Discrete-time Population Models Research Articles

Interspecific competition models and resource inequality between individuals.

Most classical discrete-time population models of interspecific competition have emerged as population-level phenomenological models with no evident basis at the individual level. This study shows that the Hassell-Comins model, a widely used discrete-time model of interspecific competition, can be derived in a bottom-up manner from a simple model of random resource competition between individuals of two species as an expression of expected population sizes in the next generation. The random competition leads to inequalities in resource allocation between individuals, which are related to the key parameters of the Hassell-Comins model that determine the density dependence of each species. The relationship between population-level parameters, such as intra- and interspecific competition coefficients, and individual-level parameters is discussed in detail, as is how the derived competition equations depend on the degree of inequality within each species. By considering limits of maximum or minimum resource inequality within each species, the derived model can describe interspecific competition for various combinations of two species exhibiting ideal scramble or ideal contest intraspecific competition.

Infection-induced increases to population size during cycles in a discrete-time epidemic model

One-dimensional discrete-time population models, such as those that involve Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Expanding the system by one dimension to incorporate epidemiological interactions causes an interesting complexity of new behaviors. Here, we examine a discrete-time two-dimensional susceptible-infectious (SI) model with Ricker growth and show that the introduction of infection can not only produce a distinctly different bifurcation structure than that of the underlying disease-free system but also lead to counter-intuitive increases in population size. We use numerical bifurcation analysis to determine the influence of infection on the location and types of bifurcations. In addition, we examine the appearance and extent of a phenomenon known as the ‘hydra effect,’ i.e., increases in total population size when factors, such as mortality, that act negatively on a population, are increased. Previous work, primarily focused on dynamics at fixed points, showed that the introduction of infection that reduces fecundity to the SI model can lead to a so-called ‘infection-induced hydra effect.’ Our work shows that even in such a simple two-dimensional SI model, the introduction of infection that alters fecundity or mortality can produce dynamics can lead to the appearance of a hydra effect, particularly when the disease-free population is at a cycle.

Stock patterns in a class of delayed discrete-time population models

Stock patterns in a class of delayed discrete-time population models

Exploring dynamics of plant-herbivore interactions: bifurcation analysis and chaos control with Holling type-II functional response.

In this study, we examine the plant-herbivore discrete model of apple twig borer and grape vine interaction, with a particular emphasis on the extended weak-predator response to Holling type-II response. We explore the dynamical and qualitative analysis of this model and investigate the conditions for stability and bifurcation. Our study demonstrates the presence of the Neimark-Sacker bifurcation at the interior equilibrium and the transcritical bifurcation at the trivial equilibrium, both of which have biological feasibility. To avoid unpredictable outcomes due to bifurcation, we employ chaos control methods. Furthermore, to support our theoretical and mathematical findings, we develop numerical simulation techniques with examples. In summary, our research enhances the comprehension of the dynamics pertaining to interactions between plants and herbivores in the context of discrete-time population models.

STRONG ALLEE EFFECT AND EVOLUTIONARY DYNAMICS IN A SINGLE-SPECIES RICKER POPULATION MODEL

In this paper, a discrete-time Ricker population model with the strong Allee effect is proposed, and its complex dynamic behavior is analyzed. First, the existence and asymptotic stability of the unique positive equilibrium are studied. Second, Neimark–Sacker and period-doubling bifurcations of this discrete model were carried out, and corresponding bifurcation conditions were obtained. Third, the pole placement method and the hybrid control strategy have been used to control the chaos produced by these bifurcations. Finally, we use MATLAB software to carry out some numerical simulations to analyze the rich dynamics of the system as well as to verify our theoretical results.

Continuity of the spectral radius, applied to structured semelparous two-sex population models

If a structured discrete-time population model x n = F ( x n − 1 ) , n ∈ N , takes account of two-sexes, the year-to-year turnover operator F often has a first-order approximation B that is homogeneous and order-preserving rather than linear and positive. Still, one can define the spectral radius T = r ( B ) , the basic turnover number, which plays the role of a threshold parameter between extinction and persistence. If T < 1 , the extinction state is locally stable, and if T > 1 the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. This threshold property raises interest in the continuous dependence of the basic turnover number on the parameters in the population model. In this paper, we concentrate on the continuous dependence of T = r ( B ) on mating functions.

Dynamical analysis and chaos control of a fractional-order Leslie-type predator–prey model with Caputo derivative

In this paper, the dynamical behaviors of a discrete-time fractional-order population model are considered. The stability analysis and the topological classification of the model at the fixed point have been investigated. It is shown that the model undergoes flip and Neimark–Sacker bifurcations around the co-existence fixed point by using the bifurcation and the normal form theory. These bifurcations lead to chaos when the parameter changes at critical point. In order to control chaotic behavior in the model result from Neimark–Sacker bifurcation, the OGY feedback method has been used. Furthermore, some numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the presented model are plotted to support the correctness of the analytical results. The positive Lyapunov exponents demonstrate that chaotic behavior exists in the considered model.

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.

A Stage-Structured Continuous-/Discrete-Time Population Model: Persistence and Spatial Spread.

Population persistence and spatial propagation and their dependence on demography and dispersal are of great importance in spatial ecology. Many species with highly structured life cycles invade new habitats through the dispersal of organisms in their early life stages (e.g., seeds, larvae, etc.). We develop a stage-structured continuous/discrete-time hybrid model to describe the spatiotemporal dynamics of such species, in which a reaction-diffusion equation describes the random movement of dispersing individuals, while two difference equations describe the demography of sedentary individuals. We obtain a formula for the spreading speed of the population in terms of model parameters. We show that the spreading speed can be characterized as the slowest wave speed of a class of traveling wave solutions. We provide an explicit formula for the critical domain size that separates population persistence from extinction. By comparing our stage-structured model with a physically unstructured model, we find that the structured model reduces to the unstructured one in some special cases. Accordingly, the results about the spreading speed and the critical domain size for the unstructured model represent some special cases of those for the structured one. This highlights the significance of including stage structure in studying the spatial dynamics of species with complex life cycles.

THE EFFECT OF ALLEE FACTOR ON A NONLINEAR DELAYED POPULATION MODEL WITH HARVESTING

The Allee effect may have a stabilizing or destabilizing effect on population dynamics, or reaching stability may be delayed. Also this effect can push populations to extinction, especially under sharp harvesting effect. Compensatory or overcompensatory population maps are situations that change the effect of the Allee factor on population dynamics. The harvesting factor has also the effect of changing the equilibrium levels differently in theoretical population models. This study includes the local stability of the equilibrium point of the delayed discrete-time population model exposed to the harvest effect with and without Allee effect. We also review the effect of Allee factor on the local stability of equilibrium point of the discrete-time population model involving delay [1]. The dynamics of this population models are given together with the enriched dynamics.

Discrete-time models for interactive wild and sterile mosquitoes with general time steps

Different from the discrete-time population models based on evolution of generations or life cycles, we formulate discrete-time homogeneous and stage-structured models with time steps in more general settings such that survivals are included at each time step. We assume that sterile mosquitoes are released and their number in the field is kept at a constant level. We study the interactive dynamics of wild and sterile mosquitoes where only sexually active sterile mosquitoes are considered. We determine threshold values of releases and investigate the interactive dynamics for both homogeneous and stage-structured populations. Numerical examples are provided to confirm and demonstrate the obtained theoretical results.

Keeping random walks safe from extinction and overpopulation in the presence of life-taking disasters

ABSTRACT Recurrence and transience conditions are made explicit in discrete-time Markov chain population models for which random stationary growth alternates with disastrous random life-taking events. These events either have moderate stationary magnitudes or lead to an abrupt population decline. The probability of their occurrence may or may not depend on the population size. These conditions are based on the existence or not of a “weak” carrying capacity, where “weak” means that the carrying capacity can be exceeded, temporarily. In this framework, the population is threatened with extinction, an event whose probability is expressed, as well as the law of the time remaining until this deadline. On the other hand, the population is also threatened by overpopulation, an event whose time to reach a given threshold is expressed, as well as the difference between the population size and the carrying capacity. The theory is that of extreme values for Markov chains and is based on the control of the spectral properties of the northwest truncation of the transition matrix of the original Markov chain with life-taking disasters. The article presents an extension to the case where the process of life-taking disasters is no longer geometric and to the case where the probability of occurrence of a disaster depends on the population size. Both the time to extinction and the time to a given threshold have geometrically decaying distribution tails. The use of the extremal Markov chain in the calculation of the time to overpopulation is innovative.

Stability, bifurcations and hydra effects in a stage-structured population model with threshold harvesting

In this paper, we investigate the dynamics of a discrete-time stage-structured population model where juveniles and adults may be subject to threshold harvesting. This management policy allows the harvesting of the target population only if its size exceeds a predetermined threshold. It is commonly used for maintaining biomass, obtaining a higher yield, and preventing population collapse. We focus on the response of the structured populations to harvesting and study how they can be affected by considering different thresholds for each age class. We find all possible equilibria of the system and analyze their stability; we show that harvesting does not destabilize globally stable equilibria. We discuss when hydra effects may occur, more specifically, we determine when the adult population size can increase at equilibrium in response to an increase in its per-capita mortality rate as a consequence of threshold harvesting. A rigorous 2-parameter bifurcation diagram is given for semelparous populations, which helps to understand the general case when adult survivorship rates are low. Numerical results complement the bifurcation analysis in the general case.

Bifurcation analysis and chaos control of the population model with harvest

In this article, we investigated the dynamic behavior of a discrete-time population model with the harvest. We give numerical simulation and chaos control by using the linear feedback control method.

Discrete-time dynamics of structured populations via Feller kernels

&lt;p style='text-indent:20px;'&gt;Feller kernels are a concise means to formalize individual structural transitions in a structured discrete-time population model. An iteroparous populations (in which generations overlap) is considered where different kernels model the structural transitions for neonates and for older individuals. Other Feller kernels are used to model competition between individuals. The spectral radius of a suitable Feller kernel is established as basic turnover number that acts as threshold between population extinction and population persistence. If the basic turnover number exceeds one, the population shows various degrees of persistence that depend on the irreducibility and other properties of the transition kernels.&lt;/p&gt;

Destabilization and chaos induced by harvesting: insights from one-dimensional discrete-time models.

One-dimensional discrete-time population models are often used to investigate the potential effects of increasing harvesting on population dynamics, and it is well known that suitable harvesting rates can stabilize fluctuations of population abundance. However, destabilization is also a possible outcome of increasing harvesting even in simple models. We provide a rigorous approach to study when harvesting is stabilizing or destabilizing, considering proportional harvesting and constant quota harvesting, that are usual strategies for the management of exploited populations. We apply our results to some of the most popular discrete-time population models (quadratic, Ricker and Bellows maps). While the usual case is that increasing harvesting is stabilizing, we prove, somehow surprisingly, that increasing values of constant harvesting can destabilize a globally stable positive equilibrium in some cases; moreover, we give a general result which ensures that global stability can be shifted to observable chaotic dynamics by increasing one model parameter, and apply this result to some of the considered harvesting models.

Stability Analysis of a Discrete Time Prey-Predator Population Model with Immigration

In this paper, a discrete-time prey-predator population model with immigration which is obtained by implementing forward Euler’s scheme has been considered. The existence of fixed points of the presented model has been investigated. Moreover, the stability analysis of the fixed points of the population model has been examined and the topological classification of the fixed points of the model has been made. Moreover, the OGY feedback control method is to implement to controlchaos caused by the Flip bifurcation. Finally, Flip bifurcation,chaos control strategy, and asymptotic stability of the only positive fixed point are verifiedwith the help of numerical simulations.

Bifurcation and Chaos of a Discrete-Time Population Model

A Leslie population model for two generations is investigated by qualitative analysis and numerical simulation. For the different parameters a and b in the model, the dynamics of the system are studied, respectively. It shows many complex dynamic behavior, including several types of bifurcations leading to chaos, such as period-doubling bifurcations and Neimark–Sacker bifurcations. With the change of parameters, attractor crises and chaotic bands with periodic windows appear. The largest Lyapunov exponents are numerically computed and can verify the rationality of the theoretical analysis.

Characterizing the existence of hydra effect in spatial predator-prey models and the influence of functional response types and species dispersal

The hydra effect in ecology corresponds to the increase in the equilibrium or time-averaged density of a population when its mortality rate is augmented. The literature presents examples of the hydra effect in continuous as well as in discrete time population models described by nonlinear systems of ordinary differential and difference equations, respectively. The novelty of this work is that we show by means of numerical simulation the occurrence of the hydra effect in some stationary dynamics of one dimensional reaction-diffusion predator-prey models described by nonlinear systems of partial differential equations with distinct nonlinear functional responses. Reaction-diffusion models are a set of mathematical models that take into account population dynamics and dispersal (diffusivity) within a predefined spatial domain. We show that for a type 2 functional response an increasing diffusivity shortens the range of the hydra effect. Another interesting result concerns the fact that, together with the hydra effect, the prey density also increases with the increase of the per capita density-independent mortality rate of the predator. This result indicates that this top-down disturbance does not necessarily create a trophic cascade in a spatial predator-prey interaction at the steady-state regime.

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.