Multiple Allee Effects Research Articles

Impact of Allee effects on the establishment of reintroduction populations of endangered species: The case of the Crested Ibis

Allee effects, defined as the positive relationships between component fitness and population density (or size), play an important role in the dynamics of small or lowdensity populations. Reintroduction has become a widely applied tool with the continuous loss of biodiversity. Since reintroduced populations are initially small, Allee effects commonly exist when a species is colonizing new habitat. However, direct evidence of positive density-dependence acting in reintroduced populations is rare. To understand the role of Allee effects in regulating post-release population dynamics of reintroduced species, we analyzed time-series data collected from two spatially isolated populations of the reintroduced Crested Ibis (Nipponia nippon) in Shaanxi Province, China (Ningshan and Qianyang Counties). We examined the potential relationships between population size and (1) survival and reproductive rates, (2) per capita population growth rates for the existence of Allee effects in the reintroduced ibis populations. The results showed that the simultaneous occurrence of component Allee effects in survival and reproduction have been detected, while the reduction of adult survival and per female breeding probability led to a demographic Allee effect in the Qianyang ibis population, which may have contributed to the population decline. In parallel, mate-limitation and predation as the possible initiation mechanisms of Allee effects were presented. Our findings provided the evidence of multiple Allee effects in reintroduced populations and conservation management strategies to eliminate or reduce the strength of Allee effects in future reintroductions of endangered species were proposed, including release of large number of individuals, food supplementation, and predator control.

ROLE OF ALLEE EFFECT AND HARVESTING OF A FOOD-WEB SYSTEM IN THE PRESENCE OF SCAVENGERS

The role of scavengers, which consume the carcasses of predators along with predation of the prey, has been ignored in comparisons to herbivores and predators. It has now become a topic of high interest among researchers working with food-web systems of prey–predator interactions. The food-web considered in these works contains prey, predators, and scavengers as the third species. In this work, we attempt to study a food-web model of these species in the presence of the multiplicative Allee effect and harvesting. It is observed that this makes the model more complex in the form of multiple co-existing steady states. The conditions for the existence and local stability of all possible steady states of the proposed system are analyzed. The global stability of the steady state lying on the x-axis and the interior steady state have been discussed by choosing suitable Lyapunov functions. The existence conditions for saddle-node and Hopf bifurcations are derived analytically. The stability of Hopf bifurcating periodic solutions with respect to both Allee and harvesting constants is examined. It is also observed that multiple Hopf bifurcation thresholds occur for harvesting parameters in the case of two co-existing steady states, which indicates that the system may regain its stability. The proposed model is also studied beyond Hopf bifurcation thresholds, where we have observed that the model is capable of exhibiting period-doubling routes to chaos, which can be controlled by a suitable choice of Allee and harvesting parameters. The largest Lyapunov exponents and sensitivity to initial conditions are examined to ensure the chaotic nature of the system.

Qualitative analysis of a diffusive predator–prey model with Allee and fear effects

In this study, we consider a diffusive predator–prey model with multiple Allee effects induced by fear factors. We investigate the existence, boundedness and permanence of the solution of the system. We also discuss the existence and non-existence of non-constant solutions. We derive sufficient conditions for spatially homogeneous (non-homogenous) Hopf bifurcation and steady state bifurcation. Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.

The influence of density in population dynamics with strong and weak Allee effect

In this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

Dynamics of Leslie–Gower model with double Allee effect on prey and mutual interference among predators

In this paper, a Leslie–Gower-type model is proposed to study a diffusive predator–prey system with multiple Allee effects acting on the growth rate of the prey population. Crowley–Martin-type functional response is considered to describe mutual interference among predators. Preliminary analysis includes uniqueness, dissipation and persistence property of the proposed model system. Detailed equilibrium analysis suggests that proposed system has rich dynamics in the sense that it can have four to eight equilibrium points. Multistability behavior is observed which is an interesting phenomenon. Parametric conditions are established for stability and Hopf bifurcation in both the local and diffusive systems. We investigate the emergence of complex patterns through Turing instability phenomenon of coupled reaction–diffusion system. A variety of patterns is obtained such as flower, square, circle and rhombus. Comparative analysis of patterns between weak and strong Allee effect is the novelty of this study. It is found that prey population uses its agility in aggregation. Bond between prey populations becomes stronger when the system experiences strong Allee effect which forces the predator population to remain united despite mutual interference. A highly significant result has been found that for slow prey population, predators decide the hunting strategy in a short time, while it takes more time for the predators to form a strategy to encircle the more dynamic prey.

Analysis on a diffusive multiple Allee effects predator–prey model induced by fear factors

This paper is concerned with a diffusive predator–prey system with multiple Allee effects induced by fear factors subject to Neumann boundary conditions. Firstly, we introduce a priori estimates of non-negative and non-trivial solutions. Moreover, nonexistence of non-constant positive solutions are shown for certain parameters ranges. Secondly, we study the stability of non-negative constant solutions by the linearized theory. Then, by choosing the Allee threshold θ as the bifurcation parameter, we investigate the spatially homogeneous and non-homogeneous periodic solutions as well as non-constant steady-state solutions. Finally, some theoretical results are illustrated by numerical simulations.

A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

We study a predator-prey model with Holling type Ⅰ functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.

Investigating the interactions between Allee effect and harvesting behaviour of a single species model: An evolutionary dynamics approach

In this study, we propose an ecological model, which is based on the Smith growth model and subject to multiplicative Allee effect. We then investigate the equilibrium bifurcation induced by the harvesting behaviour and the Allee effect, and pitch fork bifurcation and saddle node bifurcation may occur. Assuming that the carrying capacity and harvesting rate obey a Gaussian distribution, we then consider the evolutionary dynamics by using continuous strategy evolutionary game theory. Our study shows that strong Allee effect and low harvesting rate can induce bistability. It also suggests that evolution prevents extinction even if population has strong Allee effect. Numerical analysis shows that the evolutionary model can admit up to nine equilibrium points including up to four Evolutionary Stable Strategy equilibria.

Population dynamics with multiple Allee effects induced by fear factors – A mathematical study on prey-predator interactions

• Mathematical models on effect of fear and strong Allee mechanism have been proposed and analysed. • We have shown how fear can significantly reduce the per-capita growth rate of prey with generalised predator. • We have shown that fear has no role in the stability of prey-predator system, with linear functional response. • Fear effect can stabilize the eco-epidemiological system and promote the coexistence of all the three populations. • Finally, fear may be a cause of the Allee effect/multiple Allee effects at low population density. In the present, ecologists generally consider the interactions, which are directly related to the density effects, that species have on each other, like predation, mutualism, refuge, etc. However, some experimental studies showed that apart from the direct killing, predation fears itself can reduce the prey growth rate by 40%. Therefore, in the present study, we have considered a trait effect, which is characterized by the reduction of prey growth rate due to fear of predator, where the prey is already suffered by the mating induced strong Allee effects, in the reproduction process. First, we developed and analyzed the single-species model and showed that how the fear effect can significantly reduce the per-capita growth rate (pgr) and may be a possible cause of the multiple Allee effects at low population density. Next, we consider a prey-predator model with linear functional response and showed that fear does not affect the equilibrium stability, but the time scale difference among two populations has a positive effect. It will help the system to converge to the stable steady states faster than before. Finally, we study an eco-epidemiological model with the same assumptions, where prey is being suffered by the disease. We showed that in eco-epidemiological systems fear can greatly affect the system stability. Fear can stabilize the system at the interior equilibrium, where all the three population coexists, or it can create the oscillatory coexistence of all the three populations. In the presence of fear, system shows bi-stability among different equilibria. We have also shown the basin stability at multiple stable parameter regions, which yields the probability of convergence of each equilibrium for a given set of different initial conditions. All these findings may have potential impacts in population management and conservation biology.

A Michaelis–Menten Predator–Prey Model with Strong Allee Effect and Disease in Prey Incorporating Prey Refuge

Here, we have proposed a predator–prey model with Michaelis–Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh–Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson–Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.

Dynamics and pattern formation in a modified Leslie–Gower model with Allee effect and Bazykin functional response

In this paper, we study the dynamics of a diffusive modified Leslie–Gower model with the multiplicative Allee effect and Bazykin functional response. We give detailed study on the stability of equilibria. Non-existence of non-constant positive steady state solutions are shown to identify the rage of parameters of spatial pattern formation. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.

Interaction between Allee effects caused by organism-environment feedback and by other ecological mechanisms.

Understanding Allee effect has crucial importance for ecological conservation and management because it is strongly related to population extinction. Due to various ecological mechanisms accounting for Allee effect, it is necessary to study the influence of multiple Allee effects on the dynamics and persistence of population. We here focus on organism-environment feedback which can incur strong, weak, and fatal Allee effect (AE-by-OEF), and further examine their interaction with the Allee effects caused by other ecological mechanisms (AE-by-OM). The results show that multiple Allee effects largely increase the extinction risk of population either due to the enlargement of Allee threshold or the change of inherent characteristic of Allee effect, and such an increase will be enhanced dramatically with increasing the strength of individual Allee effects. Our simulations explicitly considering spatial structure also demonstrate that local interaction among habitat patches can greatly mitigate such superimposed Allee effects as well as individual Allee effect. This implies that spatially structurized habitat could play an important role in ecological conservation and management.

CONSEQUENCES OF MULTIPLE ALLEE EFFECT IN AN OPEN ACCESS FISHERY MODEL

This work deals with the dynamics of a bioeconomic continuous time model, where the combined action of the fishing effort exerted by men (as a predator) and multiple Allee effect or depensation on the growth rate of a self-regenerating resource (the prey) are considered. It has been recently established that a depensation phenomenon appears by diverse causes and new functions have been proposed to describe multiple Allee effects. One of these formalizations is here incorporated in the well-known Smith's model, one of the simplest models to open access fisheries. We prove that this new and complex expression is topologically equivalent to a simpler form. Then, we postulate that the parsimony principle must be used to describe this phenomenon. It is also shown that in the phase plane of biomass-effort on the proposed model, the origin is an attractor equilibrium for all parameters values as a consequence of the Allee effect. Moreover, there is a subset of the parameter values, for which two limit cycles exist surrounding the unique positive equilibrium point of the system, one of them being asymptotically stable (the non damped oscillatory tragedy of the commons); hence, multiestability exists, particularly three-stability.

A general class of predation models with multiplicative Allee effect

A class of models of predator–prey interaction with Allee effect on the prey population is presented. Both the Allee effect and the functional response are modelled in the most simple way by means of general terms whose conveniently chosen mathematical properties agree with, and generalise, a number of concrete Leslie–Gower-type models. We show that this class of models is well posed in the sense that any realistic solution is bounded and remains non-negative. By means of topological equivalences and desingularization techniques, we find specific conditions such that there may be extinction of both species. In particular, the local basin boundaries of the origin are found explicitly, which enables one to determine the extinction or survival of species for any given initial condition near this equilibrium point. Furthermore, we give conditions such that an equilibrium point corresponding to a positive steady state may undergo saddle-node, Hopf and Bogdanov–Takens bifurcations. As a consequence, we are able to describe the dynamics governed by the bifurcated limit cycles and homoclinic orbits by means of carefully sketched bifurcation diagrams and suitable illustrations of the relevant invariant manifolds involved in the overall organisation of the phase plane. Finally, these findings are applied to concrete model vector fields; in each case, the particular relevant functions that define the conditions for the associated bifurcations are calculated explicitly.

Bio-economics of a renewable resource subjected to strong Allee effect

In this article bio-economics of a renewable resource that is subjected to strong Allee effect (multiplicative Allee effect) is investigated from sole owner perspective. The considered optimal harvesting problem has been solved using Pontryagin maximum principle. The control problem admits multiple singular equilibrium solutions in contrast to the case where the growth of the resource is of compensatory nature. Thus the choice of optimal singular solution and the nature of associated approach paths make the problem pertinent and interesting.

Optimal Management of a Renewable Resource Subjected to Strong Allee effect in a Periodically Fluctuating Environment

In this article we present the qualitative properties of solutions of a periodic differential equation that describes dynamics of a renewable resource subjected to strong Allee effect (multiplicative Allee effect) with periodic harvesting in a periodically fluctuating environment. We also present the bio-economic aspects of the considered renewable resource studied from sole owner perspective. An optimal harvesting problem with periodic dynamic constraints has been solved using Pontryagin’s maximum principle. It is interesting to find that the considered optimal control problem admits multiple periodic singular solutions. Techniques are developed to identify the optimal singular solution and the associated optimal singular periodic effort function. Optimal, suboptimal harvest policies and their corresponding approach paths are constructed. It is found that the optimal harvest policies are periodic after some time.

Consequences of double Allee effect on the number of limit cycles in a predator–prey model

The main goal of this work is to show a comparative analysis of simple continuous time predator–prey models considering the Allee effect affecting the prey population, also known as depensation in fisheries sciences. This phenomenon may be expressed by different mathematical forms, yielding a distinct number of limit cycles surrounding a positive equilibrium point, when two of these different formalizations are considered in the same system. It is known that the Volterra predation model, using the most usual form to express the Allee effect, has a unique limit cycle. In this work, considering a more complex mathematical expression, the existence of two limit cycles is proved, by means of the Lyapunov quantities. We argue that the second equation explains the existence of two Allee effects affecting the same population, which could justify the difference observed between the models. These results imply that the election of mathematical formulation can have consequences on the fit of the observed data, thus leading to mistakes for ecologists. We conclude that the oscillatory behaviors and overall dynamics depend strongly on the algebraic expression of the Allee effect, making difficult the proposition of general results. Nevertheless, the techniques reviewed in this paper emerge as key tools to analyze the existence of limit cycles in the presence of multiple Allee effects.

Dynamical analysis of fractional-order modified logistic model

In this paper, we study a fractional differential equation model of the single species multiplicative Allee effect. First we study the stability of equilibrium points. Further we give some sufficient conditions ensuring the existence and uniqueness of integral solution. In the last section we perform several numerical simulations to validate our analytical findings.

Multiple Allee effects and population management

Allee effects, strongly related to the extinction vulnerability of populations and gradually becoming acknowledged by both theoretically oriented and applied ecologists, have already been shown to have important roles in the dynamics of many populations. Although not yet widely recognized, two or more Allee effects can occur simultaneously in the same population. Here, we review the evidence for multiple Allee effects and show that their interactions can take several forms, many of which are far from inconsequential. We suggest that more research is needed to assess the prevalence and interactions of multiple Allee effects, as failing to take them into account could have adverse consequences for the management of threatened or exploited populations.