Discrete-time Population Models Research Articles

Title: Recruitment is key to understanding amphibian’s different population-level responses to chytrid fungus infection

Understanding why related species have differing population-level responses to threats can be key to identifying conservation options for declining populations. However, this is difficult when multiple threats are implicated. Chytrid fungus (Batrachochytrium dendrobatidis: Bd) is implicated in at least 500 amphibian population declines globally, although few tangible options exist to mitigate pathogen impacts. Other threatening processes also invariably operate on most amphibians. Non-native fish, for example, can contribute to amphibian declines and may exacerbate Bd impacts. We disentangled the impacts of Bd and non-native fish upon two stream breeding frog species with differing conservation statuses to reveal vital rates that are crucial for species persistence. Litoria spenceri are threatened and historically occurred at elevations between 300–1100m asl in southeastern Australia. Litoria lesueurii are non-threatened and sympatric with L. spenceri at some sites. Using mark-recapture derived demographic rates known to correlate with climate and elevation, discrete-time deterministic population models were constructed for each species at high, moderate and low elevation sites, and multiple management scenarios. Our study reveals that age to maturation, clutch size and egg-year 1 survival influenced interspecific and intraspecific population-level responses of the two frogs to Bd impacts. Importantly, our results highlight that an amphibian population’s capacity to counteract Bd-mediated adult mortality is clearly constrained by other threats and environment interactions that moderate species recruitment. Furthermore, understanding limits to recruitment may help identify Bd mitigation strategies. In our study, disease mitigation may be best achieved at some sites by enhancing recruitment through non-native fish management.

Proportional threshold harvesting in discrete-time population models

Threshold-based harvesting strategies tend to give high yields while protecting the exploited population. A significant drawback, however, is the possibility of harvesting moratoria with their socio-economic consequences, if the population size falls below the threshold and harvesting is not allowed anymore. Proportional threshold harvesting (PTH) is a strategy, where only a fraction of the population surplus above the threshold is harvested. It has been suggested to overcome the drawbacks of threshold-based strategies. Here, we use discrete-time single-species models and rigorously analyze the impact of PTH on population dynamics and stability. We find that the population response to PTH can be markedly different depending on the specific population model. Reducing the threshold and allowing more harvest can be destabilizing (for the Ricker and Hassell map), stabilizing (for the quadratic map), or both (for the generalized Beverton-Holt map). Similarly, management actions in the form of increasing the threshold do not always improve population stability-this can also be due to bistability. Our results therefore emphasize the importance of a rigorous analysis in investigating the impact of PTH on population stability.

Global Asymptotic Stability of a Generalization of the Pielou Difference Equation

The Pielou equation is a well-known discrete-time population model in which the per capita growth rate depends on the population size, but the density dependence operates with a delay of d generations. Thus, the between-year dynamics are governed by a difference equation of order $$d+1$$ . The main result in this paper establishes the global stability of the unique positive equilibrium for a generalization of the two-dimensional Pielou equation. Our proof is based on a rather natural combination of two techniques which could be, in principle, applicable to obtain global asymptotic stability in other problems: some dominance conditions and the determination of a first integral for a related equation, which turns out to be a quasi-Lyanupov function for the generalized Pielou equation. We provide additional results on the global dynamics of the generalized Pielou equation for dimensions higher than two, and discuss its relationship with other families of difference equations traditionally employed for modelling population dynamics.

Fractional coalescent

An approach to the coalescent, the fractional coalescent (f-coalescent), is introduced. The derivation is based on the discrete-time Cannings population model in which the variance of the number of offspring depends on the parameter α. This additional parameter α affects the variability of the patterns of the waiting times; values of [Formula: see text] lead to an increase of short time intervals, but occasionally allow for very long time intervals. When [Formula: see text], the f-coalescent and the Kingman's n-coalescent are equivalent. The distribution of the time to the most recent common ancestor and the probability that n genes descend from m ancestral genes in a time interval of length T for the f-coalescent are derived. The f-coalescent has been implemented in the population genetic model inference software Migrate Simulation studies suggest that it is possible to accurately estimate α values from data that were generated with known α values and that the f-coalescent can detect potential environmental heterogeneity within a population. Bayes factor comparisons of simulated data with [Formula: see text] and real data (H1N1 influenza and malaria parasites) showed an improved model fit of the f-coalescent over the n-coalescent. The development of the f-coalescent and its inclusion into the inference program Migrate facilitates testing for deviations from the n-coalescent.

A refined mean field approximation of synchronous discrete-time population models

Mean field approximation is a popular method to study the behaviour of stochastic models composed of a large number of interacting objects. When the objects are asynchronous, the mean field approximation of a population model can be expressed as an ordinary differential equation. When the objects are (clock-) synchronous the mean field approximation is a discrete time dynamical system. We focus on the latter.We study the accuracy of mean field approximation when this approximation is a discrete-time dynamical system. We extend a result that was shown for the continuous time case and we prove that expected performance indicators estimated by mean field approximation are O(1∕N)-accurate. We provide simple expressions to effectively compute the asymptotic error of mean field approximation, for finite time-horizon and steady-state, and we use this computed error to propose what we call a refined mean field approximation. We show, by using a few numerical examples, that this technique improves the quality of approximation compared to the classical mean field approximation, especially for relatively small population sizes.

Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection

A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of \emph{potential} parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using sampling- and moment-duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a two-types $\Xi$-Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process' ergodic properties.

A Global Picture of the Gamma-Ricker Map: A Flexible Discrete-Time Model with Factors of Positive and Negative Density Dependence.

The gamma-Ricker model is one of the more flexible and general discrete-time population models. It is defined on the basis of the Ricker model, introducing an additional parameter [Formula: see text]. For some values of this parameter ([Formula: see text], population is overcompensatory, and the introduction of an additional parameter gives more flexibility to fit the stock-recruitment curve to field data. For other parameter values ([Formula: see text]), the gamma-Ricker model represents populations whose per-capita growth rate combines both negative density dependence and positive density dependence. The former can lead to overcompensation and dynamic instability, and the latter can lead to a strong Allee effect. We study the impact of the cooperation factor in the dynamics and provide rigorous conditions under which increasing the Allee effect strength stabilizes or destabilizes population dynamics, promotes or prevents population extinction, and increases or decreases population size. Our theoretical results also include new global stability criteria and a description of the possible bifurcations.

A unifying approach to discrete single-species populations models

In this article, we write the recruitment function \begin{document}$f$\end{document} for the discrete-time density-dependent population model \begin{document}$p_{n+1} =f(p_n)$ \end{document} as \begin{document}$f(p) =p+r(p)p$\end{document} where \begin{document}$r$\end{document} is the per capita growth rate. Making reasonable assumptions about the intraspecies relationships for the population, we develop four conditions that the function \begin{document}$r$\end{document} should satisfy. We then analyze the implications of these conditions for the recruitment function \begin{document}$f$\end{document} . In particular, we compare our conditions to those of Cull [2007], finding that the Cull model, with two additional conditions, is equivalent to our model. Studying the per capita growth rate when satisfying our four conditions gives insight into contest and scramble competition. In particular, depending on the properties of \begin{document}$r$\end{document} and \begin{document}$f$\end{document} , we have two different types of contest and scramble competitions, depending on the size of the population. We finally extend our approach to develop new models for discontinuous recruitment functions and for populations exhibiting Allee effects.

On bifurcations of delay single species discrete time population models

Our goal is an study on bifurcations of delay single species discrete time population models. Generally we transform the fixed point to origin. Conditions for existence of transcritical bifurcation, period doubling bifurcation, Neimark-Sacker bifurcation, resonance 1 : 3 bifurcation and resonance 1 : 4 bifurcation are studied. These bifurcations are analyzed by the centre manifold theorem and the normal form theory. In fact we investigate the non degeneracy conditions of generic bifurcations by computing the critical coefficients of the normal form of reduced system to the centre manifold. Then we apply our results for two kinds of population models. First we apply the results for the model and we will obtain the conditions for transcritical, period doubling, Neimark-Sacker, resonance 1 : 3 and resonance 1 : 4 bifurcations. Then we apply the results for two examples delay Logistic and delay Ricker models. Second we apply the results for Clark’s delayed recruitment model and we will obtain the conditions for mentioned bifurcations. Then we apply the results for the Clark’s delayed recruitment model of Ricker type.

The Local Stability Analysis of a Nonlinear Discrete-Time Population Model with Delay and Allee Effect

Bu calismada, genel lineer olmayan ayrik zamanli gecikmeli bir populasyon modeli, dusuk populasyon yogunlugunda ortaya cikan Allee etkisiyle ve Allee etkisiz olarak ele alinmistir. Her iki modelin, denge noktasinda lokal kararlilik sartlari incelendik ve bu iki modelin lokal kararliliklarini karsilastirdik. Elde edilen tum teorik sonuclar numerik simulasyonlar ile desteklenmistir.

Semi-global persistence and stability for a class of forced discrete-time population models

We consider persistence and stability properties for a class of forced discrete-time difference equations with three defining properties: the solution is constrained to evolve in the non-negative orthant, the forcing acts multiplicatively, and the dynamics are described by so-called Lur’e systems, containing both linear and non-linear terms. Many discrete-time biological models encountered in the literature may be expressed in the form of a Lur’e system and, in this context, the multiplicative forcing may correspond to harvesting, culling or time-varying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique non-zero equilibrium is stable with respect to the forcing in a sense which is reminiscent of input-to-state stability, a concept well-known in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by so-called proportional feedback control.

Global stability of discrete-time competitive population models

We develop practical tests for the global asymptotic stability of interior fixed points for discrete-time competitive population models. Our method constitutes the extension to maps of the Split Lyapunov method developed for differential equations. We give ecologically-motivated sufficient conditions for global stability of an interior fixed point of a map of Kolmogorov form. We introduce the concept of a principal reproductive mode, which is linked to a normal at the interior fixed point of a hypersurface of vanishing weighted-average growth. Our method is applied to establish new global stability results for 3-species competitive systems of May-Leonard type, where previously only parameter values for local stability was known.

From homogeneous eigenvalue problems to two-sex population dynamics.

Enclosure theorems are derived for homogeneous bounded order-preserving operators and illustrated for operators involving pair-formation functions introduced by Karl-Peter Hadeler in the late 1980s. They are applied to a basic discrete-time two-sex population model and to the relation between the basic turnover number and the basic reproduction number.

Controlling Chaos and Bifurcations in Discrete-Time Population Models

Controlling Chaos and Bifurcations in Discrete-Time Population Models

Validation of a Mathematical Model for Green Algae (Raphidocelis Subcapitata) Growth and Implications for a Coupled Dynamical System with Daphnia Magna

Toxicity testing in populations probes for responses in demographic variables to anthropogenic or natural chemical changes in the environment. Importantly, these tests are primarily performed on species in isolation of adjacent tropic levels in their ecosystem. The development and validation of coupled species models may aid in predicting adverse outcomes at the ecosystems level. Here, we aim to validate a model for the population dynamics of the green algae Raphidocelis subcapitata, a planktonic species that is often used as a primary food source in toxicity experiments for the fresh water crustacean Daphnia magna. We collected longitudinal data from three replicate population experiments of R. subcapitata. We used this data with statistical model comparison tests and uncertainty quantification techniques to compare the performance of four models: the Logistic model, the Bernoulli model, the Gompertz model, and a discretization of the Logistic model. Overall, our results suggest that the logistic model is the most accurate continuous model for R. subcapitata population growth. We then implement the numerical discretization showing how the continuous logistic model for algae can be coupled to a previously validated discrete-time population model for D. magna.

A Mechanistic Stochastic Ricker Model: Analytical and Numerical Investigations

The Ricker model is one of the simplest and most widely-used ecological models displaying complex nonlinear dynamics. We study a discrete-time population model, which is derived from simple assumptions concerning individual organisms’ behavior, using the “site-based” approach, developed by Brännström, Broomhead, Johansson and Sumpter. In the large-population limit the model converges to the Ricker model, and can thus be considered a mechanistic version of the Ricker model, derived from basic ecological principles, and taking into account the demographic stochasticity inherent to finite populations. We employ several analytical and precise numerical methods to study the model, showing how each approach contributes to understanding the model’s dynamics. Expressing the model as a Markov chain, we employ the concept of quasi-stationary distributions, which are computed numerically, and used to examine the interaction between complex deterministic dynamics and demographic stochasticity, as well as to calculate mean times to extinction. A Gaussian Markov chain approximation is used to obtain quantitative asymptotic approximations for the size of fluctuations of the stochastic model’s time series around the deterministic trajectory, and for the correlations between successive fluctuations. Results of these approximations are compared to results obtained from quasi-stationary distributions and from direct simulations, and are shown to be in good agreement.

Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective.

The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes with using the per capita invasion speed of mutants as their fitness. In this paper we develop a concept of frequency independence that attempts to capture the practical use of the term by ecologists, which although inspired by population genetics rarely fits its strict definition. We propose to call the invasion fitnesses of an eco-evolutionary model frequency independent when the phenotypes can be ranked by competitive strength, measured by who can invade whom. This is equivalent to the absence of weak priority effects, protected dimorphisms and rock–scissor–paper configurations. Our concept differs from that of Heino et al. (TREE 13:367–370, 1998) in that it is based only on the signs of the invasion fitnesses, whereas Heino et al. based their definitions on the structure of the feedback environment, summarising the effect of all direct and indirect interactions between individuals on fitness. As it turns out, according to our new definition an eco-evolutionary model has frequency independent fitnesses if and only if the effect of the feedback environment on the fitness signs can be summarised by a single scalar with monotonic effect. This may be compared with Heino et al.’s concept of trivial frequency dependence defined by the environmental feedback influencing fitness, and not just its sign, in a scalar manner, without any monotonicity restriction. As it turns out, absence of the latter restriction leaves room for rock–scissor–paper configurations. Since in ‘realistic’ (as opposed to toy) models frequency independence is exceedingly rare, we also define a concept of weak frequency dependence, which can be interpreted intuitively as almost frequency independence, and analyse in which sense and to what extent the restrictions on the potential model outcomes of the frequency independent case stay intact for models with weak frequency dependence.

Border Collision Bifurcations in a Generalized Model of Population Dynamics

We analyze the dynamics of a generalized discrete time population model of a two-stage species with recruitment and capture. This generalization, which is inspired by other approaches and real data that one can find in literature, consists in considering no restriction for the value of the two key parameters appearing in the model, that is, the natural death rate and the mortality rate due to fishing activity. In the more general case the feasibility of the system has been preserved by posing opportune formulas for the piecewise map defining the model. The resulting two-dimensional nonlinear map is not smooth, though continuous, as its definition changes as any border is crossed in the phase plane. Hence, techniques from the mathematical theory of piecewise smooth dynamical systems must be applied to show that, due to the existence of borders, abrupt changes in the dynamic behavior of population sizes and multistability emerge. The main novelty of the present contribution with respect to the previous ones is that, while using real data, richer dynamics are produced, such as fluctuations and multistability. Such new evidences are of great interest in biology since new strategies to preserve the survival of the species can be suggested.

An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius

Persistence and local stability of the extinction state are studied for discrete-time population models $x_n = F(x_{n-1})$, $n \in \mathbb{N}$, with a map $F$ on the cone $X_+$ of an ordered normed vector space $X$. Since sexual reproduction is accounted for, the first order approximation of $F$ at 0 is an order-preserving homogeneous map $B$ on $X_+$ that is not additive. The cone spectral radius of $B$ acts as the threshold parameter that separates persistence from local stability of 0, the extinction state. An important ingredient of the persistence theory for the induced semiflow is a homogeneous order-preserving eigenfunctional $\theta:X_+ \to \mathbb{R}_+$ of $B$ that is associated with the cone spectral radius and interacts with an appropriate persistence function $\rho$. Applications are presented for spatially distributed or rank-structured populations that reproduce sexually.

Population models with partial migration

Populations exhibiting partial migration consist of two groups of individuals: Those that migrate between habitats, and those that remain fixed in a single habitat. We propose several discrete-time population models to investigate the coexistence of migrants and residents. The first class of models is linear, and we distinguish two scenarios. In the first, there is a single egg pool to which both populations contribute. A fraction of the eggs is destined to become migrants, and the remainder become residents. In a second model, there are two distinct egg pools to which the two types contribute, one corresponding to residents and another to migrants. The asymptotic growth or decline in these models can be phrased in terms of the value of the basic reproduction number being larger or less than one respectively. A second class of models incorporates density dependence effects. It is assumed that increased densities in the various life history stages adversely affect the success of transitioning of individuals to subsequent stages. Here too we consider models with one or two egg pools. Although these are nonlinear models, their asymptotic dynamics can still be classified in terms of the value of a locally defined basic reproduction number: If it is less than one, then the entire population goes extinct, whereas it settles at a unique fixed point consisting of a mixture of residents and migrants, when it is larger than one. Thus, the value of the basic reproduction number can be used to predict the stable coexistence or collapse of populations exhibiting partial migration.