Improving the false nearest neighbors method with graphical analysis

We introduce a graphical presentation for the false nearest neighbors (FNN) method. In the original method only the percentage of false neighbors is computed without regard to the distribution of neighboring points in the time-delay coordinates. With this new presentation it is much easier to distinguish deterministic chaos from noise. The graphical approach also serves as a tool to determine better conditions for detecting low dimensional chaos, and to get a better understanding on the applicability of the FNN method.Comment: 4 pages, with 5 PostScript figure

How Should We Define Fitness in Structured Metapopulation Models? Including an Application to the Calculation of Evolutionarily Stable Dispersal Strategies

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches, and provide means for calculating its numerical value. In addition we introduce a more easily calculated quantity RM that relates to fitness in the same manner as RO relates to fitness in ordinary population dynamics: RM of a mutant is only defined when the resident population dynamics converges to an equilibrium, and RM is larger (smaller) than one if and only if mutant fitness is positive (negative). RM corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. As an example of the usefulness of these concepts we calculate the ES conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions

Evolutionary Suicide and Evolution of Dispersal in Structured Metapopulations

In this article we study the evolution of dispersal in a structured metapopulation model. The metapopulation consists of a large (infinite)number of local populations living in patches of habitable environment. Dispersal between patches is modeled by a disperser pool and individuals in transit between patches are exposed to a risk of mortality. Occasionally, local catastrophes eradicate a local population: all individuals in the affected patch die, yet the patch remains habitable. The rate at which such disasters occur can depend on the local population size of a patch. We prove that, in the absence of catastrophes, the strategy not to migrate is evolutionarily stable. It is also convergence stable unless there is no mortality during dispersal. Under a given set of environmental conditions, a metapopulation may be viable and yet selection may favor dispersal rates that drive the metapopulation to extinction. This phenomenon is known as evolutionary suicide. We show that in our model evolutionary suicide can occur for certain types of size-dependent catastrophes. Evolutionary suicide can also happen for constant catastrophe rates, if local growth within patches shows an Allee effect. We study the evolutionary bifurcation towards evolutionary suicide and show that a discontinuous transition to extinction is a necessary condition for evolutionary suicide to occur. In other words, if population size smoothly approaches zero at a boundary of viability in parameter space, this boundary is evolutionarily repelling and no suicide can occur

On the Formulation and Analysis of General Deterministic Structured Population Models. I. Linear Theory

We define a linear physiologically structured population model by two rules, one for reproduction and one for "movement" and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio Ro and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step

Dynamics of Similar Populations: The Link Between Population Dynamics and Evolution

We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of "similar" populations. Similarity implies that the "relative" dynamics of the populations is slow compared to, and decoupled from, their "aggregated" dynamics. The relative dynamics is simple, and captured by a Taylor expansion in the difference between the populations. The emerging evolution is directional, except at the "singular" points of the evolutionary state space, where "evolutionary branching" may happen

Competitive Exclusion and Limiting Similarity: A Unified Theory

Robustness of coexistence against changes of parameters is investigated in a model-independent manner through analyzing the feed-back loop of population regulation. We define coexistence as fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears when the Jacobian of the dynamics decreases to zero. A general notion of regulating factors/variables is introduced. For each population, its 'impact' and 'sensitivity' niches a re defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either similarity of the impact niches, or similarity of the sensitivity niches, result in a small Jacobian and in a reduced likelihood of coexistence. For the case of a resource continuum, this result reduces to the usual "limited niches overlap" picture for both kinds of niche. As an extension of these ideas to the coexistence of infinitely many species, we demonstrate that Roughgarden's example for coexistence of a 'continuum' of populations is structurally unstable

Evolution of Dispersal in Metapopulations with Local Density Dependence and Demographic Stochasticity

Selective pressures governing the evolution of dispersal rates are difficult to evaluate and currently poorly understood. In particular, predictions of evolutionarily stable dispersal strategies have only been derived under a number of limiting conditions regarding the ecology of of dispersing species. In this paper we predict the outcome of dispersal evolution in metapopulations based on a suit of assumptions that are more likely to be met in the field: (i) population dynamics within patches are density-regulated by realistic growth functions,(ii) demographic stochasticity resulting from finite population sizes within patches is accounted for and (iii) the transition of individuals between patches is explicitly modeled by a disperser pool. In addition we make few further changes which add to the models interest for comparison purposes; (iv) individuals can disperse between habitable patches throughout their lifetime, and (v) metapopulations are described in continuous time instead of relying on season-to-season descriptions. Extending available models in regard to these features, we demonstrate the existence of two general patterns of metapopulation adaptation. We show,first, that evolutionarily stable dispersal rates do not necessarily increase with rates for the local extinction of populations due to external disturbances in habitable patches. Instead, without demographic stochasticity, adapted dispersal rates exhibit a maximum for intermediate levels of disturbance and fall off for both higher and lower rates of local extinction. Second, we describe how the demographic stochasticity that inevitably occurs in finite populations affects the evolution of dispersal rates. Contrary to predictions from deterministic models, evolutionarily stable dispersal rates in metapopulations composed of small local populations can remain high even when rates of local extinction are low. The first pattern is shown to be robust, provided that demographic stochasticity is not too severe. under a range of local growth conditions, including logistic growth and its variants. We also demonstrate that high degrees of demographic stochasticity can enrich the behavior of adapted dispersal rates in response to varied levels of disturbance: monotonic increases or decreases can be observed as well as intermediate maxima or minima

On the Formulation and Analysis of General Deterministic Structured Population Models

We define a linear physiologically structured population model by two rules, one for reproduction and one for "movement" and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R0 and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step

Continuous coexistence or discrete species? A new review of an old question

Question: Is the coexistence of a continuum of species or ecological types possible in real-world communities? Or should one expect distinctly different species? Mathematical methods: We study whether the coexistence of species in a continuum of ecological types is (a) dynamically stable (against changes in population densities) and (b) structurally robust (against changes in population dynamics). Since most of the reviewed investigations are based on Lotka-Volterra models, we carefully explain which of the presented conclusions are model-independent. mathematical conclusions: Seemingly plausible models with dynamically stable continuous- coexistence solutions do exist. However, these models either depend on biologically unrealistic mathematical assumptions (e.g. non-differentiable ingredient functions) or are structurally unstable (i.e. destroyable by arbitrarily small modifications to those ingredient functions). The dynamical stability of a continuous-coexistence solution, if it exists, requires positive definiteness of the model's competition kernel. Biological conclusions: While the classical expectation of fixed limits to similarity is mathematically naive, the fundamental discreteness of species is a natural consequence of the basic structure of ecological interactio

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.Comment: 83 page