Scaling up the predator functional response in heterogeneous environment: when Holling type III can emerge?

Accurate parametrization of functional terms in model equations is of great importance for reproducing the dynamics of real food webs. Constructing models over large spatial and temporal scales using mathematical expressions obtained based on microcosm experiments can be erroneous. Here, using a generic spatial predator-prey model, we show that scaling up the microscale functional response of a predator can result in qualitative alterations of functional response on macroscales. In particular, a global functional response of sigmoid type (Holling type III) can emerge as a result of non-linear averaging of non-sigmoid local responses (Holling type I or II). We demonstrate that alteration between the local and the global response in the model is a result of the interplay between density-dependent dispersal of the predator across the habitat and heterogeneity of the environment. Using the method of aggregation of variables, we analytically derive the mathematical formulation of the global functional response as a function of the total amount of prey in the system, and reveal the key parameters which control the emergence of a Holling type III global response. We argue that this mechanism by which a global Holling type III emerges from a local Holling type II response has not been reported in the literature yet: in particular, Holling type III can emerge in the case of a fixed gradient of resource distribution across the habitat, which would be impossible in priorly suggested mechanisms. As a case study, we consider the interaction between phytoplankton and zooplankton grazers in the water column; and we show that the emergence of a Holling type III global response can allow for the efficient top-down regulation of primary producers and stabilization of planktonic ecosystems under eutrophic conditions.

Structural sensitivity of biological models revisited.

Enhancing the predictive power of models in biology is a challenging issue. Among the major difficulties impeding model development and implementation are the sensitivity of outcomes to variations in model parameters, the problem of choosing of particular expressions for the parametrization of functional relations, and difficulties in validating models using laboratory data and/or field observations. In this paper, we revisit the phenomenon which is referred to as structural sensitivity of a model. Structural sensitivity arises as a result of the interplay between sensitivity of model outcomes to variations in parameters and sensitivity to the choice of model functions, and this can be somewhat of a bottleneck in improving the models predictive power. We provide a rigorous definition of structural sensitivity and we show how we can quantify the degree of sensitivity of a model based on the Hausdorff distance concept. We propose a simple semi-analytical test of structural sensitivity in an ODE modeling framework. Furthermore, we emphasize the importance of directly linking the variability of field/experimental data and model predictions, and we demonstrate a way of assessing the robustness of modeling predictions with respect to data sampling variability. As an insightful illustrative example, we test our sensitivity analysis methods on a chemostat predator-prey model, where we use laboratory data on the feeding of protozoa to parameterize the predator functional response.

Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities.

Although density-dependent dispersal and relative dispersal (the difference in dispersal rates between species) have been documented in natural systems, their effects on the stability of metacommunities are poorly understood. Here we investigate the effects of intra- and interspecific density-dependent dispersal on the regional stability in a predator-prey metacommunity model. We show that, when the dynamics of the populations reach equilibrium, the stability of the metacommunity is not affected by density-dependent dispersal. However, the regional stability, measured as the regional variability or the persistence, can be modified by density-dependent dispersal when local populations fluctuate over time. Moreover these effects depend on the relative dispersal of the predator and the prey. Regional stability is modified through changes in spatial synchrony. Interspecific density-dependent dispersal always desynchronizses local dynamics, whereas intraspecific density-dependent dispersal may either synchronize or desynchronize it depending on dispersal rates. Moreover, intra- and interspecific density-dependent dispersal strengthen the top-down control of the prey by the predator at intermediate dispersal rates. As a consequence the regional stability of the metacommunity is increased at intermediate dispersal rates. Our results show that density-dependent dispersal and relative dispersal of species are keys to understanding the response of ecosystems to fragmentation.

Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations.

Deterministic population models for adaptive dynamics are derived mathematically from individual-centred stochastic models in the limit of large populations. However, it is common that numerical simulations of both models fit poorly and give rather different behaviours in terms of evolution speeds and branching patterns. Stochastic simulations involve extinction phenomenon operating through demographic stochasticity, when the number of individual 'units' is small. Focusing on the class of integro-differential adaptive models, we include a similar notion in the deterministic formulations, a survival threshold, which allows phenotypical traits in the population to vanish when represented by few 'individuals'. Based on numerical simulations, we show that the survival threshold changes drastically the solution; (i) the evolution speed is much slower, (ii) the branching patterns are reduced continuously and (iii) these patterns are comparable to those obtained with stochastic simulations. The rescaled models can also be analysed theoretically. One can recover the concentration phenomena on well-separated Dirac masses through the constrained Hamilton-Jacobi equation in the limit of small mutations and large observation times.

Evolution and persistence of obligate mutualists and exploiters: competition for partners and evolutionary immunization.

Mutualisms are ubiquitous in nature, as is their exploitation by both conspecific and heterospecific cheaters. Yet, evolutionary theory predicts that cheating should be favoured by natural selection. Here, we show theoretically that asymmetrical competition for partners generally determines the evolutionary fate of obligate mutualisms facing exploitation by third-species invaders. When asymmetry in partner competition is relatively weak, mutualists may either exclude exploiters or coexist with them, in which case their co-evolutionary response to exploitation is usually benign. When asymmetry is strong, the mutualists evolve towards evolutionary attractors where they become extremely vulnerable to exploiter invasion. However, exploiter invasion at an early stage of the mutualism's history can deflect mutualists' co-evolutionary trajectories towards slightly different attractors that confer long-term stability against further exploitation. Thus, coexistence of mutualists and exploiters may often involve an historical effect whereby exploiters are co-opted early in mutualism history and provide lasting 'evolutionary immunization' against further invasion.

Cheating and the evolutionary stability of mutualisms.

Interspecific mutualisms have been playing a central role in the functioning of all ecosystems since the early history of life. Yet the theory of coevolution of mutualists is virtually nonexistent, by contrast with well-developed coevolutionary theories of competition, predator-prey and host-parasite interactions. This has prevented resolution of a basic puzzle posed by mutualisms: their persistence in spite of apparent evolutionary instability. The selective advantage of 'cheating', that is, reaping mutualistic benefits while providing fewer commodities to the partner species, is commonly believed to erode a mutualistic interaction, leading to its dissolution or reciprocal extinction. However, recent empirical findings indicate that stable associations of mutualists and cheaters have existed over long evolutionary periods. Here, we show that asymmetrical competition within species for the commodities offered by mutualistic partners provides a simple and testable ecological mechanism that can account for the long-term persistence of mutualisms. Cheating, in effect, establishes a background against which better mutualists can display any competitive superiority. This can lead to the coexistence and divergence of mutualist and cheater phenotypes, as well as to the coexistence of ecologically similar, but unrelated mutualists and cheaters.