Allee Effect Model Research Articles

Factors affecting establishment and population growth of the invasive weed Ambrosia artemisiifolia.

Ambrosia artemisiifolia is a highly invasive weed. Identifying the characteristics and the factors influencing its establishment and population growth may help to identify high invasion risk areas and facilitate monitoring and prevention efforts. Six typical habitats: river banks, forests, road margins, farmlands, grasslands, and wastelands, were selected from the main distribution areas of A. artemisiifolia in the Yili Valley, China. Six propagule quantities of A. artemisiifolia at 1, 5, 10, 20, 50, and 100 seeds m-2 were seeded by aggregation, and dispersion in an area without A. artemisiifolia. Using establishment probability models and Allee effect models, we determined the minimum number of seeds and plants required for the establishment and population growth of A. artemisiifolia, respectively. We also assessed the moisture threshold requirements for establishment and survival, and the influence of native species. The influence of propagule pressure on the establishment of A. artemisiifolia was significant. The minimum number of seeds required varied across habitats, with the lowest being 60 seeds m-2 for road margins and the highest being 398 seeds for forests. The minimum number of plants required for population growth in each habitat was 5 and the largest number was 43 in pasture. The aggregation distribution of A. artemisiifolia resulted in a higher establishment and survival rate. The minimum soil volumetric water content required for establishment was significantly higher than that required for survival. The presence of native dominant species significantly reduced the establishment and survival rate of A. artemisiifolia. A. artemisiifolia has significant habitat selectivity and is more likely to establish successfully in a habitat with aggregated seeding with sufficient water and few native species. Establishment requires many seeds but is less affected by the Allee effect after successful establishment, and only a few plants are needed to ensure reproductive success and population growth in the following year. Monitoring should be increased in high invasion risk habitats.

Confronting population models with experimental microcosm data: from trajectory matching to state‐space models

AbstractPopulation and community ecology traditionally has a very strong theoretical foundation with well‐known dynamical models, such as the logistic and its variations, and many modifications of the classical Lotka–Volterra predator–prey and interspecific competition models. More and more, these classical models are being confronted with data via fitting to empirical time series for purposes of projections or for estimating model parameters of interest. However, using statistical models to fit theoretical models to data is far from trivial, especially for time series data where subsequent measurements are not independent. This raises the question of whether statistical inferences using pure observation error models, such as simple (non‐)linear regressions, are biased, and whether more elaborate process error models or state‐space models have to be used to address this complexity. In order to help empiricists, especially researchers working with experimental laboratory populations in micro‐ and mesocosms, make informed decisions about the statistical formalism to use, we here compare different error structures one could use when fitting classical deterministic ordinary differential equation (ODE) models to empirical data. We consider a large range of biological scenarios and theoretical models, from single species to community dynamics and trophic interactions. In order to compare the performance of different error structure models, we use both realistically simulated data and empirical data from microcosms in a Bayesian framework. We find that many model parameters can be estimated precisely with an appropriate choice of error structure using pure observation error or state‐space models, if observation errors are not too high. However, Allee effect models are typically hard to identify and state‐space models should be preferred when model complexity increases. Our work shows that, at least in the context of low environmental stochasticity and high quality observations, deterministic models can be used to describe stochastic population dynamics that include process variability and observation error. We discuss when more complex state‐space model formulations may be required for obtaining accurate parameter estimates. Finally, we provide a comprehensive tutorial for fitting these models in R.

A mathematical modelling framework for the regulation of intra-cellular OCT4 in human pluripotent stem cells.

Human pluripotent stem cells (hPSCs) have the potential to differentiate into all cell types, a property known as pluripotency. A deeper understanding of how pluripotency is regulated is required to assist in controlling pluripotency and differentiation trajectories experimentally. Mathematical modelling provides a non-invasive tool through which to explore, characterise and replicate the regulation of pluripotency and the consequences on cell fate. Here we use experimental data of the expression of the pluripotency transcription factor OCT4 in a growing hPSC colony to develop and evaluate mathematical models for temporal pluripotency regulation. We consider fractional Brownian motion and the stochastic logistic equation and explore the effects of both additive and multiplicative noise. We illustrate the use of time-dependent carrying capacities and the introduction of Allee effects to the stochastic logistic equation to describe cell differentiation. We conclude both methods adequately capture the decline in OCT4 upon differentiation, but the Allee effect model has the advantage of allowing differentiation to occur stochastically in a sub-set of cells. This mathematical framework for describing intra-cellular OCT4 regulation can be extended to other transcription factors and developed into predictive models.

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

<p style='text-indent:20px;'>We study positive solutions to steady state reaction diffusion equations of the form: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \; \; \begin{matrix} -\Delta u = \lambda f(u);\; \Omega \\ \; \; \alpha(u)\frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[1-\alpha(u)]u = 0; \; \partial \Omega\end{matrix} \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> is the population density, <inline-formula><tex-math id="M4">\begin{document}$ f(u) = \frac{1}{a}u(u+a)(1-u) $\end{document}</tex-math></inline-formula> represents a weak Allee effect type growth of the population with <inline-formula><tex-math id="M5">\begin{document}$ a\in (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \alpha(u) $\end{document}</tex-math></inline-formula> is the probability of the population staying in the habitat <inline-formula><tex-math id="M7">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> when it reaches the boundary, and positive parameters <inline-formula><tex-math id="M8">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> represent the domain scaling and effective exterior matrix hostility, respectively. In particular, we analyze the case when <inline-formula><tex-math id="M10">\begin{document}$ \alpha(s) = \frac{1}{[1+(A - s)^2 + \epsilon]} $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M11">\begin{document}$ s \in [0,1] $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M12">\begin{document}$ A\in (0,1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ \epsilon\geq 0 $\end{document}</tex-math></inline-formula>. In this case <inline-formula><tex-math id="M14">\begin{document}$ 1-\alpha(s) $\end{document}</tex-math></inline-formula> represents a U-shaped relationship between density and emigration. Existence, nonexistence, and multiplicity results for this model are established via the method of sub-super solutions.

Unpacking the Allee effect: determining individual-level mechanisms that drive global population dynamics.

We present a solid theoretical foundation for interpreting the origin of Allee effects by providing the missing link in understanding how local individual-based mechanisms translate to global population dynamics. Allee effects were originally proposed to describe population dynamics that cannot be explained by exponential and logistic growth models. However, standard methods often calibrate Allee effect models to match observed global population dynamics without providing any mechanistic insight. By introducing a stochastic individual-based model, with proliferation, death and motility rates that depend on local density, we present a modelling framework that translates particular global Allee effects to specific individual-based mechanisms. Using data from ecology and cell biology, we unpack individual-level mechanisms implicit in an Allee effect model and provide simulation tools for others to repeat this analysis.

Stress response and population dynamics: Is Allee effect hormesis?

Hormesis is a fundamental notion in ecotoxicology while competition between organisms is an essential notion in population ecology and species adaptation and evolution. Both sub-disciplines of ecology deal with the response of organisms to abiotic and biotic stresses. In ecotoxicology, the Linear-non-Threshold (LNT), Threshold and Hormetic models are used to describe the dominant responses of a plethora of endpoints to abiotic stress. In population ecology, the logistic, theta-logistic and the Allee effect models are used to describe the growth of populations under different responses to (biotic) stress induced by population density. The per capita rate of population increase (r) measures species fitness. When it is used as endpoint, the responses to population density seem to perfectly correspond to LNT, Threshold and Hormetic responses to abiotic stress, respectively. Our analysis suggests the Allee effect is a hormetic-like response of r to population density, an ultimate biotic stress. This biphasic dose-response model appears across different systems and situations (from molecules to tumor growth to population dynamics), is highly supported by ecological and evolutionary theory, and has important implications in most sub-disciplines of biology as well as in environmental and earth sciences. Joined multi-disciplinary efforts would facilitate the development and application of advanced research approaches for better understanding potential planetary-scale implications.

Generalized Allee effect model

The Allee effect consists of a positive correlation between very small population size and fitness. Offering a new view point on the weak and strong demographic Allee effect, we propose to combine them with the Richards growth model. In particular, a peculiar manifestation of the Allee effect is analytically predicted and still not validated by experiments. Model validation with ecological data is presented for some special situations.

Application of the Anthropogenic Allee Effect Model to Trophy Hunting as a Conservation Tool

Trophy hunting can provide economic incentives to conserve wild species, but it can also involve risk when rare species are hunted. The anthropogenic Allee effect (AAE) is a conceptual model that seeks to explain how rarity may spread the seeds of further endangerment. The AAE model has increasingly been invoked in the context of trophy hunting, increasing concerns that such hunting may undermine rather than enhance conservation efforts. We question the appropriateness of uncritically applying the AAE model to trophy hunting for 4 reasons. First, the AAE assumes an open-access resource, which is a poor characterization of most trophy-hunting programs and obscures the potential for state, communal, or private-property use rights to generate positive incentives for conservation. Second, study results that show the price of hunting increases as the rarity of the animal increases are insufficient to indicate the presence of AAE. Third, AAE ignores the existence of biological and behavioral factors operating in most trophy-hunting contexts that tend to regulate the effect of hunting. We argue that site-specific data, rather than aggregated hunting statistics, are required to demonstrate that patterns of unsustainable exploitation can be well explained by an AAE model. Instead, we suggest that conservation managers seeking to investigate and identify constraints that limit the potential conservation role of trophy hunting, should focus on the critical governance characteristics that shape the potential conservation role of trophy hunting, such as corruption, insecure property rights, and inadequate sharing of benefits with local people.

Limited evidence for the demographic Allee effect from numerous species across taxa

Extensive theoretical work on demographic Allee effects has led to the latent assumption that they are ubiquitous in natural populations, yet current empirical support for this phenomenon is sparse. We extended previous single-taxon analyses to evaluate the empirical support for demographic Allee effects in the per capita population growth rate of 1198 natural populations spanning all major taxa. For each population, we quantified the empirical support for five population growth models: no growth (random walk); exponential growth, with and without an Allee effect; and logistic growth, with and without an Allee effect. We used two metrics to quantify empirical support, information-theoretic and Bayesian strength of evidence, and observed top-rank frequency. The Ricker logistic model was both the most supported and most frequently top-ranked model, followed by random walk. Allee models had a combined relative support of 12.0% but were top-ranked in only 1.1% of the time series. Accounting for local climate variation and measurement error caused the loss of top-ranked Allee models, although the latter also increased their relative support. The 13 time series exhibiting Allee models were shorter and less variable than other time series, although only three were non-trending. Time series containing observations at low abundance were not more likely and did not show higher support for Allee effect models. We conclude that there is relatively high potential for demographic Allee effects in these 1198 time series but comparatively few observed cases, perhaps due to the influences of climate and measurement error.

On the mechanistic underpinning of discrete-time population models with Allee effect

The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261–269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resource-consumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience.

Periodic dynamics in a two-stage Allee effect model are driven by tension between stage equilibria

Single species difference population models can show complex dynamics such as periodicity and chaos under certain circumstances, but usually only when rates of intrinsic population growth or other life history parameter are unrealistically high. Single species models with Allee effects (positive density dependence at low density) have also been shown to exhibit complex dynamics when combined with overcompensatory density dependence or a narrow fertility window. Here we present a simple two-stage model with Allee effects which shows large amplitude periodic fluctuations for some initial conditions, without these requirements. Periodicity arises out of a tension between the critical equilibrium of each stage, i.e. when the initial population vector is such that the adult stage is above the critical value, while the juvenile stage is below the critical value. Within this area of parameter space, the range of initial conditions giving rise to periodic dynamics is driven mainly by adult mortality rates. Periodic dynamics become more important as adult mortality increases up to a certain point, after which periodic dynamics are replaced by extinction. This model has more realistic life history parameter values than most ‘chaotic’ models. Conditions for periodic dynamics might arise in some marine species which are exploited (high adult mortality) leading to recruitment limitation (low juvenile density) and might be an additional source of extinction risk.