationally expensive for even modest organism densities. In this report, we propose an algorithm to overcome some of these challenges. We first provide a thorough validation of the algorithm in one and two dimensions without flow. Next, we focus on the case of weakly compressible flows in two dimensions. This models organisms such as phytoplankton living at a specific depth in the three-dimensional, incompressible ocean experiencing upwelling and/or downwelling events. We show that organisms born at sources in a two-dimensional time-independent flow experience an increase in fixation probability.
We consider the dynamics of a three-species system incorporating the Allee Effect, focussing on its influence on the emergence of extreme events in the system. First we find that under Allee effect the regular periodic dynamics changes to chaotic. Further, we find that the system exhibits unbounded growth in the vegetation population after a critical value of the Allee parameter. The most significant finding is the observation of a critical Allee parameter beyond which the probability of obtaining extreme events becomes non-zero for all three population densities. Though the emergence of extreme events in the predator population is not affected much by the Allee effect, the prey population shows a sharp increase in the probability of obtaining extreme events after a threshold value of the Allee parameter, and the vegetation population also yields extreme events for sufficiently strong Allee effect. Lastly we consider the influence of additive noise on extreme events. First, we find that noise tames the unbounded vegetation growth induced by Allee effect. More interestingly, we demonstrate that stochasticity drastically diminishes the probability of extreme events in all three populations. In fact for sufficiently high noise, we do not observe any more extreme events in the system. This suggests that noise can mitigate extreme events, and has potentially important bearing on the observability of extreme events in naturally occurring systems.
Meso- and submesoscales (fronts, eddies, filaments) in surface ocean flow have a crucial influence on marine ecosystems. Their dynamics partly control the foraging behaviour and the displacement of marine top predators (tuna, birds, turtles, and cetaceans). In this work we focus on the role of submesoscale structures in the Mozambique Channel on the distribution of a marine predator, the Great Frigatebird. Using a newly developed dynamical concept, namely the Finite-Size Lyapunov Exponent (FSLE), we have identified Lagrangian coherent structures (LCSs) present in the surface flow in the Channel over a 2-month observation period (August and September 2003). By comparing seabirds satellite positions with LCSs locations, we demonstrate that frigatebirds track precisely these structures in the Mozambique Channel, providing the first evidence that a top predator is able to track these FSLE ridges to locate food patches. After comparing bird positions during long and short trips, and different parts of these trips, we propose several hypotheses to understand how frigatebirds can follow these LCSs. The birds might use visual and/or olfactory cues and/or atmospheric current changes over the structures to move along these biological corridors. The birds being often associated to tuna schools around foraging areas, a thorough comprehension of their foraging behaviour and movement during the breeding season is crucial not only to seabirds ecology but also to an appropriate ecosystemic approach of fisheries in the Channel.
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