Modeling Allee Effect from Beta(p, 2) Densities

In this work we develop and investigate generalized populational growth models, adjusted from Beta(p, 2) densities, with Allee effect. The use of a positive parameter leads the presented generalization, which yields some more flexible models with variable extinction rates. An Allee limit is incorporated so that the models under study have strong Allee effect.

Big Bang bifurcations and allee effect in Blumberg’s dynamics

This paper concerns dynamics and bifurcations properties of a class of continuous-defined one-dimensional maps, in a three-dimensional parameter space: Blumberg's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon, associated with the stability of a fixed point. A central point of our investigation is the study of bifurcations structure for this class of functions. We verified that under some sufficient conditions, Blumberg's functions have a particular bifurcations structure: the big bang bifurcations of the so-called "box-within-a-box" type, but for different kinds of boxes. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct attractors. This work contributes to clarify the big bang bifurcation analysis for continuous maps. To support our results, we present fold and flip bifurcations curves and surfaces, and numerical simulations of several bifurcation diagrams.

Generalized models from Beta(p,2) densities with strong allee effect: dynamical approach

A dynamical approach to study the behaviour of generalized populational growth models from Bets(p, 2) densities, with strong Allee effect, is presented. The dynamical analysis of the respective unimodal maps is performed using symbolic dynamics techniques. The complexity of the correspondent discrete dynamical systems is measured in terms of topological entropy. Different populational dynamics regimes are obtained when the intrinsic growth rates are modified: extinction, bistability, chaotic semistability and essential extinction.

From weak Allee effect to no Allee effect in Richards’ growth models

Population dynamics have been attracting interest since many years. Among the considered models, the Richards’ equations remain one of the most popular to describe biological growth processes. On the other hand, Allee effect is currently a major focus of ecological research, which occurs when positive density dependence dominates at low densities. In this chapter, we propose the dynamical study of classes of functions based on Richards’ models describing the existence or not of Allee effect. We investigate bifurcation structures in generalized Richards’ functions and we look for the conditions in the (β, r) parameter plane for the existence of a weak Allee effect region. We show that the existence of this region is related with the existence of a dovetail structure. When the Allee limit varies, the weak Allee effect region disappears when the dovetail structure also disappears. Consequently, we deduce the transition from the weak Allee effect to no Allee effect to this family of functions. To support our analysis, we present fold and flip bifurcation curves and numerical simulations of several bifurcation diagrams.

Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models

This work concerns dynamics and bifurcations properties of a new class of continuous-defined one-dimensional maps: Tsoularis-Wallace's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon of extinction. To establish this result we introduce the notions of Allee's functions, Allee's effect region and Allee's bifurcation curve. Another central point of our investigation is the study of bifurcation structures for this class of functions, in a three-dimensional parameter space. We verified that under some sufficient conditions, Tsoularis-Wallace's functions have particular bifurcation structures: the big bang and the double big bang bifurcations of the so-called "box-within-a-box" type. The double big bang bifurcations are related to the existence of flip codimension-2 points. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct kinds of boxes. This work contributes to clarify the big bang bifurcation analysis for continuous maps and understand their relationship with explosion birth and extinction phenomena.

Strong and weak Allee effects and chaotic dynamics in Richards' growths

In this paper we define and investigate generalized Richards' growth models with strong and weak Allee effects and no Allee effect. We prove the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, depending on the implicit conditions, which involve the several parameters considered in the models. New classes of functions describing the existence or not of Allee effect are introduced, a new dynamical approach to Richards' populational growth equation is established. These families of generalized Richards' functions are proportional to the right hand side of the generalized Richards' growth models proposed. Subclasses of strong and weak Allee functions and functions with no Allee effect are characterized. The study of their bifurcation structure is presented in detail, this analysis is done based on the configurations of bifurcation curves and symbolic dynamics techniques. Generically, the dynamics of these functions are classified in the following types: extinction, semi-stability, stability, period doubling, chaos, chaotic semistability and essential extinction. We obtain conditions on the parameter plane for the existence of a weak Allee effect region related to the appearance of cusp points. To support our results, we present fold and flip bifurcations curves and numerical simulations of several bifurcation diagrams.

Allee effects and population spread in patchy landscapes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Big bang bifurcations in von Bertalanffy’s dynamics with strong and weak Allee effects

The main purpose of this work was to study population dynamic discrete models in which the growth of the population is described by generalized von Bertalanffy's functions, with an adjustment or correction factor of polynomial type. The consideration of this correction factor is made with the aim to introduce the Allee effect. To the class of generalized von Bertalanffy's functions is identified and characterized subclasses of strong and weak Allee's functions and functions with no Allee effect. This classification is founded on the concepts of strong and weak Allee's effects to population growth rates associated. A complete description of the dynamic behavior is given, where we provide necessary conditions for the occurrence of unconditional and essential extinction types. The bifurcation structures of the parameter plane are analyzed regarding the evolution of the Allee limit with the aim to understand how the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is realized. To generalized von Bertalanffy's functions with strong and weak Allee effects is identified an Allee's effect region, to which is associated the concepts of chaotic semistability curve and Allee's bifurcation point. We verified that under some sufficient conditions, generalized von Bertalanffy's functions have a particular bifurcation structure: the big bang bifurcations of the so-called box-within-a-box type. To this family of maps, the Allee bifurcation points and the big bang bifurcation points are characterized by the symmetric of Allee's limit and by a null intrinsic growth rate. The present paper is also a significant contribution in the framework of the big bang bifurcation analysis for continuous 1D maps and unveil their relationship with the explosion birth and the extinction phenomena.

Source populations in coastal crabs: parameters affecting egg production

Benthic marine invertebrates may form metapopulations connected via propagule dispersal. Conservation efforts often target potential source coastlines to indirectly benefit areas depending on allochthonous offspring production. Besides population density, adult size structure, sex ratio, brooding frequency and the proportion of breeding individuals may significantly influence the reproductive output of benthic populations, but these effects have seldom been tested. We used rocky shore crabs to assess the spatial variability of such parameters at relevant scales for conservation purposes and to test their consistency over 2 consecutive years; we then used the data to address whether bottom-up processes or biological interactions might explain the patterns observed. We decomposed egg production rates into their components for the 2 most abundant brachyuran species inhabiting the intertidal rocky habitat. Adult density and brooding frequency varied consistently among shores for both species and largely explained the overall spatial trends of egg production. Temporally consistent patterns also included among-shore differences in the size of ovigerous females of the grapsid Pachygrapsus transversus and between-bay differences in the fecundity of the spider crab Epialtus brasiliensis. Sex ratio was remarkably constant in both. We found no positive or negative correlations between adult density and brooding frequency to support either the existence of a component Allee effect (lack of mate encounters) or an effect of intra-specific competition. Likewise, shore-specific potential growth in P. transversus does not negatively correlate with frequency of ovigerous individuals, as would be expected under a critical balance between these 2 processes. The patterns observed suggest that bottom-up drivers may best explain spatial trends in the reproductive output of these species.

Dynamical analysis in growth models: Blumberg’s equation

We present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to Beta(p, q) probability densities functions. Using the symmetry of the Beta(p, q) distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.

Population demography and the evolution of helping behaviors.

Limited dispersal may favor the evolution of helping behaviors between relatives as it increases their relatedness, and it may inhibit such evolution as it increases local competition between these relatives. Here, we explore one way out of this dilemma: if the helping behavior allows groups to expand in size, then the kin-competition pressure opposing its evolution can be greatly reduced. We explore the effects of two kinds of stochasticity allowing for such deme expansion. First, we study the evolution of helping under environmental stochasticity that may induce complete patch extinction. Helping evolves if it results in a decrease in the probability of extinction or if it enhances the rate of patch recolonization through propagules formed by fission of nonextinct groups. This mode of dispersal is indeed commonly found in social species. Second, we consider the evolution of helping in the presence of demographic stochasticity. When fecundity is below its value maximizing deme size (undersaturation), helping evolves, but under stringent conditions unless positive density dependence (Allee effect) interferes with demographic stochasticity. When fecundity is above its value maximizing deme size (oversaturation), helping may also evolve, but only if it reduces negative density-dependent competition.

Manipulating sex ratio to increase population growth: the example of the Lesser Kestrel

Small or decreasing populations call for emergency actions like, for example, captive breeding programs. Such programs aim at rapidly increasing population sizes in order to reduce the loss of genetic variability and to avoid possible Allee effects. The Lesser Kestrel Falco naumanni is one of the species that is currently supported in several captive breeding programs at various locations. Here, we model the demographic and genetic consequences of potential management strategies that are based on offspring sex ratio manipulation. Increased population growth could be achieved by manipulating female conditions and/or male attractiveness in the captive breeders and consequently shifting the offspring sex ratio towards more female offspring, which are then used for reintroduction. Fragmenting populations into wild-breeding and captive-breeding demes and manipulating population sex ratio both immediately increase the inbreeding coefficient in the next generation (i.e. decrease N-e) but may, in the long term, reduce the loss of genetic variability if population growth is restricted by the number of females. We use the Lesser Kestrel and the wealth of information that is available on this species to predict the long-term consequences of various kinds of sex-ratio manipulation. We find that, in our example and possibly in many other cases, a sex-ratio manipulation that seems realistic could have a beneficial effect on the captive breeding program. However, the possible long-term costs and benefits of such measures need to be carefully optimized.

Evaluating the Small Population Paradigm for Rare Large-Bodied Woodpeckers, with Implications for the Ivory-billed Woodpecker

Six large-bodied, ≥ 120 g, woodpecker species are listed as near-threatened to critically endangered by the International Union for Conservation of Nature (IUCN). The small population paradigm assumes that these populations are likely to become extinct without an increase in numbers, but the combined influences of initial population size and demographic rates, i.e., annual adult survival and fecundity, may drive population persistence for these species. We applied a stochastic, stage-based single-population model to available demographic rates for Dryocopus and Campephilus woodpeckers. In particular, we determined the change in predicted extinction rate, i.e., proportion of simulated populations that went extinct within 100 yr, to concomitant changes in six input parameters. To our knowledge, this is the first study to evaluate the combined importance of initial population size and demographic rates for the persistence of large-bodied woodpeckers. Under a worse-case scenario, the median time to extinction was 7 yr (range: 1–32). Across the combinations of other input values, increasing initial population size by one female induced, on average, 0.4%–3.2% (range: 0%–28%) reduction in extinction rate. Increasing initial population size from 5–30 resulted in extinction rates < 0.05 under limited conditions: (1) all input values were intermediate, or (2) Allee effect present and annual adult survival ≥ 0.8. Based on our model, these species can persist as rare, as few as five females, and thus difficult-to-detect, populations provided they maintain ≥ 1.1 recruited females annually per adult female and an annual adult survival rate ≥ 0.8. Athough a demographic-based population viability analysis (PVA) is useful to predict how extinction rate changes across scenarios for life-history attributes, the next step for modeling these populations should incorporate more easily acquired data on changes in patch occupancy to make predictions about patch colonization and extinction rates.

Towards a population ecology of stressed environments: the effects of zinc on the springtail Folsomia candida

1. To understand population dynamics in stressed environments it is necessary to join together two classical lines of research. Population responses to environmental stress have been studied at low density in life table response experiments. These show how the population's growth rate (pgr) at low density varies in relation to levels of stress. Population responses to density, on the other hand, are based on examination of the relationship between pgr and population density. 2. The joint effects of stress and density on pgr can be pictured as a contour map in which pgr varies with stress and density in the same way that the height of land above sea level varies with latitude and longitude. Here a microcosm experiment is reported that compared the joint effects of zinc and population density on the pgr of the springtail Folsomia candida (Collembola). 3. Our experiments allowed the plotting of a complete map of the effects of density and a stressor on pgr. Particularly important was the position of the pgr= 0 contour, which suggested that carrying capacity varied little with zinc concentration until toxic levels were reached. 4. This prediction accords well with observations of population abundance in the field. The method also allowed us to demonstrate, simultaneously, hormesis, toxicity, an Allee effect and density dependence. 5. The mechanisms responsible for these phenomena are discussed. As zinc is an essential trace element the initial increase in pgr is probably a consequence of dietary zinc deficiency. The Allee effect may be attributed to productivity of the environment increasing with density at low density. Density dependence is a result of food limitation. 6. Synthesis and applications. We illustrate a novel solution based on mapping a population's growth rate in relation to stress and population density. Our method allows us to demonstrate, simultaneously, hormesis, toxicity, an Allee effect and density dependence in an important ecological indicator species. We hope that the approach followed here will prove to have general applicability enabling predictions of field abundance to be made from estimates of the joint effects of the stressors and density on population growth rate.

Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.