A model of physiologically structured population dynamics with a nonlinear individual growth rate

In this article we consider a size structured population model with a nonlinear growth rate depending on the individual's size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.This work was partially supported by DGICYT PB90-0730-C02-01 and PB91-0497.     

Adaptive dynamics of Lotka-Volterra systems with trade-offs: the role of interspecific parameter dependence in branching

We determine the adaptive dynamics of a general Lotka-Volterra system containing an intraspecific parameter dependency--in the form of an explicit functional trade-off between evolving parameters--and interspecific parameter dependencies--arising from modelling species interactions. We develop expressions for the fitness of a mutant strategy in a multi-species resident environment, the position of the singular strategy in such systems and the non-mixed second-order partial derivatives of the mutant fitness. These expressions can be used to determine the evolutionary behaviour of the system. The type of behaviour expected depends on the curvature of the trade-off function and can be interpreted in a biologically intuitive manner using the rate of acceleration/deceleration of the costs implicit in the trade-off function. We show that for evolutionary branching to occur we require that one (or both) of the traded-off parameters includes an interspecific parameter dependency and that the trade-off function has weakly accelerating costs. This could have important implications for understanding the type of mechanisms that cause speciation. The general theory is motivated by using adaptive dynamics to examine evolution in a predator-prey system. The applicability of the general theory as a tool for examining specific systems is highlighted by calculating the evolutionary behaviour in a three species (prey-predator-predator) system.     

Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree

We present a general framework for modelling adaptive trait dynamics in which we integrate various concepts and techniques from modern ESS-theory. The concept of evolutionarily singular strategies is introduced as a generalization of the ESS-concept. We give a full classification of the singular strategies in terms of ESS-stability, convergence stability, the ability of the singular strategy to invade other populations if initially rare itself, and the possibility of protected dimorphisms occurring within the singular strategy's neighbourhood. Of particular interest is a type of singular strategy that is an evolutionary attractor from a great distance, but once in its neighbourhood a population becomes dimorphic and undergoes disruptive selection leading to evolutionary branching. Modelling the adaptive growth and branching of the evolutionary tree can thus be considered as a major application of the framework. A haploid version of Levene's soft selection model is developed as a specific example to demonstrate evolutionary dynamics and branching in monomorphic and polymorphic populations.     

The adaptive dynamics of function-valued traits

This study extends the framework of adaptive dynamics to function-valued traits. Such adaptive traits naturally arise in a great variety of settings: variable or heterogeneous environments, age-structured populations, phenotypic plasticity, patterns of growth and form, resource gradients, and in many other areas of evolutionary ecology. Adaptive dynamics theory allows analysing the long-term evolution of such traits under the density-dependent and frequency-dependent selection pressures resulting from feedback between evolving populations and their ecological environment. Starting from individual-based considerations, we derive equations describing the expected dynamics of a function-valued trait in asexually reproducing populations under mutation-limited evolution, thus generalizing the canonical equation of adaptive dynamics to function-valued traits. We explain in detail how to account for various kinds of evolutionary constraints on the adaptive dynamics of function-valued traits. To illustrate the utility of our approach, we present applications to two specific examples that address, respectively, the evolution of metabolic investment strategies along resource gradients, and the evolution of seasonal flowering schedules in temporally varying environments.     

Scale-free extinction dynamics in spatially structured host-parasitoid systems

Much of the work on extinction events has focused on external perturbations of ecosystems, such as climatic change, or anthropogenic factors. Extinction, however, can also be driven by endogenous factors, such as the ecological interactions between species in an ecosystem. Here we show that endogenously driven extinction events can have a scale-free distribution in simple spatially structured host-parasitoid systems. Due to the properties of this distribution there may be many such simple ecosystems that, although not strictly permanent, persist for arbitrarily long periods of time. We identify a critical phase transition in the parameter space of the host-parasitoid systems, and explain how this is related to the scale-free nature of the extinction process. Based on these results, we conjecture that scale-free extinction processes and critical phase transitions of the type we have found may be a characteristic feature of many spatially structured, multi-species ecosystems in nature. The necessary ingredient appears to be competition between species where the locally inferior type disperses faster in space. If this condition is satisfied then the eventual outcome depends subtly on the strength of local superiority of one species versus the dispersal rate of the other.     

Travelling wave dynamics and self-organization in a spatio-temporally structured population

We analyse spatial population dynamics showing that periodic or period-like chaotic dynamics produce self-organization structures, such as travelling waves. We suggest that self-organized patterns are associated with spatial synchrony patterns that often depend on geographical distance between subpopulations. The population dynamics also show statistical spatial autocorrelation patterns. We contrast our theoretical simulations with empirical data on annual damages in young sapling stands caused by voles. We conclude, on the basis of the periodicity, synchrony, and spatial autocorrelation patterns, and our simulation results, that vole dynamics represent travelling waves in population dynamics. We suggest that because such synchrony patterns are frequently observed in natural populations, spatial self-organization may be more common in population dynamics than reported in the literature.     

The adaptive dynamics of the evolution of host resistance to indirectly transmitted microparasites

We use adaptive dynamics and pairwise invadability plots to examine the evolutionary dynamics of host resistance to microparasitic infection transmitted indirectly via free stages. We investigate trade-offs between pathogen transmission rate and intrinsic growth rate. Adaptive dynamics distinguishes various evolutionary outcomes associated with repellors, attractors or branching points. We find criteria corresponding to these and demonstrate that a major factor deciding the evolutionary outcome is whether trade-offs are acceleratingly or deceleratingly costly. We compare and contrast two models and show how the differences between them lead to different evolutionary outcomes.     

Well-posedness of physiologically structured population models for Daphnia magna

In this paper we heuristically discuss the well-posedness of three variants of the Kooijman/Metz model. Shortcomings concerning the uniqueness and continuous dependence on data of the solutions to one of the variants are traced back to an inconsistency in the biological concept of energy allocation in this model version. The conceptional consequences are discussed and an open question concerning energy allocation is pin-pointed.     

Adaptive evolution of foraging-related traits in a predator-prey community

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.     

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.     

Adaptive evolution of anti-predator ability promotes the diversity of prey species: critical function analysis

In this paper, with the method of adaptive dynamics and critical function analysis, we investigate the evolutionary diversification of prey species. We assume that prey species can evolve safer strategies such that it can reduce the predation risk, but this has a cost in terms of its reproduction. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for continuously stable strategy and evolutionary branching in the prey strategy. It is found that if the trade-off curve is globally concave, then the evolutionarily singular strategy is continuously stable. However, if the trade-off curve is concave-convex-concave and the prey's sensitivity to crowding is not strong, then the evolutionarily singular strategy may be an evolutionary branching point, near which the resident and mutant prey can coexist and diverge in their strategies. Second, we find that after branching has occurred in the prey strategy, if the trade-off curve is concave-convex-concave, the prey population will eventually evolve into two different types, which can coexist on the long-term evolutionary timescale. The algebraical analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible for the concave-convex-concave trade-off relationship.     

Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics

The measure dynamics approach to modelling single-species coevolution with a one-dimensional trait space is developed and compared to more traditional methods of adaptive dynamics and the Maximum Principle. It is assumed that individual fitness results from pairwise interactions together with a background fitness that depends only on total population size. When fitness functions are quadratic in the real variables parameterizing the one-dimensional traits of interacting individuals, the following results are derived. It is shown that among monomorphisms (i.e. measures supported on a single trait value), the continuously stable strategy (CSS) characterize those that are Lyapunov stable and attract all initial measures supported in an interval containing this trait value. In the cases where adaptive dynamics predicts evolutionary branching, convergence to a dimorphism is established. Extensions of these results to general fitness functions and/or multi-dimensional trait space are discussed.     

A stochastic model of evolutionary dynamics with deterministic large-population asymptotics

An evolutionary birth-death process is proposed as a model of evolutionary dynamics. Agents residing in a continuous spatial environment X, play a game G, with a continuous strategy set S, against other agents in the environment. The agents’ positions and strategies continuously change in response to other agents and to random effects. Agents spawn asexually at rates that depend on their current fitness, and agents die at rates that depend on their local population density. Agents’ individual evolutionary trajectories in X and S are governed by a system of stochastic ODEs. When the number of agents is large and distributed in a smooth density on (X,S), the collective dynamics of the entire population is governed by a certain (deterministic) PDE, which we call a fitness-diffusion equation.     

Dynamical and statistical models of vertebrate population dynamics

Population dynamics methodology now powerfully combines discrete time models (with constant parameters, density dependence, random environment, and/or demographic stochasticity) and capture-recapture models for estimating demographic parameters. Vertebrate population dynamics has strongly benefited from this progress: survival estimates have been revised upwards, trade-offs between life history traits have been demonstrated, analyses of population viability and management are more and more realistic. Promising developments concern random effects, multistate and integrated models. Some biological questions (density dependence, links between individual and population levels, and diversification of life histories) can now be efficiently attacked.     

Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model

We consider a nonlinear system describing a juvenile-adult population undergoing small mutations. We analyze two aspects: from a mathematical point of view, we use an entropy method to prove that the population neither goes extinct nor blows-up; from an adaptive evolution point of view, we consider small mutations on a long time scale and study how a monomorphic or a dimorphic initial population evolves towards an Evolutionarily Stable State. Our method relies on an asymptotic analysis based on a constrained Hamilton-Jacobi equation. It allows to recover earlier predictions in Calsina and Cuadrado [A. Calsina, S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol. 48 (2004) 135; A. Calsina, S. Cuadrado, Stationary solutions of a selection mutation model: the pure mutation case, Math. Mod. Meth. Appl. Sci. 15(7) (2005) 1091.] that we also assert by direct numerical simulation. One of the interests here is to show that the Hamilton-Jacobi approach initiated in Diekmann et al. [O. Diekmann, P.-E. Jabin, S. Mischler, B. Perthame, The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol. 67(4) (2005) 257.] extends to populations described by systems.     

Rapidly converging numerical algorithms for models of population dynamics

We propose algorithms for the approximation of the age distributions of populations modeled by the McKendrick-von Foerster and the Gurtin-MacCamy systems both in one- and two-sex versions. For the one-sex model methods of second and fourth order are proposed. For the two-sex model a second order method is described. In each case the convergence is demonstrated. Several numerical examples are given.     

On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics

During an adaptive immune response, lymphocytes proliferate for five to twenty-five cell divisions, then stop and die over a period of weeks. Based on extensive flow cytometry data, Hawkins et al. (Proc Natl Acad Sci USA 104:5032–5037, 2007) introduced a cell-level stochastic model of lymphocyte population dynamics, called the Cyton Model, that accurately captures mean lymphocyte population size as a function of time. In Subramanian et al. (J Math Biol 56(6):861–892, 2008), we performed a branching process analysis of the Cyton Model and deduced from parameterizations for in vitro and in vivo data that the immune response is predictable despite each cell’s fate being highly variable. One drawback of flow cytometry data is that individual cells cannot be tracked, so that it is not possible to investigate dependencies in the fate of cells within family trees. In the absence of this information, while the Cyton Model abandons one of the usual assumptions of branching processes (the independence of lifetime and progeny number), it adopts another of the standard branching processes hypotheses: that the fates of progeny are stochastically independent. However, new experimental observations of lymphocytes show that the fates of cells in the same family tree are not stochastically independent. Hawkins et al. (2008, submitted) report on ciné lapse photography experiments where every founding cell’s family tree is recorded for a system of proliferating lymphocytes responding to a mitogenic stimulus. Data from these experiments demonstrate that the death-or-division fates of collaterally consanguineous cells (those in the same generation within a founding cell’s family tree) are strongly correlated, while there is little correlation between cells of distinct generations and between cells in distinct family trees. As this finding contrasts with one of the assumptions of the Cyton Model, in this paper we introduce three variants of the Cyton Model with increasing levels of collaterally consanguineous correlation structure to incorporate these new found dependencies. We investigate their impact on the predicted expected variability of cell population size. Mathematically we conclude that while the introduction of correlation structure leaves the mean population size unchanged from the Cyton Model, the variance of the population size distribution is typically larger. Biologically, through comparison of model predictions for Cyton Model parameterizations determined by in vitro and in vivo experiments, we deduce that if collaterally consanguineous correlation extends beyond cousins, then the immune response is less predictable than would be concluded from the original Cyton Model. That is, some of the variability seen in data that we previously attributed to experimental error could be due to intrinsic variability in the cell population size dynamics.      

INSTAR,a discrete event model for simulating zooplankton population dynamics

In this paper we discuss the basic principles of discrete event, individual oriented, data based modelling in ecology, and we present an application of this modelling strategy. The strategy is contrasted with some more conventional modelling strategies with respect to its purpose, its basic units and its heuristic properties.INSTAR applies this modelling strategy to the simulation of the fluctuations of the population structure and density of microcrustaceans through the year. The model encompasses one microcrustacean species at a time, and its interface with the rest of the ecosystem; it has been applied to several Cladocera and Copepoda species in a shallow eutrophic lake in the Netherlands (Vijverberg & Richter 1982a, b). Possibilities for extending the model are discussed.     

On the concept of attractor for community-dynamical processes II: the case of structured populations

In Part I of this paper Jacobs and Metz (2003) extended the concept of the Conley-Ruelle, or chain, attractor in a way relevant to unstructured community ecological models. Their modified theory incorporated the facts that certain parts of the boundary of the state space correspond to the situation of at least one species being extinct and that an extinct species can not be rescued by noise. In this part we extend the theory to communities of physiologically structured populations. One difference between the structured and unstructured cases is that a structured population may be doomed to extinction and not rescuable by any biologically relevant noise before actual extinction has taken place. Another difference is that in the structured case we have to use different topologies to define continuity of orbits and to measure noise. Biologically meaningful noise is furthermore related to the linear structure of the community state space. The construction of extinction preserving chain attractors developed in this paper takes all these points into account.