Fragmentation followed by desertification in water-limited resources and/or nutrient-poor ecosystems is a major risk to the biological productivity of vegetation. By using the vegetation interaction-redistribution model, we analyse the interaction between localised vegetation patches. Here we show analytically and numerically that the interaction between two or more patches is always repulsive. As a consequence, only a single localised vegetation patch is stable, and other localised bounded states or clusters of them are unstable. Following this, we discuss the impact of the repulsive nature of the interaction on the formation and the selection of vegetation patterns in fragmented ecosystems.

Combination therapy aims to prevent growth of organisms not resistant to all component drugs, making it an obvious strategy for countering the global rise of multi-drug resistance. However, success relies on preventing resistance from arising to all component drugs before full inhibition is reached during treatment. Here, we investigated whether bacterial populations can overcome combination therapy by evolving 'multi-resistance', i.e. independent resistance mutations to multiple drugs, during single-drug and combination antibiotic treatment. Using both experimental evolution and in silico stochastic simulations, we studied resistance evolution in a common laboratory strain of bacteria (Escherichia coli K-12 BW25113). Populations were exposed to either single-drug or combination treatments involving rifampicin and nalidixic acid, with concentrations increasing through time. For wild-type populations, multi-resistance was not detected in any of the experimental populations, and simulations predict its evolution should be rare.However, populations comprising mixtures of wild-type and 'mutator' strains were readily capable of evolving multi-resistance. Increasing the initial frequency of mutators resulted in a higher proportion of populations evolving multi-resistance. Experiments and simulations produced the same qualitative-and in many cases, quantitative-insights about the association between resistance, mutators and antibiotic treatment. In particular, both approaches demonstrated that multi-resistance can arise through sequential acquisition of independent resistance mutations, without a need to invoke multi-drug resistance mechanisms. Crucially, we found multi-resistance evolved even when not directly favoured by natural selection, i.e. under single-drug treatments. Simulations revealed this resulted from elevated mutation supply caused by genetic hitch-hiking of the mutator allele on single-drug resistant backgrounds. Our results suggest that combination therapy does not necessarily prevent sequential acquisition of multiple drug resistances via spontaneous mutation when mutators are present. Indeed both combination and single-drug treatments actively promoted multi-resistance, meaning that combination therapy will not be a panacea for the antibiotic resistance crisis. inference, stochastic simulations, individual-based model, Markov chains W , K. B., 2016 Pairwise interactions and the battle against combinatorics in multidrug therapies.

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The dynamics of ecological systems are often described by integrodifferential equations that incorporate nonlocal interactions associated with facilitative, competitive interactions between plants, and seed dispersion. In the weak-gradient limit, these models can be reduced to a simple partial-differential equation in the form of a nonvariational Swift-Hohenberg equation. In this contribution, we perform this reduction for any type of kernels provided that their Taylor series converge. Some parameters such as linear and nonlinear diffusion coefficients are affected by the spatial form of the kernel. In particular, Gaussian and exponential kernels are used to evaluate all coefficients of the reduced model. This weak gradient approximation is greatly useful for the investigation of periodic and localized vegetation patches, and gaps. Based on this simple model, we investigate the interaction between two-well separated patches and gaps. In the case of patches, the interaction is always repulsive. As a consequence, bounded states of patches are excluded. However, when two gaps are close to one another, they start to interact through their oscillatory tails. The interaction alternates between attractive and repulsive depending on the distance separating them. This allows for the stabilization of bounded gaps and clusters of them. The analytical formula of the interaction potential is derived for both patches and gaps interactions and checked by numerical investigation of the model equation. This volume is dedicated to Professor Ehud Meron on the occasion of his sixtieth birthday. We take this opportunity to express our warmest and most sincere wishes to him.

While facultative sex is common in sexually reproducing species, for reasons of tractability most mathematical models assume that such sex is asynchronous in the population. In this paper, we develop a model of switching environments to instead capture the effect of an entire population transitioning synchronously between sexual and asexual modes of reproduction. We use this model to investigate the evolution of the number of self-incompatible mating types in finite populations, which empirically can range from two to thousands. When environmental switching is fast, we recover the results of earlier studies that implicitly assumed populations were engaged in asynchronous sexual reproduction. However when the environment switches slowly, we see deviations from previous asynchronous theory, including a lower number of mating types at equilibrium and bimodality in the stationary distribution of mating types. We provide analytic approximations for both the fast and slow switching regimes, as well as a numerical scheme based on the Kolmogorov equations for the system to quickly evaluate the model dynamics at intermediate parameters. Our approach exploits properties of integer partitions in number theory. We also demonstrate how additional biological processes such as selective sweeps can be accounted for in this switching environment framework, showing that beneficial mutations can further erode mating type diversity in synchronous facultatively sexual populations..