Inverse problems in partial differential equations

Identification in partial differential equations by Laplace equatio

The stability of ecosystems: a brief overview of the paradox of enrichment

In theory, enrichment of resource in a predator-prey model leads to destabilization of the system, thereby collapsing the trophic interaction, a phenomenon referred to as "the paradox of enrichment". After it was first proposed by Rosenzweig (1971), a number of subsequent studies were carried out on this dilemma over many decades. In this article, we review these theoretical and experimental works and give a brief overview of the proposed solutions to the paradox. The mechanisms that have been discussed are modifications of simple predator-prey models in the presence of prey that is inedible, invulnerable, unpalatable and toxic. Another class of mechanisms includes an incorporation of a ratio-dependent functional form, inducible defence of prey and density-dependent mortality of the predator. Moreover, we find a third set of explanations based on complex population dynamics including chaos in space and time. We conclude that, although any one of the various mechanisms proposed so far might potentially prevent destabilization of the predator-prey dynamics following enrichment, in nature different mechanisms may combine to cause stability, even when a system is enriched. The exact mechanisms, which may differ among systems, need to be disentangled through extensive field studies and laboratory experiments coupled with realistic theoretical models

Long-term cyclic persistence in an experimental predator–prey system

Predator–prey cycles rank among the most fundamental concepts in ecology, are predicted by the simplest ecological models and enable, theoretically, the indefinite persistence of predator and prey1,2,3,4. However, it remains an open question for how long cyclic dynamics can be self-sustained in real communities. Field observations have been restricted to a few cycle periods5,6,7,8 and experimental studies indicate that oscillations may be short-lived without external stabilizing factors9,10,11,12,13,14,15,16,17,18,19. Here we performed microcosm experiments with a planktonic predator–prey system and repeatedly observed oscillatory time series of unprecedented length that persisted for up to around 50 cycles or approximately 300 predator generations. The dominant type of dynamics was characterized by regular, coherent oscillations with a nearly constant predator–prey phase difference. Despite constant experimental conditions, we also observed shorter episodes of irregular, non-coherent oscillations without any significant phase relationship. However, the predator–prey system showed a strong tendency to return to the dominant dynamical regime with a defined phase relationship. A mathematical model suggests that stochasticity is probably responsible for the reversible shift from coherent to non-coherent oscillations, a notion that was supported by experiments with external forcing by pulsed nutrient supply. Our findings empirically demonstrate the potential for infinite persistence of predator and prey populations in a cyclic dynamic regime that shows resilience in the presence of stochastic events

Quantitative and Molecular Genetic Analyses of Mutations Increasing Drosophila Life Span

Understanding the genetic and environmental factors that affect variation in life span and senescence is of major interest for human health and evolutionary biology. Multiple mechanisms affect longevity, many of which are conserved across species, but the genetic networks underlying each mechanism and cross-talk between networks are unknown. We report the results of a screen for mutations affecting Drosophila life span. One third of the 1,332 homozygous P–element insertion lines assessed had quantitative effects on life span; mutations reducing life span were twice as common as mutations increasing life span. We confirmed 58 mutations with increased longevity, only one of which is in a gene previously associated with life span. The effects of the mutations increasing life span were highly sex-specific, with a trend towards opposite effects in males and females. Mutations in the same gene were associated with both increased and decreased life span, depending on the location and orientation of the P–element insertion, and genetic background. We observed substantial—and sex-specific—epistasis among a sample of ten mutations with increased life span. All mutations increasing life span had at least one deleterious pleiotropic effect on stress resistance or general health, with different patterns of pleiotropy for males and females. Whole-genome transcript profiles of seven of the mutant lines and the wild type revealed 4,488 differentially expressed transcripts, 553 of which were common to four or more of the mutant lines, which include genes previously associated with life span and novel genes implicated by this study. Therefore longevity has a large mutational target size; genes affecting life span have variable allelic effects; alleles affecting life span exhibit antagonistic pleiotropy and form epistatic networks; and sex-specific mutational effects are ubiquitous. Comparison of transcript profiles of long-lived mutations and the control line reveals a transcriptional signature of increased life span

The functional costs and benefits of dietary restriction in Drosophila

Dietary restriction (DR) extends lifespan in an impressively wide array of species spanning three eukaryotic kingdoms. In sharp contrast, relatively little is known about the effects of DR on functional senescence, with most of the work having been done on mice and rats. Here we used Drosophila melanogaster to test the assumption that lifespan extension through DR slows down age-related functional deterioration. Adult virgin females were kept on one of three diets, with sucrose and yeast concentrations ranging from 7% to 11% to 16% (w/v). Besides age-specific survival and fecundity, we measured starvation resistance, oxidative stress resistance, immunity, and cold-stress resilience at ages 1, 3, 5, and 7 weeks. We confirmed that DR extends lifespan: median lifespans ranged from 38 days (16% diet) to 46 days (11% diet) to 54 days (7% diet). We also confirmed that DR reduces fecundity, although the shortest-lived flies only had the highest fecundity when males were infrequently available. The most striking result was that DR initially increased starvation resistance, but strongly decreased starvation resistance later in life. Generally, the effects of DR varied across traits and were age dependent. We conclude that DR does not universally slow down functional deterioration in Drosophila. The effects of DR on physiological function might not be as evolutionarily conserved as its effect on lifespan. Given the age-specific effects of DR on functional state, imposing DR late in life might not provide the same functional benefits as when applied at early ages

Effects of assay conditions in life history experiments with Drosophila melanogaster

Selection experiments with Drosophila have revealed constraints on the simultaneous evolution of life history traits. However, the responses to selection reported by different research groups have not been consistent. Two possible reasons for these inconsistencies are (i) that different groups used different environments for their experiments and (ii) that the selection environments were not identical to the assay environments in which the life history traits were measured. We tested for the effect of the assay environment in life history experiments by measuring a set of Drosophila selection lines in laboratories working on life history evolution with Drosophila in Basel, Groningen, Irvine and London. The lines measured came from selection experiments from each of these laboratories. In each assay environment, we measured fecundity, longevity, development time and body size. The results show that fecundity measurements were particularly sensitive to the assay environment. Differences between assay and selection environment in the same laboratory or differences between assay environments between laboratories could have contributed to the differences in the published results. The other traits measured were less sensitive to the assay environment. However, for all traits there were cases where the measurements in one laboratory suggested that selection had an effect on the trait, whereas in other laboratories no such conclusion would have been drawn. Moreover, we provide good evidence for local adaptation in early fecundity for lines from two laboratories