Mats Gyllenberg

Two General Metapopulation Models and the Core-Satellite Species Hypothesis

This article describes two general metapopulation models with spatial variation in the sizes of habitat patches. The first model is a simple, nonstructured model that includes the mainland-island and Levins models as two limiting cases. The second model is a structured model explicitly including the size distribution of habitat patches, the size distribution of local populations, and migration among local populations. The models may have up to four equilibria, including two stable, positive equilibria. We discuss the core-satellite species hypothesis in light of these models. This hypothesis predicts that the distribution of patch-occupancy frequencies is bimodal in many species assemblages. We extend the original concept by demonstrating that the bimodal distribution of patch-occupancy frequencies can be generated by structurally more complex and more realistic metapopulation models than the original one; that the bimodal distribution is predicted by deterministic models, with no or infrequent switches of species between the core and the satellite state; and that metapopulation extinctions of rare species may be compensated by migration from outside the metapopulation (from a mainland), or metapopulation extinction may be prevented by low extinction probabilities of local populations in large or high-quality habitat patches. In every case the bimodal core-satellite distribution is due to the rescue effect, that is, the increasing migration rate and hence the decreasing probability of local extinction with an increasing fraction of patches occupied. We discuss how the metapopulation dynamic mechanisms described in this article may generate the bimodal core-satellite distribution in different kinds of communities.

On the Formulation and Analysis of General Deterministic Structured Population Models. II. Nonlinear Theory

Abstract—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.

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. In addition, we introduce a more easily calculated quantity Rm that relates to fitness in the same manner as R0 relates to fitness in ordinary population dynamics: the Rm of a mutant is only defined when the resident population dynamics converges to a point equilibrium and Rm is larger (smaller) than 1 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. Efficient algorithms for calculating its numerical value are provided. As an example of the usefulness of these concepts we calculate the evolutionarily stable conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions. Below a threshold density ã, at which staying and leaving are equality profitable, everybody should stay and above ã everybody should leave, where profitability is measured as the mean number of dispersers produced through lines of descent consisting of only non–dispersers.

BODIL: a molecular modeling environment for structure-function analysis and drug design.

BODIL is a molecular modeling environment geared to help the user to quickly identify key features of proteins critical to molecular recognition, especially (1) in drug discovery applications, and (2) to understand the structural basis for function. The program incorporates state-of-the-art graphics, sequence and structural alignment methods, among other capabilities needed in modern structure–function–drug target research. BODIL has a flexible design that allows on-the-fly incorporation of new modules, has intelligent memory management, and fast multi-view graphics. A beta version of BODIL and an accompanying tutorial are available at http://www.abo.fi/fak/mnf/bkf/research/johnson/bodil.html

Single-species metapopulation dynamics: a structured model

Abstract We describe and analyse a general metapopulation model, which consists of a model of local dynamics within a habitat patch, and balance equations for dispersing individuals and the metapopulation. The model includes the effects of emigration and immigration on local dynamics. We derive the equilibrium population size distribution, which is skewed towards either small or large populations, depending on the relative magnitudes of local and metapopulation time scales. The model predicts a generally positive relationship between the fraction of occupied patches and the average local population size. Such a relationship has been commonly observed in nature. The model allows alternative stable equilibria, not found in models which ignore the effect of dispersal on local dynamics. We discuss the implications of our results for biological invasions and conservation biology.

Steady-state analysis of structured population models

Our systematic formulation of nonlinear population models is based on the notion of the environmental condition. The defining property of the environmental condition is that individuals are independent of one another (and hence equations are linear) when this condition is prescribed (in principle as an arbitrary function of time, but when focussing on steady states we shall restrict to constant functions). The steady-state problem has two components: (i). the environmental condition should be such that the existing populations do neither grow nor decline; (ii). a feedback consistency condition relating the environmental condition to the community/population size and composition should hold. In this paper we develop, justify and analyse basic formalism under the assumption that individuals can be born in only finitely many possible states and that the environmental condition is fully characterized by finitely many numbers. The theory is illustrated by many examples. In addition to various simple toy models introduced for explanation purposes, these include a detailed elaboration of a cannibalism model and a general treatment of how genetic and physiological structure should be combined in a single model.

A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators.

Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model.

A discrete model for a metapopulation consisting of two local populations connected by migration is described and analyzed. It is assumed that the local populations grow according to the logistic law, that both populations have the same emigration rate, and that migrants choose their new habitat patch at random. Mathematically this leads to a coupled system of two logistic equations. A complete characterization of fixed point and two-periodic orbits is given, and a bifurcation analysis is performed. The region in the parameter plane where the diagonal is a global attractor is determined. In the symmetric case, where both populations have the same growth rate, the analysis is rigorous with complete proofs. In the nonsymmetric case, where the populations grow at different rates, the results are obtained numerically. The results are interpreted biologically. Particular attention is given to the sense in which migration has a stabilizing and synchronizing effect on local dynamics.

Structured Metapopulation Models

Publisher Summary This chapter presents a unified treatment of a large class of deterministic, structured metapopulation models, and illustrates the mathematical framework with several examples. Being deterministic, the models continue to assume an infinite number of patches and local populations, and the results are applicable to large metapopulations. Deterministic metapopulation models with a finite number of patches are concerned with the effect of migration on local dynamics, with a special focus on how migration may synchronize and stabilize local dynamics. The chapter explains the Levins model as the simplest mathematical model of classical metapopulation dynamics with local population turnover. This simple model captures the key idea of a metapopulation of extinction prone local populations, persisting in a balance between local extinctions and recolonizations of empty habitat patches. The model predicts a threshold patch density necessary for long-term metapopulation persistence, a conclusion that is of fundamental significance for conservation. The chapter also gives a non-mathematical description of the basic principles of modeling structured populations, and shows by examples the kind of results that can be obtained by such models. An empirical example illustrates the relevance of structured models.

Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity.

In this paper, we predict the outcome of dispersal evolution in metapopulations based on the following assumptions: (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 modelled by a disperser pool. 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. Second, we describe how demographic stochasticity affects the evolution of dispersal rates: evolutionarily stable dispersal rates remain high even when disturbance‐related rates of local extinction are low, and a variety of qualitatively different responses of adapted dispersal rates to varied levels of disturbance become possible. This paper shows, for the first time, that evolution of dispersal rates may give rise to monotonically increasing or decreasing responses, as well as to intermediate maxima or minima.