Discrete-time Population Dynamics Research Articles

A generalized model for the population dynamics of a two stage species with recruitment and capture using a nonstandard finite difference scheme

The aim of this work is to formulate and analyze a new and generalized discrete-time population dynamics model for a two-stage species with recruitment and capture factors. This model is derived from a well-known continuous-time population dynamics model of a two-stage species with recruitment and capture developed by Ladino and Valverde and the nonstandard finite difference (NSFD) methodology proposed by Mickens. We establish positivity and asymptotic stability of the proposed discrete-time population dynamics model. As an important consequence, the population dynamics of the new discrete-time model is determined fully. Also, a set of numerical examples is conducted to illustrate the theoretical results and to demonstrate advantages of the new model. The theoretical results and numerical examples show that the proposed discrete-time model not only preserves correctly the population dynamics of the continuous one but is also easy to be implemented. However, some discrete-time models based on the standard Runge–Kutta methods fail to preserve the population dynamics of the continuous-time model. As a result, they generate numerical approximations which are not only non-negative but also unstable.

Discrete-time population dynamics of spatially distributed semelparous two-sex populations

Spatially distributed populations with two sexes may face the problem that males and females concentrate in different parts of the habitat and mating and reproduction does not happen sufficiently often for the population to persist. For simplicity, to explore the impact of sex-dependent dispersal on population survival, we consider a discrete-time model for a semelparous population where individuals reproduce only once in their life-time, during a very short reproduction season. The dispersal of females and males is modeled by Feller kernels and the mating by a homogeneous pair formation function. The spectral radius of a homogeneous operator is established as basic reproduction number of the population, [Formula: see text]. If [Formula: see text], the extinction state is locally stable, and if [Formula: see text] the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. Special cases exhibit how sex-biased dispersal affects the persistence of the population.

SPop: Age-structured discrete-time population dynamics model in C, Python, and R.

This article describes the sPop packages implementing the deterministic and stochastic versions of an age-structured discrete-time population dynamics model. The packages enable mechanistic modelling of a population by monitoring the age and development stage of each individual. Survival and development are included as the main effectors and they progress at a user-defined pace: follow a fixed rate, delay for a given time, or progress at an age-dependent manner. The model is implemented in C, Python, and R with a uniform design to ease usage and facilitate adoption. Early versions of the model were previously employed for investigating climate-driven population dynamics of the tiger mosquito and the chikungunya disease spread by this vector. The sPop packages presented in this article enable the use of the model in a range of applications extending from vector-borne diseases towards any age-structured population including plant and animal populations, microbial dynamics, host-pathogen interactions, infectious diseases, and other time-dependent epidemiological processes.

Discrete-time population dynamics on the state space of measures.

If the individual state space of a structured population is given by a metric space S, measures μ on the σ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: μ(T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.

Inequality in resource allocation and population dynamics models.

The Hassell model has been widely used as a general discrete-time population dynamics model that describes both contest and scramble intraspecific competition through a tunable exponent. Since the two types of competition generally lead to different degrees of inequality in the resource distribution among individuals, the exponent is expected to be related to this inequality. However, among various first-principles derivations of this model, none is consistent with this expectation. This paper explores whether a Hassell model with an exponent related to inequality in resource allocation can be derived from first principles. Indeed, such a Hassell model can be derived by assuming random competition for resources among the individuals wherein each individual can obtain only a fixed amount of resources at a time. Changing the size of the resource unit alters the degree of inequality, and the exponent changes accordingly. As expected, the Beverton–Holt and Ricker models can be regarded as the highest and lowest inequality cases of the derived Hassell model, respectively. Two additional Hassell models are derived under some modified assumptions.

Influence of technological progress and renewability on the sustainability of ecosystem engineers populations.

Overpopulation and environmental degradation due to inadequate resource-use are outcomes of human's ecosystem engineering that has profoundly modified the world's landscape. Despite the age-old concern that unchecked population and economic growth may be unsustainable, the prospect of societal collapse remains contentious today. Contrasting with the usual approach to modeling human-nature interactions, which are based on the Lotka-Volterra predator-prey model with humans as the predators and nature as the prey, here we address this issue using a discrete-time population dynamics model of ecosystem engineers. The growth of the population of engineers is modeled by the Beverton-Holt equation with a density-dependent carrying capacity that is proportional to the number of usable habitats. These habitats (e.g., farms) are the products of the work of the individuals on the virgin habitats (e.g., native forests), hence the denomination engineers of ecosystems to those agents. The human-made habitats decay into degraded habitats, which eventually regenerate into virgin habitats. For slow regeneration resources, we find that the dynamics is dominated by rounds of prosperity and collapse, in which the population reaches vanishing small densities. However, increase of the efficiency of the engineers to explore the resources eliminates the dangerous oscillatory patterns of feast and famine and leads to a stable equilibrium that balances population growth and resource availability. This finding supports the viewpoint of growth optimists that technological progress may avoid collapse.

SPop: Age-structured discrete-time population dynamics model in C, Python, and R.

This article describes the sPop packages implementing the deterministic and stochastic versions of an age-structured discrete-time population dynamics model. The packages enable mechanistic modelling of a population by monitoring the age and development stage of each individual. Survival and development are included as the main effectors and they progress at a user-defined pace: follow a fixed-rate, delay for a given time, or progress at an age-dependent manner. The model is implemented in C, Python, and R with a uniform design to ease usage and facilitate adoption. Early versions of the model were previously employed for investigating climate-driven population dynamics of the tiger mosquito and the chikungunya disease spread by this vector. The sPop packages presented in this article enable the use of the model in a range of applications extending from vector-borne diseases towards any age-structured population including plant and animal populations, microbial dynamics, host-pathogen interactions, infectious diseases, and other time-dependent epidemiological processes.

Effect of harvest timing on the dynamics of the Ricker–Seno model

The moment of intervention is a key question in harvest programmes and is currently generating an increasing interest. However, little is known about its effect on the population stability. This lack of knowledge is greater in the case of global stability, which is always desirable as it allows to predict the fate of populations regardless of the initial size. Here we use a discrete-time equation to model the dynamics of populations harvested at any time during the reproductive season. We study the effect of the time of intervention on the global stability of populations governed by the Ricker model, which is one of the most relevant models in discrete-time population dynamics. We prove that harvest timing never has a negative effect on the global stability of these populations, extending recent results in the literature. We also study the effect of delayed harvesting on the constancy stability of the controlled populations.

SPop: Age-structured discrete-time population dynamics model in C, Python, and R

This article describes the sPop packages implementing the deterministic and stochastic versions of an age-structured discrete-time population dynamics model. The packages enable mechanistic modelling of a population by monitoring the age and development stage of each individual. Survival and development are included as the main effectors and they progress at a user-defined pace: follow a fixed rate, delay for a given time, or progress at an age-dependent manner. The model is implemented in C, Python, and R with a uniform design to ease usage and facilitate adoption. Early versions of the model were previously employed for investigating climate-driven population dynamics of the tiger mosquito and the chikungunya disease spread by this vector. The sPop packages presented in this article enable the use of the model in a range of applications extending from vector-borne diseases towards any age-structured population including plant and animal populations, microbial dynamics, host-pathogen interactions, infectious diseases, and other time-dependent epidemiological processes.

SPop: Age-structured discrete-time population dynamics model in C, Python, and R.

This article describes the sPop packages implementing the deterministic and stochastic versions of an age-structured discrete-time population dynamics model. The packages enable mechanistic modelling of a population by monitoring the age and development stage of each individual. Survival and development are included as the main effectors and they progress at a user-defined pace: follow a fixed-rate, delay for a given time, or progress at an age-dependent manner. The model is implemented in C, Python, and R with a uniform design to ease usage and facilitate adoption. Early versions of the model were previously employed for investigating climate-driven population dynamics of the tiger mosquito and the chikungunya disease spread by this vector. The sPop packages presented in this article enable the use of the model in a range of applications extending from vector-borne diseases towards any age-structured population including plant and animal populations, microbial dynamics, host-pathogen interactions, infectious diseases, and other time-delayed epidemiological processes.

Anomalous Growth of Aging Populations

We consider a discrete-time population dynamics with age-dependent structure. At every time step, one of the alive individuals from the population is chosen randomly and removed with probability \(q_k\) depending on its age, whereas a new individual of age 1 is born with probability r. The model can also describe a single queue in which the service order is random while the service efficiency depends on a customer’s “age” in the queue. We propose a mean field approximation to investigate the long-time asymptotic behavior of the mean population size. The age dependence is shown to lead to anomalous power-law growth of the population at the critical regime. The scaling exponent is determined by the asymptotic behavior of the probabilities \(q_k\) at large k. The mean field approximation is validated by Monte Carlo simulations.

What life cycle graphs can tell about the evolution of life histories

We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject to density-dependent population regulation and two or more demographic parameters to be subject to evolutionary change. Our aim is to identify structural features of life cycles and modes of population regulation that correspond to specific evolutionary dynamics. Our derivations are based on a fitness proxy that is an algebraically simple function of loops within the life cycle. This allows us to phrase the results in terms of properties of such loops which are readily interpreted biologically. The following results could be obtained. First, we give sufficient conditions for the existence of optimisation principles in models with an arbitrary number of evolving traits. These models are then classified with respect to their appropriate optimisation principle. Second, under the assumption of just two evolving traits we identify structural features of the life cycle that determine whether equilibria of the monomorphic adaptive dynamics (evolutionarily singular points) correspond to fitness minima or maxima. Third, for one class of frequency-dependent models, where optimisation is not possible, we present sufficient conditions that allow classifying singular points in terms of the curvature of the trade-off curve. Throughout the article we illustrate the utility of our framework with a variety of examples.

Feedback control systems analysis of density dependent population dynamics

We use feedback control methods to prove a trichotomy of stability for nonlinear (density dependent) discrete-time population dynamics defined on a natural state space of non-negative vectors. Specifically, using comparison results and small gain techniques we obtain a computable formula for parameter ranges when one of the following must hold: there is a positive, globally asymptotically stable equilibrium; zero is globally asymptotically stable or all solutions with non-zero initial conditions diverge. We apply our results to a model for Chinook Salmon.

On the stability analysis of a general discrete-time population model involving predation and Allee effects

This paper presents the stability analysis of equilibrium points of a general discrete-time population dynamics involving predation with and without Allee effects which occur at low population density. The mathematical analysis and numerical simulations show that the Allee effect has a stabilizing role on the local stability of the positive equilibrium points of this model.

Non-unique population dynamics: basic patterns

We review the basic patterns of complex non-uniqueness in simple discrete-time population dynamics models. We begin by studying a population dynamics model of a single species with a two-stage, two-habitat life cycle. We then explore in greater detail two ecological models describing host–macroparasite and host–parasitoid interspecific interactions. In general, several types of attractors, e.g. point equilibria vs. chaotic, periodic vs. quasiperiodic and quasiperiodic vs. chaotic attractors, may coexist in the same mapping. This non-uniqueness also indicates that the bifurcation diagrams, or the routes to chaos, depend on initial conditions and are therefore non-unique. The basins of attraction, defining the initial conditions leading to a certain attractor, may be fractal sets. The fractal structure may be revealed by fractal basin boundaries or by the patterns of self-similarity. The fractal basin boundaries make it more difficult to predict the final state of the system, because the initial values can be known only up to some precision. We conclude that non-unique dynamics, associated with extremely complex structures of the basin boundaries, can have a profound effect on our understanding of the dynamical processes of nature.

A Uniqueness Result for Nontrivial Steady States of a Density-dependent Population Dynamics Model

In this paper we investigate the uniqueness of a nontrivial fixed point of a map preserving a positive cone. This work is motivated by application to age-structured density-dependent population dynamics models. The main result of this paper shows that in certain situations all the fixed points are comparable for the order induced by the positive cone. We then apply this result to prove the uniqueness of the nontrivial equilibrium solution for a discrete-time population dynamics model and its continuous-time analogue. Ž 5 5. Let K be a cone of a Banach space X, ? ; that is, K is a closed 4 convex subset of X, satisfying tK ; K, for all t G 0, and if x g K _ 0 , then yx f K. Such a cone K induces a partial order on X, denoted by F , and defined by x F y m y y x g K. In the sequel, we will also define

Population dynamics and the colour of environmental noise.

The effect of red, white and blue environmental noise on discrete-time population dynamics is analyzed. The coloured noise is superimposed on Moran-Ricker and Maynard Smith dynamics, the resulting power spectra are less than examined. Time series dominated by short- and long-term fluctuations are said to be blue and red, respectively. In the stable range of the Moran-Ricker dynamics, environmental noise of any colour will make population dynamics red or blue depending the intrinsic growth rate. Thus, telling apart the colour of the noise from the colour of the population dynamics may not be possible. Population dynamics subjected to red and blue environmental noises show, respectively, more red or blue power spectra than those subjected to white noise. The sensitivity to differences in the noise colours decreases with increasing complexity and ultimately disappears in the chaotic range of the population dynamics. These findings are duplicated with the Maynard Smith model for high growth rates when the strength of density dependence changes. However, for low growth rates the power spectra of the population dynamics with noise are red in stable, periodic and aperiodic ranges irrespective of the noise colour. Since chaotic population fluctuations may show blue spectra in the deterministic case, this implies that blue deterministic chaos may become red under any colour of the noise.

Discrete-Time Population Dynamics of Interacting Self-Oscillators

Discrete-Time Population Dynamics of Interacting Self-Oscillators