Discrete Population Model Research Articles

Dynamics of a Discrete Population Model

In order to explore the influences of the strong Allee effect on population evolution in two-patch environments, this paper constructs a population model with discrete diffusion and analyzes its dynamics. Using the center manifold theorem, we discuss the codimension-one bifurcation, such as flip bifurcation, of the model. The qualitative structures and stability of the model at the degenerate fixed point are obtained from the approximation of flow. Based on the theoretical analysis results, we investigate the influences of the strong Allee effect and diffusion on population evolution. The qualitative structures of degenerate fixed point and numerical simulation show that the strong Allee effect will make the population tend to a stable size when the two initial population patch sizes are equal and exceed a threshold. But the stable size of population does not exceed the carrying capacity, depending on the constant Allee effect and carrying capacity, which differ from the conclusion of logistic model that population size tends to the carrying capacity. However, if the two initial population patches are equal in size and below this threshold, or if the sizes of the initial population patches are unequal, the population size will decrease. The fixed point becomes degenerate with respect to the strong Allee effect causing the population tend to a stable size below carrying capacity, which exacerbates the complexity of population evolution.

Derivation and dynamics of discrete population models with distributed delay in reproduction

We introduce a class of discrete single species models with distributed delay in the reproductive process and a cohort dependent survival function that accounts for survival pressure during that delay period. These delay recurrences track the mature population for species in which individuals reach maturity after at least τ and at most τ+τM breeding cycles. Under realistic model assumptions, we prove the existence of a critical delay threshold, τ˜c. For given delay kernel length τM, if each individual takes at least τ˜c time units to reach maturity, then the population is predicted to go extinct. We show that the positive equilibrium is decreasing in both τ and τM. In the case of a constant reproductive rate, we provide an equation to determine τ˜c for fixed τM, and similarly, provide a lower bound on the kernel length, τ˜M for fixed τ such that the population goes extinct if τM≥τ˜M. We compare these critical thresholds for different maturation distributions and show that if all else is the same, to avoid extinction it is best if all individuals in the population have the shortest delay possible. We apply the model derivation to a Beverton–Holt model and discuss its global dynamics. For this model with kernels that share the same mean delay, we show that populations with the largest variance in the time required to reach maturity have higher population levels and lower chances of extinction.

Behavior of solutions of a discrete population model with mutualistic interaction

Abstract We focus on the stability analysis of two types of discrete dynamic models: a discrete dynamic equation and a discrete dynamics system consisting of two equations with mutualistic interaction given by x n + 1 = a + b x n λ − ( x n − 1 + x n − k ) c + x n − 1 + x n − k {x}_{n+1}=a+\frac{b{x}_{n}{\lambda }^{-\left({x}_{n-1}+{x}_{n-k})}}{c+{x}_{n-1}+{x}_{n-k}} and x n + 1 = a 1 + b 1 y n λ − ( y n − 1 + x n − k ) c 1 + y n − 1 + x n − k , y n + 1 = a 2 + b 2 x n λ − ( x n − 1 + y n − k ) c 2 + x n − 1 + y n − k , {x}_{n+1}={a}_{1}+\frac{{b}_{1}{y}_{n}{\lambda }^{-({y}_{n-1}+{x}_{n-k})}}{{c}_{1}+{y}_{n-1}+{x}_{n-k}},\hspace{1.0em}{y}_{n+1}={a}_{2}+\frac{{b}_{2}{x}_{n}{\lambda }^{-\left({x}_{n-1}+{y}_{n-k})}}{{c}_{2}+{x}_{n-1}+{y}_{n-k}}, respectively, where k ∈ { 2 , 3 , … } k\in \left\{2,3,\ldots \right\} , the constants a , a 1 , ≥ 0 a,{a}_{1},\ge 0 , b , b 1 > 0 b,{b}_{1}\gt 0 , and c , c 1 ≥ 0 c,{c}_{1}\ge 0 be the initial densities, finite rate of increase and the limiting constant associated with the density of the species, respectively, and a 2 ≥ 0 {a}_{2}\ge 0 , b 2 > 0 {b}_{2}\gt 0 , and c 2 ≥ 0 {c}_{2}\ge 0 be the initial densities, finite rate of increase, and the limiting constant associated with the density of the mutually interacting species, respectively. k k , a positive integer represents a time delay in the system and λ ≥ 1 \lambda \ge 1 shows a decay factor based on the sum of two past time step population densities. Our main objective is to understand the impact of mutualistic interactions on the stability of discrete dynamic systems. To illustrate the boundedness and stability of these models, we also provide animated plots and bifurcation diagrams.

Periodic solutions and stability of a discrete mosquito population model with periodic parameters

Periodic solutions and stability of a discrete mosquito population model with periodic parameters

Global behavior of a discrete population model

<abstract><p>In this work, the global behavior of a discrete population model</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} x_{n+1}& = \alpha x_n e^{-y_n}+\beta,\\ y_{n+1}& = \alpha x_n(1-e^{-y_n}), \end{cases}\quad n = 0,1,2,\dots, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>is considered, where $ \alpha\in (0, 1) $, $ \beta\in (0, +\infty) $, and the initial value $ (x_{0}, y_0)\in [0, \infty)\times [0, \infty) $. To illustrate the dynamics behavior of this model, the boundedness, periodic character, local stability, bifurcation, and the global asymptotic stability of the solutions are investigated.</p></abstract>

An intentional random mathematical model for immigration: A case study of Spain

This paper presents a random discrete mathematical population model for immigration. This model incorporates not only rational factors, such as the economic gradient between destination and origin countries, geographical factors, and regulatory laws but also hidden intentional factors, such as the political interests of governments and the involvement of migrant smuggling by criminal organizations, which exploit immigration as a strategic tool. These non-rational factors are modeled as sudden, random arrival flow waves, represented by a Poisson distribution. The study’s time frame is short to ensure the reliability of economic forecasts for the coming years. Although the study focuses on Spain, the proposed approach is applicable to other geographic areas with appropriate data. The results obtained from this model can be applied to predict the national budget necessary for host countries to address this complex social phenomenon.

Technique to derive discrete population models with delayed growth

We provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles that a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of, what we consider, the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to obtain two models, a Beverton–Holt adult model and a Ricker adult model and discuss the global dynamics of both models.

Stationary status of discrete and continuous age-structured population models

From Leonhard Euler to Alfred Lotka and in recent years understanding the stationary process of the human population has been of central interest to scientists. Population reproductive measure NRR (net reproductive rate) has been widely associated with measuring the status of population stationarity and it is also included as one of the measures in the millennium development goals. This article argues how the partition theorem-based approach provides more up-to-date and timely measures to find the status of the population stationarity of a country better than the NRR-based approach. We question the timeliness of the value of NRR in deciding the stationary process of the country. We prove associated theorems on discrete and continuous age distributions and derive measurable functional properties. The partitioning metric captures the underlying age structure dynamic of populations at or near stationarity. As the population growth rates for an ever-increasing number of countries trend towards replacement levels and below, new demographic concepts and metrics are needed to better characterize this emerging global demography.

Modeling the impacts of temperature during nesting seasons on Loggerhead (Caretta caretta) Sea Turtle populations in South Florida

Loggerhead sea turtles (Caretta caretta) are a threatened sea turtle species that nests on beaches along the northwestern Atlantic Ocean. Loggerhead nesting in Florida accounts for approximately 80% of the loggerhead population in the Atlantic Ocean, yet the underlying ecological mechanisms behind their population dynamics are understudied. C. caretta are largely inaccessible since they spend most of their early life in open ocean habitats. Consequently, their nesting and hatchling production offer quantifiable insight of their population dynamics. Emergence success, the proportion of eggs laid that produce offspring which successfully leave the nest, is the source of all later life stages. Yet the abiotic factors that drive emergence success trends have not been fully described. This study explores the relationship between air temperature and emergence success across multiple nesting seasons to better understand the potential impact of anthropogenic climate change on subsequent loggerhead populations. Here we investigate the effect of changing emergence success on the juvenile and adult life stages. After exploring numerous aspects of temperature and precipitation metrics, we determined that the mean of the maximum daily temperatures for each trimester of the eggs’ incubation periods related most closely to the variation in emergence success (the proportion of hatchlings that make it out of the nest) across nests. A novel feature of our approach to modeling the impact of temperature on the later life stages is through connecting a statistical model into a discrete mechanistic population model. Our population model has two parts, one with eggs and hatchlings on a daily time scale in the nesting season and the other with juveniles and adults on an annual time scale. The statistical model enables us to illustrate the effect of temperature changes across these life stages. The decline in the adult population follows from the decline of the hatchling emergence success. Using predicted increases in temperature due to climate change, we illustrate a corresponding prediction for the population classes.

Null recurrence and transience for a binomial catastrophe model in random environment

We consider a discrete time population model for which each individual alive at time n survives independently of everybody else at time n + 1 with probability β n . The sequence is i.i.d. and constitutes our random environment. Moreover, at every time n we add Z n individuals to the population. The sequence is also i.i.d. We find sufficient conditions for null recurrence and transience (positive recurrence has been addressed by Neuts 1994 J. Appl. Probab. 31 48–58). We apply our results to a particular distribution and deterministic β. This particular case shows a rather unusual phase transition in β in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.

Opposing effects of spatiotemporal variation in resources and temporal variation in climate on density dependent population growth in seabirds.

Understanding how ecological processes combine to shape population dynamics is crucial in a rapidly changing world. Evidence has been emerging for how fundamental drivers of density dependence in mobile species are related to two differing types of environmental variation-temporal variation in climate, and spatiotemporal variation in food resources. However, to date, tests of these hypotheses have been largely restricted to mid-trophic species in terrestrial environments and thus their general applicability remains unknown. We tested if these same processes can be identified in marine upper trophic level species. We assembled a multi-decadal data set on population abundance of 10 species of colonial seabirds comprising a large component of the UK breeding seabird biomass, and covering diverse phylogenies, life histories and foraging behaviours. We tested for evidence of density dependence in population growth rates using discrete time state-space population models fit to long time-series of observations of abundance at seabird breeding colonies. We then assessed if the strength of density dependence in population growth rates was exacerbated by temporal variation in climate (sea temperature and swell height), and attenuated by spatiotemporal variation in prey resources (productivity and tidal fronts). The majority of species showed patterns consistent with temporal variation in climate acting to strengthen density dependent feedbacks to population growth. However, fewer species showed evidence for a weakening of density dependence with increasing spatiotemporal variation in prey resources. Our findings extend this emerging theory for how different sources of environmental variation may shape the dynamics and regulation of animal populations, demonstrating its role in upper trophic marine species. We show that environmental variation leaves a signal in long-term population dynamics of seabirds with potentially important consequences for their demography and trophic interactions.

Ternary network models for disturbed ecosystems.

The complex network of interactions between species makes understanding the response of ecosystems to disturbances an enduring challenge. One commonplace way to deal with this complexity is to reduce the description of a species to a binary presence–absence variable. Though convenient, this limits the patterns of behaviours representable within such models. We address these shortcomings by considering discrete population models that expand species descriptions beyond the binary setting. Specifically, we focus on ternary (three-state) models which, alongside presence and absence, additionally permit species to become overabundant. We apply this ternary framework to the robustness analysis of model ecosystems and show that this expanded description permits the modelling of top-down extinction cascades emerging from consumer pressure or mesopredator release. Results therefore differ significantly from those seen in binary models, where such effects are absent. We also illustrate how this method opens up the modelling of ecosystem disturbances outside the scope of binary models, namely those in which species are externally raised to overabundance. Our method therefore has the potential to provide a richer description of ecosystem dynamics and their disturbances, while at the same time preserving the conceptual simplicity of familiar binary approaches.

Locked fronts in a discrete time discrete space population model.

A model of population growth and dispersal is considered where the spatial habitat is a lattice and reproduction occurs generationally. The resulting discrete dynamical system exhibits velocity locking, where rational speed invasion fronts are observed to persist as parameters are varied. In this article, we construct locked fronts for a particular piecewise linear reproduction function. These fronts are shown to be linear combinations of exponentially decaying solutions to the linear system near the unstable state. Based upon these front solutions, we then derive expressions for the boundary of locking regions in parameter space. We obtain leading order expansions for the locking regions in the limit as the migration parameter tends to zero. Strict spectral stability in exponentially weighted spaces is also established.

Embedded Estimation Sequential Bayes Parameter Inference for the Ricker Dynamical System

The dynamical systems are comprised of two components that change over time: the state space and the observation models. This study examines parameter inference in dynamical systems from the perspective of Bayesian inference. Inference on unknown parameters in nonlinear and non-Gaussian dynamical systems is challenging because the posterior densities corresponding to the unknown parameters do not have traceable formulations. Such a system is represented by the Ricker model, which is a traditional discrete population model in ecology and epidemiology that is used in many fields. This study, which deals with parameter inference, also known as parameter learning, is the central objective of this study. A sequential embedded estimation technique is proposed to estimate the posterior density and obtain parameter inference. The resulting algorithm is called the Augmented Sequential Markov Chain Monte Carlo (ASMCMC) procedure. Experiments are performed via simulation to illustrate the performance of the ASMCMC algorithm for observations from the Ricker dynamical system.

Modeling Immigration in Spain: Causes, Size and Consequences

This paper deals with the construction of a discrete dynamic population model addressed to estimate the expected size of the immigration population in a finite short period of time in Spain. By paying attention to a special subpopulation of interest, such as an irregular immigrant, unaccompanied minor immigrant and regular immigrant, a vector discrete population model is built after the discussion and introduction of proper hypotheses linked to economy, host and country of origin regulation policies, political interest and others. The model allows us to study the change of the results under variation of the parameters.

On second-order fuzzy discrete population model

Abstract This work is concerned with dynamical behavior of a second-order fuzzy discrete population model: x n = A x n − 1 1 + x n − 1 + B x n − 2 , n = 1 , 2 , … , {x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}},\hspace{1em}n=1,2,\ldots , where A , B A,B are positive fuzzy numbers. x n {x}_{n} is a positive fuzzy number and represents the population size at the observation instant n. According to a generalization of division ( g g -division) of fuzzy number, we study the dynamical behaviors including boundedness, global asymptotical stability, and persistence of positive fuzzy solution. Finally, two examples are given to demonstrate the effectiveness of the results obtained.

The First Population Simulation for the Zalophus japonicus (Otariidae: Sea Lions) on Dokdo, Korea

The Japanese sea lion (Z. japonicus) has been regarded as an extinct species since the last report on Dokdo in 1951. Not much ecological information on the Z. japonicus on Dokdo (hereafter Dokdo sea lion) is currently available. Using a discrete time stage-structured population model, we reconstructed the Dokdo sea lion population to explore the effect of human hunting pressure on them. This study provides the first estimate for the Dokdo sea lion population from 1900 to 1951. The reconstructed capture numbers of the Dokdo sea lion and the parameters estimated in this study were well matched with the recorded numbers and ecological parameters reported previously for the Californian sea lion. Based on the reconstructed population, their number rapidly declined after hunting started and it took less than 10 years for a 70% decline of the initial population, which would be considered to be an extinction risk. Since some caveats exist in this study, some caution about our results is necessary. However, this study demonstrates how rapidly human over-hunting can cause the extermination of a large local population. This study will be helpful to raise people’s awareness about endangered marine animals such as local finless porpoises in Korea.

Bifurcation analysis of a piecewise-smooth Ricker map with proportional threshold harvesting

Proportional threshold harvesting (PTH) refers to some control rules employed in fishing policies, which specify a biomass level below which no fishing is permitted (the threshold), and a fraction of the surplus above the threshold is removed every year. When these rules are applied to a discrete population model, the resulting map governing the harvesting model is piecewise smooth, so border-collision bifurcations play an essential role in the dynamics. In this paper, we carry out a bifurcation analysis of a PTH model, providing a thorough picture of the 2-parameter bifurcation diagram in the plane for a case study. Here, T is the threshold and q is the harvest proportion. Our results explain some numerical bifurcation diagrams in previous work for PTH, and uncover new features of the dynamics with interesting consequences for population management.

Quantifying the nature and strength of intraspecific density dependence in Arctic mosquitoes.

Processes that change with density are inherent in all populations, yet quantifying density dependence with empirical data remains a challenge. This is especially true for animals recruiting in patchy landscapes because heterogeneity in habitat quality in combination with habitat choice can obscure patterns expected from density dependence. Mosquitoes (Diptera: Culicidae) typically experience strong density dependence when larvae compete for food, however, effects vary across species and contexts. If populations experience intense intraspecific density-dependent mortality then overcompensation can occur, where the number of survivors declines at high densities producing complex endogenous dynamics. To seek generalizations about density dependence in a widespread species of Arctic mosquito, Aedes nigripes, we combined a laboratory experiment, field observations, and modeling approaches. We evaluated alternative formulations of discrete population models and compared best-performing models from our lab study to larval densities from ponds in western Greenland. Survivorship curves from the lab were the best fit by a Hassell model with compensating density dependence (equivalent to a Beverton-Holt model) where peak recruitment ranged from 8 to 80 mosquitoes per liter depending on resource supply. In contrast, our field data did not show a signal of strong density dependence, suggesting that other processes such as predation may lower realized densities in nature, and that expected patterns may be obscured because larval abundance covaries with resources (cryptic density dependence). Our study emphasizes the importance of covariation between the environment, habitat choice, and density dependence in understanding population dynamics across landscapes, and demonstrates the value of pairing lab and field studies.

Optimal control of harvest timing in discrete population models

AbstractHarvest plays an important role in management decisions, from fisheries to pest control. Discrete‐time models enable us to explore the importance of timing of management decisions, including the order of events of particular actions. We derive novel mechanistic models featuring explicit within‐season harvest timing and level. We explore optimization of within‐season harvest level and timing through optimal control of these population models. With a fixed harvest level, harvest timing is taken as the control. Then both harvest timing and level are used as controls. We maximize an objective functional which includes management goals of maximizing yield, maximizing stock, and minimizing costs associated with both harvest intensity and harvest timing. While standard models with compensatory population dynamics predict it is best to harvest as early as possible in the season, we find instances where harvesting later in the season is optimal. Furthermore, we discover interesting oscillations in the population size, which would be unexpected in the model without time‐varying controls.Recommendations for Resource Managers: This paper presents a novel model with variable harvest timing, which can be used as a tool to manage populations. While the usual population models identify harvesting at the beginning of the season as the best timing to maximize yield and/or population size, accounting for harvest costs related to timing can give rise to optimal harvest time at some intermediate point in the season. Controlling the level and the timing of harvest separately when maximizing the yield while minimizing the costs of management can lead to oscillation patterns in the population. Such oscillations would not be seen in these models without these time‐varying controls. The level of harvest intensity in a population can affect the optimal time to implement the harvest.