Swarnendu Banerjee

Indian Statistical Institute, Kolkata

"Chemical contamination mediated regime shifts in planktonic systems"

Increasing chemical contamination is a growing concern worldwide. Although, regime shifts leading to algal blooms are quite well known in aquatic ecosystems, the effect of contamination on such regime shift is not particularly well understood. Motivated by this, we studied the effect of copper enrichment on planktonic system using a minimal phytoplankton-zooplankton model. Interestingly, our results suggest that both the toxic and deficient concentration of copper in water bodies can lead to catastrophic transition of the ecosystem to an alternative stable state. Further, on adding stochasticity to the system the region of bistability is diminished and the system switches from zooplankton dominated state to the phytoplankton dominated state much prior to the tipping point. The bistability is further weakened on increasing noise intensity and redness. However, in case of systems with high nutrient enrichment, the bimodality in the probability density can be very prominent. Nevertheless, generic early warning signals may fails to predict an impending state shift due to contamination. Our study provides important perspective to regime shifts in the context of eco-toxicology.

Jody Reimer

University of Utah

"Long transient dynamics in the presence of noise"

Recent theoretical work has highlighted several mechanisms giving rise to so-called ``long transient'' dynamics. These long transients tantalizingly appear to replicate dynamics seen in real systems-with one critical difference: ecological data is noisy, a reality theoretical work often ignores. In general, stochasticity is known to have important consequences: it can qualitatively alter model dynamics as well as impact our ability to infer underlying processes through statistical analysis. To explore the effect of stochasticity on qualitative model behavior and the implications for our ability to infer underlying mechanisms, we generated time series from a simple model of long transient behavior with additive noise. We then examined if noise qualitatively changes the expected dynamics of the system and how well phenomenological and mechanistic statistical models could recover the underlying model. We found that long transients such as those generated by even the simplest stochastic model of a ghost attractor are highly sensitive to noise, and that the mean behavior of the stochastic model differs substantially from that of the deterministic model. In spite of this, we illustrate that statistical inference on a single realization may still provide insight into model parameters, and highlight that inference improves for an increasing number of realizations of the process. All approaches saw improved results with increasing data realizations. We suggest methods to increase our ability to draw inference from real ecological time series with suspected long transient dynamics.

Carlos A. Braumann

Universidade de Evora

"General autonomous fishing models with Allee effects in a randomly varying environment"

In a randomly varying environment, a general fishing model is the stochastic differential equation dX(t) = f(X(t))X(t)dt+σX(t)dW(t)−qE(t,X(t))X(t)dt, where X(t) is the fished population size at time t, f (of class C1) is the per capita arithmetic average natural growth rate, σdW(t)/dt describes the effect of environmental fluctuations on the growth rate (with W(t) a standard Wiener process and σ > 0), E(t,X(t)) is the harvesting effort applied and q > 0 is the catchability. Here, we will consider autonomous models for which E(t, X(t)) ≡ E(X(t)) non-negative of class C1. The usual density-dependence case with f strictly decreasing and f(+∞) < 0 was studied in [1] w.r.t. conditions for population extinction or for existence of a stationary density. In [5, 6], for the particular cases of f being logistic or Gompertz, profit optimization was studied comparing variable effort E(t, X(t)) fishing policies with constant effort E(t, X(t)) ≡ E policies. Sometimes, however, the population under fishing is affected by Allee effects, with an unexpected depression (accompanied by growth) of f(x) for small values of x, due, for instance, to the difficulty of finding mating partners or of putting together an effective collective defence from predators. [4] made a comparative study between variable and constant effort policies for the particular case of f being logistic-like with Allee effects. In [2, 3], for populations not subjected to fishing and for general growth models f with Allee effects, conditions for extinction or for existence of a stationary density were studied. We are now generalizing this study, under appopriate conditions, to fished populations with general growth models f(·) with Allee effects and with also general autonomous harvesting efforts E(·). Again, we found out that the deciding factor between extinction or existence of a stationary density is the sign, when the population size is small, of the net (i.e., discounting the mortality rate due to fishing) geometric average per capita growth rate. The cases previously studied, as well as the gear saturation phenomenon, can be treated as particular cases.

Toni Klauschies

Potsdam University

"Ecological and evolutionary causes of intermittent predator-prey cycles"

The presence of trait variation in prey or predator populations may affect the stability and the shape, i.e. amplitude and phase, of predator-prey dynamics. However, while previous studies have shown how trait variation can alter the overall amplitude of the predator-prey oscillations, this altered amplitude remained constant over time. This strongly contrasts with empirically observed predator-prey dynamics and recent theoretical work, showing that several mechanisms may lead to so called intermittent predator- prey cycles where the amplitude of the predator-prey dynamics varies temporally. For instance, trait differences that determine the functional responses of two predators may provoke temporal fluctuations in the amplitudes of the predator-prey cycles due to recurrent changes in the relative abundance of the two predator types: a predator with a relatively linear functional response promotes small-amplitude oscillations whereas a predator with a more strongly non-linear functional response stimulates larger amplitudes. We analysed various models that incorporate trait variation within prey, predators, or both, and identified three general conditions that are necessary for intermittent cycles to occur. First, the predator-prey system comprises at least two subsystems that exhibit substantial differences in the amplitude of their population dynamics and thus tendency to promote stable or unstable population dynamics. Second, these subsystems recurrently alternate in their dominance, leading to a second “trait” cycle superimposed on the population dynamics. Finally, the time scale of the trait dynamics must be significantly slower than that of the population dynamics. For instance, co-evolution may promote intermittent cycles in predator-prey dynamics by inducing a lag in the predators’ trait adjustment in response to altered trait values of the prey. The resulting temporal variation in the interaction strength between predator and prey is associated with temporal changes in the amplitudes of the population dynamics since e.g., a dominance of defended prey dampens the population dynamics whereas a high abundance of undefended prey enhances it. Our results highlight that intermittent cycles may frequently occur in simple predator-prey systems allowing for trait variation.