Multiple attractors in a Leslie–Gower competition system with Allee effects

We investigate asymptotic dynamics of the classical Leslie–Gower competition model when both competing populations are subject to Allee effects. The system may possess four interior steady states. It is proved that for certain parameter regimes both competing populations may either go extinct, coexist or one population drives the other population to extinction depending on initial conditions.     

Weak Allee effects and species coexistence

In this article, we study the population dynamics of a two-species discrete-time competition model where each species suffers from either predator saturation induced Allee effects and/or mate limitation induced Allee effects. We focus on the following two possible outcomes of the competition: 1. one species goes to extinction; 2. the system is permanent. Our results indicate that, even if one species’ intra-specific competition is less than its inter-specific competition, weak Allee effects induced by predation saturation can promote coexistence of the two competing species. This is supported by the outcome of two-species competition models without Allee effects. Also, we discuss our results and future work on multiple attractors in competition models with Allee effects.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

Competitive exclusion and coexistence in a Leslie–Gower competition model with Allee effects

The classical competitive exclusion principle states that two populations competing for a limited resource cannot coexist, one of the populations will drive the other to extinction. We prove in this work that when one population is subject to Allee effects, then for certain parameter regimes both competing populations may either coexist or one population may drive the other to extinction depending on initial conditions.     

Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey

In this work, a bidimensional differential equation system obtained by modifying the well-known predator–prey Rosenzweig–MacArthur model is analyzed by considering prey growth influenced by the Allee effect.One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence.The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predator–prey model, thereby increasing the risk of ecological extinction.     

Optimal control of swinging alliances in a parabolic competition model

A system of parabolic partial differential equations describes the interaction of three populations, modeling a dynamic competition/cooperation scenario. More precisely, two populations are always competing with each other, but the third population can switch the mode of alliance with the other two populations between cooperation and competition. The control is a function measuring the strength and nature of the alliance and the goal is to maximize the population with the swinging alliance while keeping the other two populations close to each other and minimizing the cost of the alliance action. Various scenarios are illustrated with numerical results.     

Modeling herd behavior in population systems

In this paper, we show that under suitable simple assumptions the classical two populations system may exhibit unexpected behaviors. Considering a more elaborated social model, in which the individuals of one population gather together in herds, while the other one shows a more individualistic behavior, we model the fact that interactions among the two occur mainly through the perimeter of the herd. We account for all types of populations’ interactions, symbiosis, competition and the predator–prey interactions. There is a situation in which competitive exclusion does not hold: the socialized herd behavior prevents the competing individualistic population from becoming extinct. For the predator–prey case, sustained limit cycles are possible, the existence of Hopf bifurcations representing a distinctive feature of this model compared with other classical predator–prey models. The system’s behavior is fully captured by just one suitably introduced new threshold parameter, defined in terms of the original model parameters.     

A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.     

Discrete-time host–parasitoid models with Allee effects: Density dependence versus parasitism

We present two general discrete-time host–parasitoid models with Allee effects on the host. In the first model, it is assumed that parasitism occurs prior to density dependence, while in the second model we assume that density dependence operates first followed by parasitism. It is shown that both models have similar asymptotic behaviour. The parasitoid population will definitely go extinct if the maximal growth rate of the host population is less than or equal to one, independent of whether density dependence or parasitism occurs first. The fate of the population is initial condition dependent if this maximal growth rate exceeds one. In particular, there exists a host population threshold, the Allee threshold, below which the host population goes extinct and so does the parasitoid. This threshold is the same for both models. Numerical examples with different functions are simulated to illustrate our analytical results.     

ALLEE EFFECTS: POPULATION GROWTH,CRITICAL DENSITY,AND THE CHANCE OF EXTINCTION

This paper develops mathematical models to describe the growth, critical density, and extinction probability in sparse populations experiencing Allee effects. An Allee effect (or depensation) is a situation at low population densities where the per-individual growth rate is an increasing function of population density. A potentially important mechanism causing Allee effects is a shortage of mating encounters in sparse populations. Stochastic models are proposed for predicting the probability of encounter or the frequency of encounter as a function of population density. A negative exponential function is derived as such an encounter function under very general biological assumptions, including random, regular, or aggregated spatial patterns. A rectangular hyperbola function, heretofore used in ecology as the functional response of predator feeding rate to prey density, arises from the negative exponential function when encounter probabilities are assumed heterogeneous among individuals. These encounter functions produce Allee effects when incorporated into population growth models as birth rates. Three types of population models with encounter-limited birth rates are compared: (1) deterministic differential equations, (2) stochastic discrete birth-death processes, and (3) stochastic continuous diffusion processes. The phenomenon of a critical density, a major consequence of Allee effects, manifests itself differently in the different types of models. The critical density is a lower unstable equilibrium in the deterministic differential equation models. For the stochastic discrete birth-death processes considered here, the critical density is an inflection point in the probability of extinction plotted as a function of initial population density. In the continuous diffusion processes, the critical density becomes a local minimum (antimode) in the stationary probability distribution for population density. For both types of stochastic models, a critical density appears as an inflection point in the probability of attaining a small population density (extinction) before attaining a large one. Multiplicative (“environmental”) stochastic noise amplifies Allee effects. Harvesting also amplifies those effects. Though Allee effects are difficult to detect or measure in natural populations, their presence would seriously impact exploitation, management, and preservation of biological resources.     

Weak Allee effect in a predator–prey model involving memory with a hump

In this paper we consider a predator–prey system which has a factor that allows for a reduction in fitness due to declining population sizes, often termed an Allee effect. We study the influence of the weak Allee effect which is included in the prey equation and we determine conditions for the occurrence of Hopf bifurcation. The prey population is limited by the carrying capacity of the environment, and the predator growth rate depends on past quantities of the prey which is represented by a weight function that specifies a moment in the past when the quantity of food is the most important from the point of view of the present growth of the predator. The stability properties of the system and the biological issues of the memory and Allee effect on the coexistence of the two species are studied. Finally we present some simulations to verify the veracity of the analytical conclusions.     

具有HollingⅡ型功能反应和Allee效应的捕食系统模型

利用计算机模拟方法研究一类离散种群相互作用模型的动态复杂性.通过理论推导建立食饵具有Allee效应和HollingⅡ型功能反应的自治捕食系统模型,用Matlab软件模拟离散种群的生长状态,探索研究参数的变化对种群大小的影响,阐释Allee效应及HollingⅡ型功能反应在种群间相互作用模型中的重要性.研究结果表明:1)当处理时间处于有效区间内时,处理时间越大种群的稳定共存参数域越大;2)Allee效应的引入使种群的动态行为更为复杂,从而增加了捕食者种群的灭绝风险;3)系统受强Allee效应的影响,种群会出现提前分叉现象,如果继续增加Allee效应就会导致种群灭绝;4)强Allee效应更容易使种群趋向灭绝.所得结论在丰富生态学理论的同时,提出了保护生态学的重要依据.     

Asymptotic behavior of reaction–advection–diffusion population models with Allee effect

In this work, a qualitative analysis is carried out for reaction–advection–diffusion (RAD) systems modeling the interactions between two species with Allee effect. In particular, we study different scenarios: mutualism, competition, and a predator–prey relationship in order to investigate the survival or extinction of both populations. Global existence and uniqueness of positive solutions of the proposed RAD problems are demonstrated. Equilibrium states and asymptotic behavior of solutions are obtained using the monotone method and the upper and lower solutions technique. Numerical simulations by a Crank–Nicolson monotone iterative method of the different asymptotic solution dynamics are shown to illustrate our theoretical results.     

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.     

Stabilization in a 3D eco-epidemiological model: From the complete extinction of a predator population to their self-healing

In this paper, using the localization method of compact invariant sets, we examine the ultimate dynamics of the 3D prey–predator model containing two subpopulations of susceptible and infected predators. Our attention is focused to finding ultimate sizes of interacting populations, and, in addition, we show the existence of a global attracting set. Then, we derive various global conditions of ultimate extinction of at least one of the predators subpopulations and describe conditions under which all types of internal bounded dynamics are ruled out. In particular, we describe convergence conditions to omega-limit sets located (1) in the intersection of the prey-free plane with the infected predators-free plane and (2) in the infected predators-free plane. Based on the dynamical analysis of the 2D infection-free subsystem, we obtain conditions of global attraction to (i) the prey-only disease-free equilibrium point, (ii) the disease-free prey-predator equilibrium point (self-healing of the predator population), and (iii) the omega-limit set containing an equilibrium point or a periodic orbit. Main theoretical results are illustrated by numerical simulation. Tools and techniques developed in this work can be appropriated in the studies within predictive population ecology of more complex eco-epidemiological models.     

Necessary optimality conditions for three species reaction–diffusion system

An optimal control problem is studied for an ecosystem composed by one predator and two prey populations. Its dynamics is modelled by a reaction–diffusion system of Volterra type. Two control variables are introduced in the system; their meaning is the mixture rates between predator and each prey population. The goal of this paper is to maximize the total density of the three populations at a fixed time moment. The existence of the optimal control is established and necessary optimality conditions are found with the aid of a maximum principle.     

Coexistence in seasonally varying predator–prey systems with Allee effect

A general seasonally-varying predator–prey model with Allee effect in the prey growth is investigated. The analysis is performed only on the basis of some properties determining the shape of the prey growth rate and the functional responses. General conditions for coexistence are determined, both in the case of weak and strong Allee effect. Finally, a modified Leslie–Gower predator–prey model with Allee effect is investigated. Numerical results illustrate the qualitative behaviors of the system, in particular the presence of periodic orbits.     

Dynamics of a single species evolutionary model with Allee effects

We investigate the evolutionary outcomes of a single species population subject to Allee effects within the framework of a continuous strategy evolutionary game theory (EGT) model. Our model assumes a single trait creates a phenotypic trade-off between carrying capacity (i.e., competition) and predator evasion ability following a Gaussian distribution. This assumption contributes to one of our interesting findings that evolution prevents extinction even when population exhibits strong Allee effects. However, the extinction equilibrium can be an ESS under some special distributions of anti-predation phenotypes. The ratio of variation in competition and anti-predation phenotypes plays an important role in determining global dynamics of our EGT model: (a) evolution may suppress strong Allee effects for large values of this ratio; (b) evolution may preserve strong Allee effects for small values of this ratio by generating a low density evolutionary stable strategy (ESS) equilibrium which can serve as a potential Allee threshold; and (c) intermediate values of this ratio can result in multiple ESS equilibria.     

Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.