W. C. Allee brought attention to the possibility of a positive relationship between aspects of fitness and population size over fifty gears ago. This phenomenon, frequently termed the Allee effect, has been the focus of increased interest over the past two decades in the light of concerns over conservation and the problems of rarity, Use of the term suffers from the absence of a clear definition however, with the result that Allee effects are frequently thought to involve only a narrow range of phenomena and are often overlooked altogether. We propose a definition for the effect and attempt to resolve the major issues underlying the confusion surrounding this term.
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... For a small population size Allee effect plays a crucial role in elimination of the species. There are several reasons responsible for Allee effect such as mate finding, inbreeding depression, environmental conditioning etc [37,38,40,41]. Also, predation driven Allee effect is an important mechanism as it can promote negative density dependent growth [39]. ...
We propose and analyze a single-species population model subject to fear and its carry-over effect with the help of evolutionary game theory (EGT). We incorporate fear and carry-over cost in the growth of a single species resource population and the extensive analysis of our non-evolutionary model suggests that it can exhibit both weak and strong Allee effects. From the game theoretical viewpoint, we assume that the intrinsic growth rate r of the resource population and the attack rate a of the consumer population are functions of a mean phenotypic trait (u) of the resource, following a Normal distribution. Evolutionary stable strategies (ESS) are determined by using ESS maximum principle. Our study of ESS suggests that species extinction may be avoided as a result of evolution, though the extinct equilibrium can also be an ESS under certain conditions. The ratio of variation in the intrinsic growth rate and the attack rate plays a significant role in the ESS conditions of different equilibria as well as the global dynamics of our EGT model. Numerical simulations are performed to support our theoretical analysis.
... ematicians. Since Lotka [27] and Volterra [42] firstly proposed a prey-predator model, many mathematical models have been presented to study the wide range of complex phenomena between prey and predators, including Allee effects [39], group defense [41], hunting cooperation [2], intra-species competition [5,6], etc. Furthermore, based on some researches [9,10,40], environmental degradation has magnified various factors that aggravate the intra-species competition of prey or predators, which include increasing population densities, pollution, habitat degradation, urbanization and so on. ...
In this paper, we study a Bazykin’s prey–predator model with piecewise-smooth Holling type I functional response and small predator’s competition rate. By using non-dimensional transformation, the model can be rewritten as a multi-scale system which is a regularly perturbed system for x<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x<1$$\end{document} and a singularly perturbed system for x>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>1$$\end{document}. We are keen on the complex dynamics when the system has a focus in the region x<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x<1$$\end{document}. Using geometric singular perturbation theory, we show that the system has a relaxation oscillation cycle, a homoclinic cycle and a heteroclinic cycle under different parameter conditions, which separately enclose a small-amplitude hyperbolically unstable limit cycle near the focus. Meanwhile, we also prove that the system undergoes saddle-node bifurcation and boundary equilibrium bifurcation. Furthermore, we present some phase portraits with different parameter values by numerical simulation, which support our theoretical analysis. These results reveal far richer and much more complex dynamics compared to the model without different time scales or with smooth Holling type I functional response.
... Both species aggregate in larger numbers during certain life cycle stages and density-dependent effects on breeding success have been observed (Hartman et al. 2013;Mineev et al. 2023). This points to Allee effects that come into play when population density decreases below a certain threshold to be a particular concern (Stephens, Sutherland, and Freckleton 1999;Hutchings 2015). Moreover, if declines continue, Allee effects could interact with genetic factors, such as emerging inbreeding depression or traits that represent adaptations for aggregating in large numbers but become maladaptive at lower population densities, further increasing risk of extinction (Murray et al. 2017). ...
Anthropogenic impact has transitioned from threatening already rare species to causing significant declines in once numerous organisms. Long‐tailed duck (Clangula hyemalis) and velvet scoter (Melanitta fusca) were once important quarry sea duck species in NW Europe, but recent declines resulted in their reclassification as vulnerable on the IUCN Red List. We sequenced and assembled genomes for both species and resequenced 15 individuals of each. Using analyses based on site frequency spectra and sequential Markovian coalescence, we found C. hyemalis to show more historical demographic stability, whereas M. fusca was affected particularly by the Last (Weichselian) Glaciation. This likely reflects C. hyemalis breeding continuously across the Arctic, with cycles of glaciation primarily shifting breeding areas south or north without major population declines, whereas the more restricted southern range of M. fusca would lead to significant range contraction during glaciations. Both species showed evidence of declines over the past thousands of years, potentially reflecting anthropogenic pressures with the recent decline indicating an accelerated process. Analysis of runs of homozygosity (ROH) showed low but nontrivial inbreeding, with FROH from 0.012 to 0.063 in C. hyemalis and ranging from 0 to 0.047 in M. fusca. Lengths of ROH suggested that this was due to ongoing background inbreeding rather than recent declines. Overall, despite demographically important declines, this has not yet led to strong inbreeding and genetic erosion, and the most pressing conservation concern may be the risk of density‐dependent (Allee) effects. We recommend monitoring of inbreeding using ROH analysis as a cost‐efficient method to track future developments to support effective conservation of these species.
... Afterward, we present stability results for each type of stability inspired from the works [6,10], to extend some classical results from [14,16]. In the last section of this paper, we suggest two applications to dynamics of population to study the asymptotic stability of some critical equilibria: in the first application, we model the dynamics of a population with Allee's effect [3,32] negatively impacted by train vibrations, and another application related to the dynamics of a population of Cyanobacteria in a cultured environment, keeping track of ammonia levels in the process. ...
In this article, we introduce Lyapunov-type results to investigate the stability of the trivial solution of a Stieltjes dynamical system. We utilize prolongation results to establish the global existence of the maximal solution. Using Lyapunov's second method, we establish results of (uniform) stability and (uniform) asymptotic stability by employing a Lyapunov function. Additionally, we present examples and real-life applications to study asymptotic stability of equilibria in two population dynamics models.
... where y(t) stands for the density of species at time t; d denotes the constant mortality of the populaton; α 1+βy is a Beverton-Holt-like growth function, the per capita reproduction rate (maximum α) decreases with density, and β is the strength of density-dependence. In addition, the Allee effect, an ecological phenomena, reflects a positive relationship between any component of indi-vidual fitness and either numbers or density of conspecifics [11] and may be caused by difficulties in mate finding, inbreeding depression, demographic stochasticity and predator defense [12,13]. The Allee effect is usually modelled by adding a factor to the logistic function, namely, ...
In this paper, an amensalism system with strong Allee effect and nonlinear growth rate is proposed. We examined the existence and stability of the equilibrium, and subsequently carried out a sensitivity analysis based on statistical techniques to determine the significance of the parameters. The saddle-node bifurcation was explored by considering the Allee effect coefficient as the bifurcation parameter. Along with the qualitative analysis of all possible equilibria and infinite singularities, as well as non-existence of closed orbits, we implemented the global dynamics of the model. Next, a new dynamic phenomenon of noise-induced transition is observed by incorporating noise into the reproduction rate of the deterministic model. The critical noise value that caused the transition was estimated by constructing the confidence ellipse for the equilibrium. Besides, numerical simulations suggest that the dynamics of the stochastic model are also dependent upon the initial condition.
... This phenomenon, driven by the survival instinct of the prey population, plays a crucial role in reproduction, cooperative breeding, reducing the risk of predation, and ensuring a stable food supply within the ecosystem. Scientists refer to this positive relationship between the fitness of prey species and their population density as the Allee effect [37][38][39][40]. The Allee effect can be categorized into two types in the field of ecology. ...
The present article deals with the employment of parameter-dependent center manifold reduction in order to comprehend the structure of codimension one and two bifurcations from two different stand points of hyperbolic and ratio-dependent functional response in the proposed system. At the onset one may explore the number of equilibria of the system including their characteristics by means of the equations of equilibria and the distribution of eigenvalues. The system appears to exhibit a multi-stable structure with refinement of the bifurcation parameters, the stable-unstable manifolds of saddles and limit cycles resulting from the respective Hopf and Bogdanov-Takens bifurcations so as to constitute the attractive domains of various attractors. The complex algebraic treatment generates the Bogdanov-Takens normal form and three different types of bifurcation curves. The structure of codimension one and two bifurcations outlines several phase portraits corresponding to the parameters in proximity to different local bifurcating points. The theoretical outcomes of the present system are finally corroborated by the analytical findings together with its numerical simulations counterpart.
Fear prompts prey to adopt risk-averse behaviors, such as reduced foraging activity, increased vigilance, and avoidance of areas with high predator presence, which affects its reproduction. In a real scenario, a population requires a minimum density to avoid extinction, known as an Allee threshold. In light of these biological factors, we propose a predator–prey model with (i) a fear effect in a prey population, (ii) an Allee effect in a predator population, and (iii) a non-constant attack rate that modifies the functional response. We ensured the non-negativity and boundedness of the solutions and examined the local and global stability status for each existing steady state solutions. We investigated some deep dynamical properties of the system by varying different parameters, such as cost of fear in prey and strength of the Allee effect in predators and their mortality rate. In codimension one bifurcations, we observed saddle node, Hopf, homoclinic, and coalescence of two limit cycles. Additionally, codimension two bifurcations were observed, including Bautin and Bogdanov Takens bifurcations. To provide a clearer understanding of these bifurcations, we conducted biparametric analysis involving the fear and Allee parameters, as well as the fear parameter and predator mortality rate. Our investigation shows that cost of fear and strength of Allee strongly influences the survival status of the predator. Furthermore, bistability and tristability reveal that the survival and extinction of predator are dependent on the initial population level. Numerical simulations and graphical illustrations are provided to support and validate our theoretical findings.
We study the effect of phase-separating diffusive dynamics on the mean time to extinction (MTE) in several reaction-diffusion models with slow reactions. We consider a continuum theory similar to model AB, and a simple model where individual particles on two sites undergo on-site reactions and hopping between the sites. In the slow-reaction limit, we project the models’ dynamics onto suitable one-dimensional reaction coordinates, which allows the derivation of quasi-equilibrium effective free energies. For weak noise, this enables characterisation of the MTE. This time can be enhanced or suppressed by the addition of phase separation, compared with homogeneous reference cases. We discuss how Allee effects can be affected by phase separation, including situations where the tendency to phase-separate renders an otherwise stable population unstable.
The bush cricket, Metrioptera roeseli (Orthoptera, Tettigoniidae), occurs in patchy and heterogeneous agricultural landscapes. Such a mosaic of different types of grassland habitats causes spatial variation in local population density. Low population densities may result in fewer mating opportunities that can give rise to an Allee effect, possibly leading to a population decline or extinction. This experimental study shows that individuals can avoid an Allee effect by adjusting their movement behaviour in sparse populations. Even at a population density of Eve individuals per hectare, i.e. approximately one percent of normal population densities, no reduction of mating frequencies was detected. Observed net displacements made by different individuals in high and low population densities could successfully be predicted by a simple model of animal movement.
Provides information that can aid the interpretation of rates of increase estimated for whale populations that currently are thought to be at small fractions of their historical or natural levels. Specially, the work was conducted to evaluate the use of such estimates to determine a lower bound to the population growth rate of expected for the population level at which maximum production (in numbers) would be expressed. -from Authors
1. In formation on the approximate number of individuals released is available for 47 of the 133 exotic bird species introduced to New Zealand in the late 19th and early 20th centuries. Of these, 21 species had populations surviving in the wild in 1969-79. The long interval between introduction and assessment of outcome provides a rare opportunity to examine the factors correlated with successful establishment without the uncertainty of long-term population persistence associated with studies of short duration. 2. The probability of successful establishment was strongly influenced by the number of individuals released during the main period of introductions. Eight-three per cent of species that had more than 100 individuals released within a 10-year period became established, compared with 21% of species that had less than 100 birds released. The relationship between the probability of establishment and number of birds released was similar to that found in a previous study of introductions of exotic birds to Australia. 3. It was possible to look for a within-family influence on the success of introduction of the number of birds released in nine bird families. A positive influence was found within seven families and no effect in two families. This preponderance of families with a positive effect was statistically significant. 4. A significant effect of body weight on the probability of successful establishment was found, and negative effects of clutch size and latitude of origin. However, the statistical significance of these effects varied according to whether comparison was or was not restricted to within-family variation. After applying the Bonferroni adjustment to significance levels, to allow for the large number of variables and factors being considered, only the effect of the number of birds released was statistically significant. 5. No significant effects on the probability of successful establishment were apparent for the mean date of release, the minimum number of years in which birds were released, the hemisphere of origin (northern or southern) and the size and diversity of latitudinal distribution of the natural geographical range.
(1) A trained male goshawk was used to arrange attacks on single pigeons and flocks, at brassica feeding sites and elsewhere. Captured pigeons were compared with shot samples from the same sites, using dry weight of the pectoralis minor muscle as an index of condition. Attacks on pigeon flocks at brassica sites were more successful than when the birds were feeding on grasslands and stubbles, possibly because of variation in pigeon condition. (2) Attacks were more successful in the hour before sunset than in the four previous hours. Pre-roost crop filling may have made the pigeons more vulnerable in the last hour, and the hawk might have been trying harder to obtain food as dusk approached. (3) Attacks on single pigeons, and on birds in small flocks, were more successful than those on flocks of more than ten pigeons. This occurred partly because single birds were in poor condition and partly because the hawk achieved less surprise as flock size increased. The hawk may also have been (a) less likely to encounter weak pigeons, (b) more confused, and (c) `less confident' in attacks on large flocks than on small ones. (4) Unless the hawk surprised pigeons feeding in flocks it was usually outflown. Pigeons captured from flocks which did not fly until the hawk reached them were in relatively good condition, but selection for poor condition became more marked if the birds took off when the hawk was further away and it had to chase them. (5) The predation was selective, partly because single pigeons tended to be both worse in quality and more vulnerable to attack than birds in flocks, and partly because pigeons captured from flocks were below average in condition when the hawk did not achieve complete surprise. There was selection for diseased and defective pigeons, but not for those of one particular age or sex. Goshawk predation could also select for behaviour which delays crop filling until as late in the day as possible, and for flocking.
The Allee effect - a decrease in population growth rate at low density - may be produced by several mechanisms including difficulty in finding mates at low densities. We present evidence for the Allee effect in natural populations of an endangered butterfly, the Glanville fritillary (Melitaea cinxia) and for two mechanisms that cause it. Specifically, we show that emigration rate increases and the fraction of mated females decreases with decreasing local density. We then demonstrate a connection between mating success and population growth rate and more generally, slightly reduced reproductive success in the smallest butterfly populations in an extensive data set gathered over four years. The Allee effect increases the significance of the rescue effect in metapopulations, and thereby the potential for alternative stable states in metapopulation as well as in local dynamics.
The evolution of sociality in many communally breeding birds has been ascribed to a lack of optimum habitat, that is, insufficient high quality resources to allow dispersal and independent breeding of young. In these species, offspring do not disperse immediately, but instead remain as helpers in their natal territory until a breeding opportunity occurs. We examined the habitat saturation model for a communally breeding South American species, the Guira Cuckoo. This species breeds in large groups composed of both related and unrelated individuals. Nestlings also exhibit varying degrees of relatedness, or they may be completely unrelated. Thirteen ecological variables were measured in 14 sites occupied by groups and compared with data from nine vacant sites that appeared similar. Data concerning breeding traits were collected for each reproductive event. Guira Cuckoos exhibit tactics that reduce the reproductive success of others within the group, mainly egg-tossing behavior and, occasionally, infanticide. Potential nesting sites within territories, in the form of araucaria trees, varied greatly in number, but there was no relation between their availability and the size of the group controlling the area. Group size was significantly positively correlated with grass cover within territories, but unrelated to the other 12 vegetation variables measured. Empty sites had a significantly greater abundance of prey items than did areas occupied by Guira Cuckoos, which may indicate depletion of insects due to the cuckoos' foraging activities. No other significant differences were found between occupied and empty sites. These results suggest that habitat saturation may not be an important factor in the maintenance of group-living in Guira Cuckoos. However, this does not exclude the possibility that habitat quality may have an important effect on reproduction. The basis for group-living in this species is most likely related to intrinsic characteristics of sociality that increase survival and lifetime reproductive fitness.
