When can a population spreading across sink habitats persist?

We consider populations with time-varying growth rates living in sinks. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. We provide a mathematical analysis of this surprising phenomenon, in the context of a deterministic model with periodic variation of growth rates and non-symmetric migration which are assumed to be piecewise continuous. We also consider a stochastic model with random variation of growth rates and migration. This work extends existing results of the literature on the DIG effects obtained for periodic continuous growth rates and time independent symmetric migration.

Untangling the role of temporal and spatial variations in persistence of populations

We consider a population distributed between two habitats, in each of which it experiences a growth rate that switches periodically between two values, 1-ɛ>0 or -(1+ɛ)<0. We study the specific case where the growth rate is positive in one habitat and negative in the other one for the first half of the period, and conversely for the second half of the period, that we refer as the (±1) model. In the absence of migration, the population goes to 0 exponentially fast in each environment. In this paper, we show that, when the period is sufficiently large, a small dispersal between the two patches is able to produce a very high positive exponential growth rate for the whole population, a phenomena called inflation. We prove in particular that the threshold of the dispersal rate at which the inflation appears is exponentially small with the period. We show that inflation is robust to random perturbation, by considering a model where the values of the growth rate in each patch are switched at random times: we prove that inflation occurs for low switching rate and small dispersal. We also consider another stochastic model, where after each period of time T, the values of the growth rates in each patch is chosen randomly, independently from the other patch and from the past. Finally, we provide some extensions to more complicated models, especially epidemiological and density dependent models.

Asymmetric dispersal in the multi-patch logistic equation

The standard model for the dynamics of a fragmented density-dependent population is built from several local logistic models coupled by migrations. First introduced in the 1970s and used in innumerable articles, this standard model applied to a two-patch situation has never been fully analyzed. Here, we complete this analysis and we delineate the conditions under which fragmentation associated with dispersal is either favorable or unfavorable to total population abundance. We pay special attention to the case of asymmetric dispersal, i.e., the situation in which the dispersal rate from patch 1 to patch 2 is not equal to the dispersal rate from patch 2 to patch 1. We show that this asymmetry can have a crucial quantitative influence on the effect of dispersal.

A density-dependent model of competition for one resource in the chemostat

This paper deals with a two-microbial species model in competition for a single-resource in the chemostat including general intra- and interspecific density-dependent growth rates with distinct removal rates for each species. In order to understand the effects of intra- and interspecific interference, this general model is first studied by determining the conditions of existence and local stability of steady states. With the same removal rate, the model can be reduced to a planar system and then the global stability results for each steady state are derived. The bifurcations of steady states according to interspecific interference parameters are analyzed in a particular case of density-dependent growth rates which are usually used in the literature. The operating diagrams show how the model behaves by varying the operating parameters and illustrate the effect of the intra- and interspecific interference on the disappearance of coexistence region and the occurrence of bi-stability region. Concerning the small enough interspecific interference terms, we would shed light on the global convergence towards the coexistence steady state for any positive initial condition. When the interspecific interference pressure is large enough this system exhibits bi-stability where the issue of the competition depends on the initial condition.

Discrete Versus Continuous Controlled System

Because decision making processes involve costly mechanisms, they seldom are implemented continuously but instead take place at discrete time intervals. The question is then: what is the impact of this process discreteness on the systems dynamics? This problem is set here in the context of fisheries. Through a very simple though realistic model representing a shrimp fishery, we show that a discrete decision making process may lead to dynamics that differ completely from those obtained with a continuous process. For this purpose, we consider the interaction between a shrimp stock and the exploiting fleet. We then focus on the decision process that governs the allocation of the fleet fishing effort between the two stages of this stock: the young adults living near the coast and the mature adults located offshore. We first analyze the behaviour of the system when the discreteness in the decision making is not accounted for. In that case, the system turns out to be globally stable. We then identify the behaviour of the system when the decision process is discretised. In that case the solutions of the system yield sustained periodic oscillations. Our conclusion is that discreteness which is known to occur in decision making processes of most anthropic systems should be taken into account, especially in studies aiming precisely at investigating the dynamics of such systems.

Association between competition and obligate mutualism in a chemostat

In this paper, we consider a simple chemostat model involving two obligate mutualistic species feeding on a limiting substrate. Systems of differential equations are proposed as models of this association. A detailed qualitative analysis is carried out. We show the existence of a domain of coexistence, which is a set of initial conditions in which both species survive. We demonstrate, under certain supplementary assumptions, the uniqueness of the stable equilibrium point which corresponds to the coexistence of the two species.

Denaturing gradient electrophoresis (DGE) and single-strand conformation polymorphism (SSCP) molecular fingerprintings revisited by simulation and used as a tool to measure microbial diversity

The exact extent of microbial diversity remains unknowable. Nevertheless, fingerprinting patterns [denaturing gradient electrophoresis (DGE), single-strand conformation polymorphism (SSCP)] provide an image of a microbial ecosystem and contain diversity data. We generated numerical simulation fingerprinting patterns based on three types of distribution (uniform, geometric and lognormal) with a range of units from 10 to 500,000. First, simulated patterns containing a diversity of around 1000 units or more gave patterns similar to those obtained in experiments. Second, the number of bands or peaks saturated quickly to about 35 and were unrelated to the degree of diversity. Finally, assuming lognormal distribution, we used an estimator of diversity on in silico and experimental fingerprinting patterns. Results on in silico patterns corresponded to the simulation inputs. Diversity results in experimental patterns were in the same range as those obtained from the same DNA sample in molecular inventories. Thus, fingerprinting patterns contain extractable data about diversity although not on the basis of a number of bands or peaks, as is generally assumed to be the case.

A new hypothesis to explain the coexistence of n species in the presence of a single resource

This paper presents a hypothesis allowing us to explain the coexistence of several species (here micro-organisms) in competition on a single resource (called a substrate) in a chemostat. We introduce a new class of kinetics that does not only depend on the substrate concentration in the medium, but also on the biomass concentration. From the study of elementary interactions (i) between micro-organisms, (ii) between micro-organisms and their environment in which they grow and from simulations, we show that this modelling approach can be interpreted in terms of substrate diffusion phenomena. A rigorous study of this new class of models allows us to hypothesize that abiotic parameters can explain the fact that an arbitrarily large number of species can coexist in the presence of a unique substrate.

[On a density-dependent model of competition for one resource]

We use the concept of steady-state characteristic of a population using a single limiting resource, in order to discuss the issue of the competition of many species for the same resource. The steady-state characteristic is a curve that is associated to each species, likely to be determined empirically. Once one knows the steady-state characteristics and the dynamic of the renewal of the resource, it is possible to predict to some extent the issue of the competition and to give sufficient conditions for coexistence.

Combinatorial properties of some cellular automata related to the mosaic cycle concept

A cellular automaton that is related to the "mosaic cycle concept" is considered. We explain why such automata sustain very often, but not always, n-periodic trajectories (n being the number of states of the automaton). Our work is a first step in the direction of a theory of these type of automata which might be useful in modeling mosaic successions.

Evolution of DNA base composition under no-strand-bias conditions when the substitution rates are not constant

The evolution of DNA base composition evolution is simplified to a six-parameter model when there are no strand biases for mutation and selection. We analyzed the dynamics of this model with special attention to the influence of a change in substitution rates. The G + C content of the DNA sequence tends to an equilibrium value that is controlled by four parameters of the model. When the substitution rates are not constant, the G + C equilibrium position is not constant. The DNA sequence base frequencies always tend to a state in which A = T and G = C within a strand, regardless of substitution rates. This is true even when the substitution rates are not constant over time. This provides a simple way of rejecting the model from inspection of present-day DNA base composition.