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.     

Effects of ecological differentiation on Lotka-Volterra systems for species with behavioral adaptation and variable growth rates

We study the properties of a n2-dimensional Lotka-Volterra system describing competing species that include behaviorally adaptive abilities. We indicate as behavioral adaptation a mechanism, based on a kind of learning, which is not viewed in the evolutionary sense but is intended to occur over shorter time scales. We consider a competitive adaptive n species Lotka-Volterra system, n > or = 3, in which one species is made ecologically differentiated with respect to the others by carrying capacity and intrinsic growth rate. The symmetry properties of the system and the existence of a certain class of invariant subspaces allow the introduction of a 7-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the complete system. The reduced model is effective also in describing the time-dependent regimes for a large range of parameter values. The case in which one species has a strong ecological advantage (i.e. with a carrying capacity higher than the others), but with a varying growth rate, has been analyzed in detail, and time-dependent behaviors have been investigated in the case of adaptive competition among four species. Relevant questions, as species survival/exclusion, are addressed focusing on the role of adaptation. Interesting forms of species coexistence are found (i.e. competitive stable equilibria, periodic oscillations, strange attractors).     

Coevolutionary dynamics of adaptive radiation for food-web development

To investigate how complex food-webs can develop through repeated evolutionary diversification, a predator–prey model was analyzed. In the model, each individual has two traits: trait x as a predator and trait y as a prey. These traits constitute a two-dimensional phenotype space, in which the whole group of individuals are represented as a phenotype distribution. Predator–prey interactions among the phenotypes are determined by their relative positions in the phenotype space. Each phenotypic cluster was treated as a species. Each species evolves in y to escape from predation, while it evolves in x to chase their prey. Analytical investigation provided two predictions. First, coupled evolutionary diversifications of y and x may occur when the x of predators have caught up with their prey’s y, which may be repeated. Second, complex food-webs may develop when species’ competitive strengths are kept similar within the communities. If the functional response is close to the ratio-dependent response, the competitive strengths of all species are similar when the relationship between predators and prey corresponds to the ideal free distribution (IFD). These predictions were confirmed by numerical simulations. Electronic supplementary material  The online version of this article doi:() contains supplementary material, which is available to authorized users.     

Global stability in Lotka-Volterra systems with diffusion

Summary Sufficient conditions for global stability inn-species Lotka-Volterra systems with diffusion are derived. Both continuous environments with 0-flux boundary conditions and environments consisting of discrete patches are considered.     

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.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.     

The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs

This article studies the effects of adaptive changes in predator and/or prey activities on the Lotka-Volterra predator-prey population dynamics. The model assumes the classical foraging-predation risk trade-offs: increased activity increases population growth rate, but it also increases mortality rate. The model considers three scenarios: prey only are adaptive, predators only are adaptive, and both species are adaptive. Under all these scenarios, the neutral stability of the classical Lotka-Volterra model is partially lost because the amplitude of maximum oscillation in species numbers is bounded, and the bound is independent of the initial population numbers. Moreover, if both prey and predators behave adaptively, the neutral stability can be completely lost, and a globally stable equilibrium would appear. This is because prey and/or predator switching leads to a piecewise constant prey (predator) isocline with a vertical (horizontal) part that limits the amplitude of oscillations in prey and predator numbers, exactly as suggested by Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey systems. Prey and predator activities in a long-term run are calculated explicitly. This article shows that predictions based on short-term behavioral experiments may not correspond to long-term predictions when population dynamics are considered.     

Coexistence for systems governed by difference equations of Lotka-Volterra type

The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.     

The adaptive dynamics of altruism in spatially heterogeneous populations

Abstract.— We study the spatial adaptive dynamics of a continuous trait that measures individual investment in altruism. Our study is based on an ecological model of a spatially heterogeneous population from which we derive an appropriate measure of fitness. The analysis of this fitness measure uncovers three different selective processes controlling the evolution of altruism: the direct physiological cost, the indirect genetic benefits of cooperative interactions, and the indirect genetic costs of competition for space. In our model, habitat structure and a continuous life cycle makes the cost of competing for space with relatives negligible. Our study yields a classification of adaptive patterns of altruism according to the shape of the costs of altruism (with decelerating, linear, or accelerating dependence on the investment in altruism). The invasion of altruism occurs readily in species with accelerating costs, but large mutations are critical for altruism to evolve in selfish species with decelerating costs. Strict selfishness is maintained by natural selection only under very restricted conditions. In species with rapidly accelerating costs, adaptation leads to an evolutionarily stable rate of investment in altruism that decreases smoothly with the level of mobility. A rather different adaptive pattern emerges in species with slowly accelerating costs: high altruism evolves at low mobility, whereas a quasi-selfish state is promoted in more mobile species. The high adaptive level of altruism can be predicted solely from habitat connectedness and physiological parameters that characterize the pattern of cost. We also show that environmental changes that cause increased mobility in those highly altruistic species can beget selection-driven self-extinction, which may contribute to the rarity of social species.     

一类竞争Lotka-Volterra系统的持续生存和灭绝

对一类非自治竞争Lotka-Volterra系统,其相互作用系数为非负常数,S.Ahmad和A.C.Lazer研究了该系统并得到结果在一定条件下,系统的第n个种群趋于灭绝而前n-1个种群强持续生存.本文我们讨论一类较一般的非自治竞争Lotka-Volterra系统,其相互作用系数为一类连续函数,得到同样结果.     

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.     

The public health implications of maternal care trade-offs

The socioeconomic and ethnic characteristics of parents are some of the most important correlates of adverse health outcomes in childhood. However, the relationships between ethnic, economic, and behavioral factors and the health outcomes responsible for this pervasive finding have not been specified in child health epidemiology. The general objective of this paper is to propose a theoretical approach to the study of maternal behaviors and child health in diverse ethnic and socioeconomic environments. The specific aims are: (a) to describe a causal pathway between the utility that women obtain through work outside the home and through child care and disease hazard rates in childhood using an optimization model; (b) to specify the influence of ethnic and socioeconomic factors on model constraints; (c) to use the model as a tool to learn about how different combinations of maternal wage labor and child care time might influence child health outcomes in diverse social contexts; (d) to identify parameters that will require measurement in future research; (e) to discuss research strategies that will enable us to obtain these measurements; and (f) to discuss the implications of the model for biostatistical modeling and public health intervention. Optimization models are powerful heuristic tools for understanding how ethnic, environmental, family, and personal characteristics can place important constraints on both the quality and quantity of care that women can provide to their children. They provide a quantitative appreciation for the difficult trade-offs that most women face between working in order to purchase basic goods that children cannot do without (e.g., food, clothing, shelter, health insurance), and increasing offspring well-being through child care (e.g., training in social skills, affection, protection from environmental hazards, help with homework). The research was funded by a Faculty Scholars Award from the William T. Grant Foundation to A. Magdalena Hurtado. A. Magdalena Hurtado, Ph.D., is an associate professor in the Department of Anthropology, University of New Mexico. Her research interests include the origins of the sexual division of labor, epidemiology of indigenous peoples, disease susceptibility, the development and intergenerational transmission of antigens and immune defense, immune function and allergic sensitization, and trauma. She also works on public health interventions, biological capital and poverty, and land tenure and human rights in native communities of South America. Carol Lambourne, M.Sc., is a doctoral candidate in the Department of Anthropology, University of New Mexico. Her research interests are evolutionary models of child and adolescent development, life history theory, family composition and investment patterns, pubertal timing and psychosexual maturation, juvenile stress, and infanticide. Kim Hill, Ph.D., is a professor in the Department of Anthropology, University of New Mexico. His research interests are modern hunter-gatherers, including extensive fieldwork in lowland South America. Current topics of interest include human evolution, economic strategies, life history theory, the evolution of cooperation, and the emergence of social norms enforced by punishment. He is also involved in economic development, health and education projects with lowland South American native populations. Karen Kessler received her M.S. in Anthropology from the University of New Mexico in 1996. Her research interests are the application of mathematical modeling to the prevention of diabetes and other causes of morbidity and mortality in historical populations.     

一个具有竞争关系的Lotka-Volterra模型全局结构分析

本文利用微分方程定性理论,对一类具有竞争关系的两种群Lotka-Volterra模型进行了系统全局结构分析,并给出了相应的结论.     

The evolution of the Indian Ocean parrots (Psittaciformes): extinction, adaptive radiation and eustacy

Parrots are among the most recognisable and widely distributed of all bird groups occupying major parts of the tropics. The evolution of the genera that are found in and around the Indian Ocean region is particularly interesting as they show a high degree of heterogeneity in distribution and levels of speciation. Here we present a molecular phylogenetic analysis of Indian Ocean parrots, identifying the possible geological and geographical factors that influenced their evolution. We hypothesise that the Indian Ocean islands acted as stepping stones in the radiation of the Old-World parrots, and that sea-level changes may have been an important determinant of current distributions and differences in speciation. A multi-locus phylogeny showing the evolutionary relationships among genera highlights the interesting position of the monotypic Psittrichas, which shares a common ancestor with the geographically distant Coracopsis. An extensive species-level molecular phylogeny indicates a complex pattern of radiation including evidence for colonisation of Africa, Asia and the Indian Ocean islands from Australasia via multiple routes, and of island populations ‘seeding’ continents. Moreover, comparison of estimated divergence dates and sea-level changes points to the latter as a factor in parrot speciation. This is the first study to include the extinct parrot taxa, Mascarinus mascarinus and Psittacula wardi which, respectively, appear closely related to Coracopsis nigra and Psittacula eupatria.     

Asymptotic behaviour of solutions of the diffusive Lotka-Volterra equations

A spatially discrete version of the diffusive Lotka-Volterra equations is considered. Asymptotical spatial homogeneity of solutions of the equations with equilibrium, periodic or zero flux boundary conditions is proved without regard to crowding effects. The proof does not require the assumption of equal diffusion coefficients and the restrictions on the dimension of space and on the initial data, which are necessary in the spatially continuous model.     

Evolution of cannibalistic traits: scenarios derived from adaptive dynamics

The evolution of cannibalistic traits in consumer populations is studied in this paper with the approach of adaptive dynamics theory. The model is kept at its minimum complexity by eliminating some environmental characteristics, like heterogeneity and seasonalities, and by hiding the size-structure of the population. Evolutionary dynamics are identified through numerical bifurcation analysis, applied both to the ecological (resident-mutant) model and to the canonical equation of adaptive dynamics. The result is a rich catalog of evolutionary scenarios involving evolutionary stable strategies and branching points both in the monomorphic and dimorphic dynamics. The possibility of evolutionary extinction of highly cannibalistic populations is also ascertained. This allows one to explain why cannibalism can be a transient stage of evolution.     

种群Lotka-Volterra捕食系统多个极限环的存在性

通过降维把高维系统平衡点的稳定必及根限环的构造用低维系统来判定和实现,给出了一个三种群Lotka-Volterra捕食系统具有两个小扰动极限环的例子。     

The diffusive Lotka-Volterra system with three species can have a stable non-constant equilibrium solution

Three examples of the diffusive 3-species Lotka-Volterra system with constant interaction parameters are given, and by bifurcation techniques shown to have stable spatially non-constant equilibrium solutions. One example is competitive; the second one predator-two-competing prey and the third involves two predators and a single prey.     

Transmission-virulence trade-offs in vector-borne diseases

Though it is commonly supposed that there is a trade-off between virulence and transmission, there is little data and little insight into what it should look like. Here, we consider the specific case of vector-borne parasites (inspired by human malaria) and analyse an embedded model to understand how specific life-cycle aspects may affect this trade-off. First, we find that, for such parasites, the transmission function may have an S-shape. Second, we find that the trade-off obtained for vector-borne parasites is less sensitive to parameter variations than the trade-off obtained for directly transmitted parasites. Third, we find that other parasite traits, such as the conversion from replicative to infective stages, could have important epidemiological implications. Finally, we compare the effect of treatments targeting either the asexual or the sexual parasite life-stage.     

The dynamics of bacteria-plasmid systems

We introduce a general model for the dynamics of a single plasmid type and a single bacterial cell type, following Stewart and Levin (1977) in subdividing the population into plasmid-bearing and plasmid-free cells. For the particular case of mortality being a linear function of population sizes, we demonstrate the existence of multiple stable states and threshold values, thus illustrating that, in some cases, the establishment of a population of plasmids may depend on introduction of plasmids at sufficiently high levels. We also analyze in detail, for a general monotonically increasing mortality function, the epidemiological case in which plasmids confer a net cost on their hosts, and demonstrate that it is possible for such plasmids to become established. Stewart and Levin previously demonstrated this effect in a more restricted model.