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Population ecology theory is replete with density-dependent processes. However, trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems, where the non-consumptive behavioral fear effect and the Allee effect, a density-dependent process, are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter. Interestingly, we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear. There are three parametric regimes of interest in the fear parameter. For small and intermediate amounts of fear, the Allee + fear effect opposes dynamics driven by the fear effect. However, for large amounts of fear the Allee + fear effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end, the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity. [ABSTRACT FROM AUTHOR]

BACKGROUND: Controlling the spread of arboviral diseases remains a considerable challenge due to the rapid development of insecticide resistance in Aedes mosquitoes. This study evaluated the effects of boric acid‐containing toxic sugar bait (TSB) on field populations of resistant Aedes aegypti mosquitoes. In addition, this study examined the flight activity and wing beat frequency and amplitude of males and the flight activity, fecundity, and insemination of females after pairing with males exposed to TSB. The population dynamics of Aedes mosquitoes under imbalanced sex ratios were examined to simulate realistic field conditions for male suppression under the effect of TSB. RESULTS: The mortality of male mosquitoes was consistently high within 24 h after exposure. By contrast, the mortality of female mosquitoes was inconsistent, with over 70% mortality observed at 168 h. The flight activity and wing beat amplitude of treated males were significantly lower than those of controls, but no significant difference in wing beat frequency was detected. The fecundity and insemination of treated female mosquitoes were lower than those of controls. A simulation study indicated that considerably low male population densities led to mating failures, triggering a mate‐finding Allee effect and resulting in persistently low population levels. CONCLUSION: Boric acid‐containing TSB could effectively complement current chemical intervention approaches to control resistant mosquito populations. TSB is effective in reducing field male populations and impairing male flight activity and female‐seeking behavior, resulting in decreased fecundity and insemination. Male suppression due to TSB potentially results in a small mosquito population. © 2024 Society of Chemical Industry. [ABSTRACT FROM AUTHOR]

This study aims to explore the complexity of a discrete-time predator–prey system with a weak Allee effect. The existence and stability of fixed points, as well as period-doubling and Neimark–Sacker bifurcations, are all investigated. The system’s bifurcating and fluctuating behavior is controlled using feedback and hybrid control techniques. Additionally, numerical simulations are performed as evidence to support theoretical results. From an ecological perspective, these findings suggest that the Allee effect plays a pivotal role in shaping predator–prey dynamics. The moderate Allee effect fosters stability in both predator and prey populations, promoting coexistence and persistence within ecosystems. However, the disproportionate impact on predator populations underscores predators’ vulnerability to changes in prey behavior and availability, highlighting the importance of considering indirect effects in ecological modeling and conservation efforts. [ABSTRACT FROM AUTHOR]

In this paper, a single-species logistic model with both fear effect-type feedback control and additive Allee effect is developed and investigated using the new coronavirus as a feedback control variable. When the system introduces additive Allee effect and fear effect-type feedback control, more complicated dynamical behavior is obtained. The system can undergo transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation. By numerical simulations, the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as parameters are altered. Remarkably, it is the first time that two limit cycles have been discovered in a single-species logistic model with the Allee effect. Further, stronger Allee effect or stronger fear effect can lead to the extinction of the species population. [ABSTRACT FROM AUTHOR]

In long‐distance dispersal events, colonising species typically begin with a small number of founding individuals. A growing body of research suggests that establishment success of small founding populations can be determined by the context of the colonisation event and the new environment. Here, we illuminate the importance of these sources of context dependence. Using a spatially explicit, temporally dynamic, mechanistic, individual‐based simulator of a model amphibian species, the cane toad (Rhinella marina), we simulated colonisation scenarios to investigate how (1) the number of founding individuals, (2) the number of dispersal events, (3) landscape's spatial composition and configuration of habitats ('spatially heterogeneous landscapes') and (4) the timing of arrival with regards to dynamic environmental conditions ('dynamic environmental conditions') influence the establishment success of small founding populations. We analysed the dynamic effects of these predictors on establishment success using running‐window logistic regression models. We showed establishment success increases with the number of founding individuals, whereas the number of dispersal events had a weak effect. At ≥ 20 founding individuals, propagule size swamps the effects of other factors, to whereby establishment success is near‐certain (≥ 90%). But below this level, confidence in establishment success dramatically decreases as number of founding individuals decreases. At low numbers of founding individuals, the prominent predictors are landscape spatial heterogeneity and dynamic environmental conditions. For instance, compared to the annual mean, founding populations with ≤ 5 individuals have up to 18% higher establishment success when they arrive in 'packed' landscapes with relatively limited and clustered essential habitats and right before the breeding season. Accounting for landscape spatial heterogeneity and dynamic environmental conditions is integral in understanding and predicting population establishment and species colonisation. This additional complexity is necessary for advancing biogeographical theory and its application, such as in guiding species reintroduction efforts and invasive alien species management. [ABSTRACT FROM AUTHOR]

We develop a spatially-explicit discrete-time model that describes tick population dynamics and incorporates the developmental stages for an individual tick. The model allows for two types of movements across patches: (1) active tick movement and (2) movement of ticks through a host. We analyse the dynamics of the model under a nonlinear fecundity term that exhibits an Allee effect. We show that the system has bounded solutions and, under certain conditions on the parameters, is monotone. Using the next generation matrix approach, we define the net reproductive number of the multi-patch model, $ R(0) $ R (0) , and establish conditions under which $ R(0) $ R (0) determines the local and global asymptotic stability of the extinction equilibrium. We then show that the model exhibits rich dynamics including a monostable equilibrium, multi-stable equilibria, and cycles. Finally, we perform some numerical studies to further examine how dispersal and Allee effects impact tick survival and system stability. [ABSTRACT FROM AUTHOR]

In this paper, the predator–prey model with strong Allee and fear effects is considered. The existence of the equilibria and their stability are established. Especially it is found that there is an interesting degenerate point, which is a cusp point with codimension 2 or higher codimension, or an attracting (repelling)-type saddle-node, subject to different conditions. Then the Hopf bifurcation and its direction, the saddle-node bifurcation and the Bogdanov–Tankens bifurcation are further explored. Afterwards, with the help of the energy estimates and the Leray–Schauder degree, the nonexistence and existence of the nonconstant steady states of the model are presented. From the obtained results, we find that strong Allee effect will cause the per capita growth rate of prey species from negative to positive; both the fear and Allee effects could affect the existence of equilibria and bifurcations; meanwhile, the diffusion rates will affect the existence of the nonconstant steady states. [ABSTRACT FROM AUTHOR]

We introduce a diffusive predator-prey system with the double Allee effect, focusing on the stability and sufficient conditions for the coexistence of prey and predator. Subsequently, we derived the amplitude equation and explore secondary-order dynamic properties using methods such as Taylor series expansion and multiscaling. The novel approach outlined above provides a precise means to thoroughly analyze the predator-prey model. Through this analysis, we demonstrated that the inclusion of the Allee effect and diffusion leads to the system exhibiting more intricate dynamic behaviors compared to systems lacking these factors. On one hand, in the diffusive system without the Allee effect, the pattern formation regarding the distribution of species was relatively scattered, whereas in the diffusive system with the Allee effect, it is more intensive. On the other hand, the system with the Allee effect transitioned from unstable to stable when the diffusion parameter in prey increased, and the aggregation degree of pattern formation in the system with the Allee effect was higher than in the system without it. These findings highlight the significant roles played by the Allee effect and diffusion in determining the dynamic behaviors of prey and predator within the system. [ABSTRACT FROM AUTHOR]

This work investigates the complex dynamics of a discrete-time predator–prey system with a nonlinear Allee effect. We obtain the discrete system using the piecewise constant argument method. The piecewise argument approach produces a more dynamically consistent discrete system than other numerical techniques for discretization. We investigate the presence and stability of fixed points. Furthermore, we have demonstrated that the system undergoes Neimark–Sacker bifurcation at the positive fixed point by utilizing the Allee effect constant as the bifurcation parameter. To reduce bifurcation and chaos, we use feedback and hybrid control strategies. Our numerical simulations demonstrate the importance of the Allee effect in determining the system's behavior. The findings indicate that an adequate Allee effect might improve social connections and cooperation across populations. However, a significant Allee effect on prey can destabilize the positive fixed point, thus resulting in the extinction of predator and prey populations. [ABSTRACT FROM AUTHOR]

Direct observation of tumor-immune interactions is unlikely in tumors with currently available technology, but computational simulations based on clinical data can provide insight to test hypotheses. It is hypothesized that patterns of collagen evolve as a mechanism of immune escape, but the exact nature of immune-collagen interactions is poorly understood. Spatial data quantifying collagen fiber alignment in squamous cell carcinomas indicates that late-stage disease is associated with highly aligned fibers. Our computational modeling framework discriminates between two hypotheses: immune cell migration that moves (1) parallel or (2) perpendicular to collagen fiber orientation. The modeling recapitulates immune-extracellular matrix interactions where collagen patterns provide immune protection, leading to an emergent inverse relationship between disease stage and immune coverage. Here, computational modeling provides important mechanistic insights by defining a kernel cell-cell interaction function that considers a spectrum of local (cell-scale) to global (tumor-scale) spatial interactions. Short-range interaction kernels provide a mechanism for tumor cell survival under conditions with strong Allee effects, while asymmetric tumor-immune interaction kernels lead to poor immune response. Thus, the length scale of tumor-immune interaction kernels drives tumor growth and infiltration. [ABSTRACT FROM AUTHOR]

Infection duration affects individual host fitness and between-host transmission. Whether an infection is cleared or becomes chronic depends on the complex interaction between host immune responses and parasite growth. Empirical and theoretical studies have suggested that there are critical thresholds of parasite dose that can determine clearance versus chronicity, driven by the ability of the parasite to manipulate host immunity. However, the mammalian immune response is characterized by strong positive and negative feedback loops that could generate duration thresholds even in the absence of direct immunomodulation. Here, we derive and analyse a simple model for the interaction between T-cell subpopulations and parasite growth. We show that whether an infection is cleared or not is very sensitive to the initial immune state, parasite dose and strength of immunological feedbacks. In particular, chronic infections are possible even when parasites provoke a strong and effective immune response and lack any ability to immunomodulate. Our findings indicate that the initial immune state, which often goes unmeasured in empirical studies, is a critical determinant of infection duration. This work also has implications for epidemiological models, as it implies that infection duration will be highly variable among individuals, and dependent on each individual's infection history. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a predator–prey model with Allee effect and double time delays. This research examines the dynamics of the model, with a focus on positivity, existence, stability and Hopf bifurcations. The stability of the periodic solution and the direction of the Hopf bifurcation are elucidated by applying the normal form theory and the center manifold theorem. To validate the correctness of the theoretical analysis, numerical simulations were conducted. The results suggest that a weak Allee effect delay can promote stability within the model, transitioning it from instability to stability. Nevertheless, the competition delay induces periodic oscillations and chaotic dynamics, ultimately resulting in the population's collapse. [ABSTRACT FROM AUTHOR]

In this paper, a two-patch model with additive Allee effect, nonlinear dispersal and commensalism is proposed and studied. The stability of equilibria and the existence of saddle-node bifurcation, transcritical bifurcation are discussed. Through qualitative analysis of the model, we know that the persistence and the extinction of population are influenced by the Allee effect, dispersal and commensalism. Combining with numerical simulation, the study shows that the total population density will increase when the Allee effect constant a increases or m decreases. In addition to suppress the Allee effect, nonlinear dispersal and commensalism are crucial to the survival of the species in the two patches. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Incorporating a weak Allee effect in a two-dimensional biological model in ℜ 2 , the study delves into the application of bifurcation theory, including center manifold and Ljapunov–Schmidt reduction, normal form theory, and universal unfolding, to analyze nonlinear stability issues across various engineering domains. The focus lies on the qualitative dynamics of a discrete-time system describing the interaction between prey and predator. Unlike its continuous counterpart, the discrete-time model exhibits heightened chaotic behavior. By exploring a biological Mmdel with linear functional prey response, the research elucidates the local asymptotic properties of equilibria. Additionally, employing bifurcation theory and the center manifold theorem, the analysis reveals that, for all α 1 (i.e., intrinsic growth rate of prey), ð 1 ˙ (i.e., parameter that scales the terms y n), and m (i.e., Allee effect constant), the model exhibits boundary fixed points A 1 and A 2 , along with the unique positive fixed point A ∗ , given that the all parameters are positive. Additionally, stability theory is employed to explore the local dynamic characteristics, along with topological classifications, for the fixed points A 1 , A 2 , and A ∗ , considering the impact of the weak Allee effect on prey dynamics. A flip bifurcation is identified for the boundary fixed point A 2 , and a Neimark–Sacker bifurcation is observed in a small parameter neighborhood around the unique positive fixed point A ∗ = (m ð 1 ˙ − 1 , α 1 − 1 − α 1 m ð 1 ˙ − 1). Furthermore, it implements two chaos control strategies, namely, state feedback and a hybrid approach. The effectiveness of these methods is demonstrated through numerical simulations, providing concrete illustrations of the theoretical findings. The model incorporates essential elements of population dynamics, considering interactions such as predation, competition, and environmental factors, along with a weak Allee effect influencing the prey population. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish new Maximum Principles for parabolic equations in the framework of mixed local/nonlocal operators. In particular, these results apply to the case of mixed local/nonlocal Neumann boundary conditions, as introduced by Dipierro, Proietti Lippi, and Valdinoci [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), pp. 1093–1166]. Moreover, they play an important role in the analysis of population dynamics involving the so-called Allee effect, which is performed by Dipierro, Proietti Lippi, and Valdinoci [J. Math. Biol. 89 (2024), Paper No. 19]. This is particularly relevant when studying biological populations, since the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. [ABSTRACT FROM AUTHOR]

Embarking on a captivating journey of discovery, our research centers around unraveling the fascinating interaction between fish and plankton. Employing a predator–prey model, we delve into the dynamic interplay of Daphnia and Algae, while scrutinizing the profound influence of fish as a top predator. Our analyses suggest that in most situations, the plankton should show transitions in response to predation pressure by fish. Thus, there exist two distinct stable states, one in which Daphnia is controlled by fish and phytoplankton biomass is high and another in which Daphnia is relatively unaffected by planktivores and algae are controlled by Daphnia. Switches from one regime to the other occur abruptly at a critical fish density. Beyond deterministic exploration, we delve into the influence of the Allee parameter, which significantly increases the number of Daphnia in a free state. To investigate the complete global dynamics of the deterministic model, we present a two-parametric bifurcation diagram. We have also analyze all possible local and global bifurcations that the system could go through. We explore the corresponding stochastic system and investigate the critical transitions with the presence of environmental noise. To understand the probabilistic mechanism behind noise-induced outbreaks, we utilize advanced techniques such as stochastic sensitivity functions and the confidence domain method. These tools grant us unique insights into the captivating world of noise-induced transitions—where stochastic trajectories skillfully navigate from one stable equilibrium point to another. [ABSTRACT FROM AUTHOR]

The nabla fractional-order uncertain difference equation with Caputo-type was analyzed in this article. To begin, the existence and uniqueness theorem of solutions for nabla Caputo uncertain difference equations with almost surely bounded uncertain variables was presented. Furthermore, the uncertainty distributions of the solutions for the proposed equations were obtained by establishing a connection between the solutions of equations and their α-paths based on new comparison theorems. Finally, an application of the uncertain difference equations in a logistic population model involving Allee effect was provided and examples were performed to demonstrate the validity of the theoretical results presented. [ABSTRACT FROM AUTHOR]

We introduce a diffusive predator-prey system with the double Allee effect, focusing on the stability and sufficient conditions for the coexistence of prey and predator. Subsequently, we derived the amplitude equation and explore secondary-order dynamic properties using methods such as Taylor series expansion and multiscaling. The novel approach outlined above provides a precise means to thoroughly analyze the predator-prey model. Through this analysis, we demonstrated that the inclusion of the Allee effect and diffusion leads to the system exhibiting more intricate dynamic behaviors compared to systems lacking these factors. On one hand, in the diffusive system without the Allee effect, the pattern formation regarding the distribution of species was relatively scattered, whereas in the diffusive system with the Allee effect, it is more intensive. On the other hand, the system with the Allee effect transitioned from unstable to stable when the diffusion parameter in prey increased, and the aggregation degree of pattern formation in the system with the Allee effect was higher than in the system without it. These findings highlight the significant roles played by the Allee effect and diffusion in determining the dynamic behaviors of prey and predator within the system.

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. [ABSTRACT FROM AUTHOR]

The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this article, we study a discrete predator-prey system with Holling type II functional response and Allee effect. First, the number of fixed points of the system, local stability, and global stability is discussed. The population changes of predator and prey under strong or weak Allee effects are proved using the nullclines and direction field, respectively. Second, using the bifurcation theory, the bifurcation conditions for the system to undergo transcritical bifurcation and Neimark-Sacker bifurcation at the equilibrium point are obtained. Finally, the dynamic behavior of the system is analyzed by numerical simulation of bifurcation diagram, phase diagram, and maximum Lyapunov exponent diagram. The results show that the system will produce complex dynamic phenomena such as periodic state, quasi-periodic state, and chaos.

This paper investigates the speed selection mechanism for traveling wave fronts of a reaction-diffusion-advection lattice stream-population model with the Allee effect. First, the asymptotic behaviors of the traveling wave solutions are given. Then, sufficient conditions for the speed determinacy of the traveling wave are successfully obtained by constructing appropriate upper and lower solutions. We examine the model with the reaction term $ f (\psi) = \psi(1-\psi)(1+\rho\psi) $, with $ \rho $ being a nonnegative constant, as a specific example. We give a novel conjecture that there exists a critical value $ \rho_c > 1 $, such that the minimal wave speed is linearly selected if and only if $ \rho\leq\rho_c $. Finally, our speculation is verified by numerical calculations.

In this study, we proposed a Leslie-Gower prey-predator model, whose dynamics includes a constant effort harvesting rate in predators and an additive Allee impact on prey. A system of stochastic differential equations is also used to study its behaviour, with the assumption that each population's exposure to environmental unpredictability is represented by noise terms. This kind of randomness is considerably more reasonable and realistic in the proposed paradigm. Due to the paucity of studies on the dynamics of this kind of model, this investigation is being believed to be a way to advance the subject of literature. First, we establish the system's positivity and boundlessness. Next, we look into the dynamics of every one of the stable states, the form of the positive equilibrium point, and the continued existence of every species in the system.It is established that the equilibrium levels of prey and predator are impacted by the Allee effect parameter as well as the impact of harvesting. The positive steady state point's global stability criterion is derived. By selecting the Allee effect and harvesting effort as the bifurcation parameters, it has been established that a Hopf bifurcation exists close to the interior steady state.This study is novel since it incorporates various ecological factors into a single model, potentially opening up new perspectives on predator-prey relationships. To support the mathematical conclusions, rigorous numerical visualisations of the key parameters are provided below using specific hypothetical data. In conclusion, we may state that our model is a project that aims to preserve the ecological equilibrium of the natural world.

Capturing complicated dynamics and understanding the underlying controlling ecological processes is one of the major ecological issues. The Allee effect is an essential component in ecology, and considering it can have a substantial impact on system dynamics. In the present investigation, we analysed a prey-predator scenario in which the predator is a generalist since it feeds on prey populations and the Allee phenomenon impacts the prey population's growth. The influence of the Allee effect on the changing nature of the system is investigated. The stability and boundedness of the model's equilibria are extensively investigated. We found that including the Allee effect enhances the system's local and global behaviours through a detailed bifurcation analysis. The chaotic nature of the system is strongly impacted by the Allee effect, particularly once a specific threshold value is reached. In the study of bifurcation analysis, we looked into bifurcations such the presence of transcritical bifurcation and Hopf-bifurcation to chaos. We added stochastic perturbation to this problem by including random fluctuations in the sensitive parameters. Finally, we analysed the system's mean-square stochastic stability towards the internal equilibrium. As a result, it is discovered that the Allee effect and stochastic perturbation considerably influence the behaviour of the prey-predator system.

The commensal symbiosis system with the Allee effect and single feedback control is proposed and analyzed in this paper. The stability analysis of all possible equilibrium points is discussed, and the sufficient conditions for global stability of the interior equilibrium points are obtained. The occurrence of transcritical bifurcation and saddle-node bifurcation around the equilibrium points is investigated. Finally, the main results of the model are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

The capacity to lay eggs inside healthy fruits rather than on decaying plant matter differentiates insect fruit pests from domestic species. This niche differentiation has been previously proposed to be an adaptation to avoid competition. We hypothesize that pest species will be more strongly affected by competition. We compare the impact of larvae density on fitness traits between Drosophila pests (Drosophila suzukii (Matsumura, 1931); Zaprionus indianus Gupta, 1970) and domestic species (Drosophila immigrans Sturtevant, 1921; Drosophila melanogaster Meigen, 1830). We assessed the effect of crowding on adult emergence and development time. Viability decreased gradually with density for D. immigrans and D. suzukii, while for D. melanogaster and Z. indianus it remained high. Development time increased with density; this was stronger on Z. indianus and D. immigrans than on D. suzukii, which had a moderate increase, and D. melanogaster, which did not change. Contrary to expectations, the distinct patterns observed were not related to each species' domestic or pest lifestyle. In fact, patterns consistent with either scramble or contest type of competition were observed on both pest and domestic species, respectively. These findings challenge prior beliefs regarding competition effects among Drosophila pest species and provide information relevant to integrated pest management. [ABSTRACT FROM AUTHOR]

An Allee effect is a density-dependent phenomenon in which population growth or individual components of fitness increase as population density increases. Understanding the density-dependent effects is vital to elucidate how populations evolve and to investigate evolutionary stability. Partial migration, where a proportion of a population migrates while other individuals remain resident, is widespread across most migratory lineages. However, the mechanism still needs to be better understood in most taxa, especially those experiencing positive density-dependent effects. Here we investigate the evolutionary stability of a partial migration population with only the migrant population experiencing Allee effects. Using the Evolutionary Game Theoretic (EGT) approach, we prove the existence and uniqueness of an evolutionary stable strategy (ESS). EGT provides a mathematical framework for understanding and modelling Darwinian evolution by natural selection. We also show that the ESS is the only Ideal Free Distribution (IFD) that arises in the context of a partially migrating population in a two-habitat environment. [ABSTRACT FROM AUTHOR]

In the study of biological populations, the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. This effect supersedes the classical logistic model, in which low densities are favorable due to lack of competition, and includes situations related to deficit of genetic pools, inbreeding depression, mate limitations, unavailability of collaborative strategies due to lack of conspecifics, etc. The goal of this paper is to provide a detailed mathematical analysis of the Allee effect. After recalling the ordinary differential equation related to the Allee effect, we will consider the situation of a diffusive population. The dispersal of this population is quite general and can include the classical Brownian motion, as well as a Lévy flight pattern, and also a “mixed” situation in which some individuals perform classical random walks and others adopt Lévy flights (which is also a case observed in nature). We study the existence and nonexistence of stationary solutions, which are an indication of the survival chance of a population at the equilibrium. We also analyze the associated evolution problem, in view of monotonicity in time of the total population, energy consideration, and long-time asymptotics. Furthermore, we also consider the case of an “inverse” Allee effect, in which low density populations may access additional benefits. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1 , and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$). We show the existence of two critical values $0 , and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$ ; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$ ; only spreading happens when the length of protection zone is larger than $L^*$. Based on the properties of $L_*$ , we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$), the protection zone should start from somewhere near $0$ ; while if $b$ is small (i.e. $b), then the protection zone should start from somewhere away from $0$ , and as far away from $0$ as possible. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complex dynamics of a predator-prey model, specifically the Leslie-Gower model, with additive Allee effect and simplified Holling Ⅲ functional response. The model has been analyzed for various bifurcations, including the nilpotent cusp singularity of codimension 3, Bogdanov-Takens bifurcation of codimension 3, and Hopf bifurcation of codimension 2. Additionally, a codimension 2 cusp of limit cycles and the coexistent acute angle region of three limit cycles have been identified. Notably, the isola bifurcation of limit cycles, which indicates a new mechanism of sustained oscillation, has been observed for the first time in a Leslie-Gower predator-prey model with additive Allee effect. One-parameter and two-parameter bifurcation diagrams and corresponding phase portraits have been presented to verify the theoretical results. From a biological perspective, the additive Allee effect may result in system collapse, leading to the extinction of the predator population and survival of the prey. The finding of the isola bifurcation of limit cycles is of significant interest, highlighting a novel mechanism of sustained oscillation in this complex system. [ABSTRACT FROM AUTHOR]

A three‐species food chain model with Allee and fear effects on both prey and predator is investigated in this work. The model consists of ODEs describing the evolution and interaction of prey, predator, and top predator populations in a biological system. We assume that the intrinsic growth rate of the prey is much smaller than a threshold value determined by other biological parameters of the model. Consequently, the model is transformed into a singularly perturbed system that acts on two different time scales, ranging from slow for both prey and predator to fast for the top predator. Positivity and boundedness of the solutions of the model along with the stability for its equilibria are investigated. The model possesses the exact invariant manifold with changing stability (black swan) and exhibits the canard phenomenon. Some numerical simulations are performed to validate our analysis. [ABSTRACT FROM AUTHOR]

A large number of herbivorous mammals and reptiles in many terrestrial ecosystems across the globe are presently in the receiving end of extinction. Over-exploitation by its immediate predator and anthropogenic actions is one of the main reasons. Reintroduction of apex predator or top predator at some instances has proven to be a successful strategy in restoring ecological balance. In this paper, we conceptualize the role of top predator in enriching the density of vulnerable species of lower trophic level, with the help of mathematical modeling. First, the dynamical behavior of two species system (prey and mesopredator) is studied, where growth of prey is subject to strong Allee effect. Also, the cost of predation induced fear is incorporated in the growth term. Parametric regions, for which the species perceive extinction risk are analyzed and depicted numerically. We consider that whenever density of the vulnerable species reach a certain threshold, minimum viable population, top predator is introduced in the habitat. Our obtained results show that a species population can be restored from the verge of extinction to a stable state with much higher population density with the introduction of top predator and even it stabilizes an oscillatory system. [ABSTRACT FROM AUTHOR]

During the last two decades there has been growing recognition of the importance of Allee effects in population dynamics and applied ecology. The Allee effect, that is decreased fitness at lower population densities, has been recognized as potentially playing an important role in the conservation of endangered species, in the practice of biological control, and the eradication of invasive species. Although a number of theoretical studies have been devoted to the role of Allee effects in the population dynamics of insects and other terrestrial arthropods, experimental evidence documenting Allee effects is still scarce. Here, we reviewed the literature reporting on density-dependent relationships in low-density populations and conducted a meta-analysis of 191 case studies to identify the occurrence of Allee effects and associated species traits. Allee effects are not rare in terrestrial arthropods, as they were reported in 47% of the cases we reviewed, comprising 46 out of 68 species. Ample examples exist for both demographic Allee effects (28 out of 74 cases cases), and component Allee effects (61 out of 117 cases). Insufficient mating success, cooperative feeding, and enemy escape were the three main mechanisms associated with Allee effects in terrestrial arthropods. Insufficient reproductive success was the mechanism with the highest proportion of related Allee effects (71%). Voltinism and host specialization were common species traits behind demographic Allee effects. Host specialists with univoltine life cycles tended to have stronger Allee effects. The high frequency of Allee effects in terrestrial arthropods reported here and the identified mechanisms behind them have important implications for the selection of management strategies. [ABSTRACT FROM AUTHOR]

The effect of fear, the hunting cooperative process, and Allee's impact on the behavior of an ecological system are investigated and discussed. The impact of the delay of the prey's response to the predation risk is included. The Leslie-Gower growth is used to describe the growth of the predator population. Firstly, the solutions' existence, positivity, and boundedness within the limits of a suitable region in the parametric space for all time are studied. The stability of all equilibrium points under the surrounding environmental effects is established. Moreover, the occurrence of a Hopf bifurcation is discovered. The stability of the bifurcating periodic dynamic and their dynamical properties are studied. Finally, the obtained theoretical results are confirmed and validated utilizing numerical simulation. It is observed that the system possesses a bi-stable behavior and a Hopf bifurcation. [ABSTRACT FROM AUTHOR]

The Allee effect is a density‐dependent phenomenon in which individual fitness increases as population density increases at low population densities. Over the past few decades, a growing number of studies have identified Allee effects in populations using experimental approaches and statistical modeling techniques. These studies have investigated multiple Allee mechanisms (e.g. mate‐finding, predation, resource limitation), across a range of systems and taxa (e.g. plants, vertebrates, invertebrates). This meta‐analysis aims to synthesize studies that experimentally manipulated population density and measured either per capita population growth or fitness components, with the goal of determining whether the 'magnitude' of the Allee effect (defined here as the positive correlation between population density and population growth or fitness) varies with Allee mechanism across taxonomic groups. A total of 2305 studies were screened, and 62 of these studies met our meta‐analysis inclusion criteria. Within these 62 studies, 155 effect sizes encompassing nine different Allee mechanisms were identified across five broad taxa. When grouped by Allee mechanism and taxa, the magnitude of the Allee effect differed across mechanisms, whereas taxonomic group was less useful at explaining variation in the magnitude of Allee effects. Of the nine Allee mechanisms identified, interspecific competition was associated with the largest Allee effects, followed by fear, pollen limitation and mate limitation. These findings suggest that Allee effects may be more dependent on mechanism than taxa and may function similarly within different taxonomic groups. However, as the majority of experimental Allee effect studies included in this meta‐analysis focused on plants and invertebrates, more research is needed on Allee effects in other taxonomic groups to confirm this conclusion. This first quantitative synthesis of Allee effect research in ecology offers novel insight into how Allee mechanisms affect the manifestation of Allee effects in populations, providing important information for ecologists and conservationists. [ABSTRACT FROM AUTHOR]

In this study, we discuss the global dynamics of the Holling-II amensalism model for a strong Allee effect of harmful species. We discuss the existence and stabilization of the extinction equilibria, exclusion equilibria, coexistence equilibria, and infinite singularities by analyzing the presence and stabilization of the system characteristics in terms of the possibilities and correspondences in the model when the death rate of the injured species is used as a threshold value. Also, we find that the two equilibrium points in the first quadrant are effective in proving that the model does not have globally stabilizing features and obtain two critical conditions and their corresponding global phase diagrams. Finally, we explore the weak Allee effect of the victim species, and using the analysis from numerical simulations, we recapitulate the analysis and dynamics of the model in equilibrium. [ABSTRACT FROM AUTHOR]

This study discusses the dynamic behaviors of the prey–predator model subject to the Allee effect and the harvesting of prey species. The existence of fixed points and the topological categorization of the co-existing fixed point of the model are determined. It is shown that the discrete-time prey–predator model can undergo Flip and Neimark–Sacker bifurcations under some parametric assumptions using bifurcation theory and the center manifold theorem. A chaos control technique called the feedback-control method is utilized to eliminate chaos. Numerical examples are given to support the theoretical findings and investigate chaos strategies' effectiveness and feasibility. Additionally, bifurcation diagrams, phase portraits, maximum Lyapunov exponents, and a graph showing chaos control are demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a Leslie-type predator-prey model that incorporates prey harvesting and group defense, leading to a modified functional response. Our analysis focuses on the existence and stability of the system's equilibria, which are essential for the coexistence of predator and prey populations and the maintenance of ecological balance. We identify the maximum sustainable yield, a critical factor for achieving this balance. Through a thorough examination of positive equilibrium stability, we determine the conditions and initial values that promote the survival of both species. We delve into the system's dynamics by analyzing saddle-node and Hopf bifurcations, which are crucial for understanding the system transitions between various states. To evaluate the stability of the Hopf bifurcation, we calculate the first Lyapunov exponent and offer a quantitative assessment of the system's stability. Furthermore, we explore the Bogdanov-Takens (BT) bifurcation, a co-dimension 2 scenario, by employing a universal unfolding technique near the cusp point. This method simplifies the complex dynamics and reveals the conditions that trigger such bifurcations. To substantiate our theoretical findings, we conduct numerical simulations, which serve as a practical validation of the model predictions. These simulations not only confirm the theoretical results but also showcase the potential of the model for predicting real-world ecological scenarios. This in-depth analysis contributes to a nuanced understanding of the dynamics within predator-prey interactions and advances the field of ecological modeling. [ABSTRACT FROM AUTHOR]

We study canard cycles and their cyclicity of a fast–slow Leslie–Gower predator–prey system with Allee effect. More specifically, we find necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its location (or their locations), and then we further completely determine its (or their) dynamics under explicit conditions. Besides, by geometric singular perturbation theory and the slow–fast normal form, we find explicit sufficient conditions to characterize singular Hopf bifurcation and canard explosion of the system. Additionally, the cyclicity of canard cycles is completely solved, and of particular interest is that we show the existence and uniqueness of a canard cycle, whose cyclicity is at most two, under corresponding precise explicit conditions. [ABSTRACT FROM AUTHOR]

In this paper, the impact of the strong Allee effect and ratio-dependent Holling–Tanner functional response on the dynamical behaviors of a predator–prey system is investigated. First, the positivity and boundedness of solutions of the system are proved. Then, stability and bifurcation analysis on equilibria is provided, with explicit conditions obtained for Hopf bifurcation. Moreover, global dynamics of the system is discussed. In particular, the degenerate singular point at the origin is proved to be globally asymptotically stable under various conditions. Further, a detailed bifurcation analysis is presented to show that the system undergoes a codimension- 1 Hopf bifurcation and a codimension- 2 cusp Bogdanov–Takens bifurcation. Simulations are given to illustrate the theoretical predictions. The results obtained in this paper indicate that the strong Allee effect and proportional dependence coefficient have significant impact on the fundamental change of predator–prey dynamics and the species persistence. [ABSTRACT FROM AUTHOR]

The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator's food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles. [ABSTRACT FROM AUTHOR]

The rarity of species increases its market price, consequently leading to the overexploitation of the species and even the extinction of the species. We study how the harvest intensity and the additive Allee effect impact on the Gordon–Schaefer model. In addition, by Sotomayor's theorem and Poincaré–Andronov theorem, we prove the existence of Hopf bifurcation, saddle-node bifurcation and transcritical bifurcation, respectively. Finally, we illustrate our results by numerical simulations. We find that both the cost per unit of harvest and the additive Allee effect have a significant impact on human exploitation of the population. As the additive Allee effect reduces to the weak Allee effect, the lower harvest cost encourages humans to increase the exploitation of species. This threshold is a switch that controls the strong Allee effect. If it exceeds its threshold, then the motivation of humans to exploit the species increases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the use of Lambert W functions in the analysis of nonlinear dynamics and bifurcations of a new two-dimensional γ -Ricker population model. Through the use of such transcendental functions, it is possible to study the fixed points and the respective eigenvalues of this exponential diffeomorphism as analytical expressions. Consequently, the maximum number of fixed points is proved, depending on whether the Allee effect parameter γ is even or odd. In addition, the analysis of the bifurcation structure of this γ -Ricker diffeomorphism, also taking into account the parity of the Allee effect parameter, demonstrates the results established using the Lambert W functions. Numerical studies are included to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In attempting to develop a coherent description of the irruptive dynamics of mountain pine beetle (MPB) in North America, we examined a range of cases where authors described the dependency of intrinsic population growth rates on population attack levels. In some cases, detailed population data were used, but in most cases, investigators relied on operational aerial detection survey data. We found that study conclusions varied significantly depending on the type of data analysed and the spatial and temporal dimensions of the monitoring program. The ability to detect an intrinsic Allee effect (i.e., positive density‐dependent growth for low and rising population densities) depended on the type and the intensity of the sampling effort and the extent of sampling in space and time. Consequently, not all studies were able to quantify the irruption threshold (i.e., the population density at which endemic populations may transition towards the epidemic state). Notably, in every study where an Allee effect was demonstrated, investigators also identified at least one extrinsic environmental factor (e.g., winter weather, summer drought, microclimatic effect) that was regulating its strength. Our results suggest that population surveys conducted from the ground are a necessary complement to aerial survey data if the goal is to make inferences about the irruptive potential for MPB populations and the role of environmental factors in shaping that irruptive potential. [ABSTRACT FROM AUTHOR]

Allee effects are density-dependent barriers that can impact species establishment and population growth, such as through reduced mating success at low population densities. The emerald ash borer, Agrilus planipennis Fairmaire, has been extremely successful at rapidly expanding its North American range. The impact of mate-finding Allee effects (an important type of component Allee effect) early in the invasion period of the emerald ash borer remains unknown. We measured mating success in females as a function of beetle abundance in Halifax, Canada, where the emerald ash borer was recently discovered, and in Connecticut USA, where it has been established for over a decade. We measured relative population abundance and sampled beetles using different strategies. In Halifax, we placed clusters of prism traps along an invasion gradient of emerald ash borer abundance, and in Connecticut, we collected beetles from foraging Cerceris fumipennis females. We dissected female reproductive tracts to measure mating success. We fit a linear regression to the mating success of females as a function of beetle abundance. We found that emerald ash borer did not present a pronounced mate-finding Allee effect as there was no positive relationship between female mating success and abundance. Lack of pronounced component Allee effects that impede population growth may explain rapid range expansion in species that are highly invasive, such as the emerald ash borer. [ABSTRACT FROM AUTHOR]

An individual's access to mates (i.e., its "mating potential") can constrain its reproduction but may also influence its fitness through effects on offspring survival. For instance, mate proximity may correspond with relatedness and lead to inbreeding depression in offspring. While offspring production and survival might respond differently to mating potential, previous studies have not considered the simultaneous effects of mating potential on these fitness components. We investigated the relationship of mating potential with both production and survival of offspring in populations of a long-lived herbaceous perennial, Echinacea angustifolia. Across 7 years and 14 sites, we quantified the mating potential of maternal plants in 1,278 mating bouts and followed the offspring from these bouts over 8 years. We used aster models to evaluate the relationship of mating potential with the number of offspring that emerged and that were alive after 8 years. Seedling emergence increased with mating potential. Despite this, the number of offspring surviving after 8 years showed no relationship to mating potential. Our results support the broader conclusion that the effect of mating potential on fitness erodes over time because of demographic stochasticity at the maternal level. [ABSTRACT FROM AUTHOR]

Sterile Insect Technique (SIT) is a biological insect (or pest) control tool aiming to reduce or eliminate wild insect (or pest) populations by releasing sterile insects (or pests). In this paper, we propose and study a stage- and sex-structured entomological model describing the dynamics of wild-type mosquito population and observed that the extinction equilibrium of the model is globally asymptotically stable when the basic offspring number is less than unity. However, when the basic offspring number is greater than unity, the extinction equilibrium becomes unstable, followed by the emergence of the stable interior equilibrium. We extend the model by introducing sterile male mosquitoes as a biological control agent against wild-type mosquito species. We have considered the Allee effect in the fertile female mosquito population due to the presence of non-egg-laying females in the mosquito population. While the wild mosquito-free equilibrium of the SIT model is always locally asymptotically stable, there exists either no interior equilibrium or a pair of interior equilibria, among which one is always unstable, and the other is always locally asymptotically stable. We observed that the wild mosquito population of the SIT system goes to extinction, followed by a saddle-node bifurcation when the supply rate of sterile males increases through some critical threshold value. As an alternative to the eradication policy, we formulated an optimal control problem to suppress the wild mosquito population, which suggests increasing the investment in awareness campaigns to suppress the mosquito population. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a tritrophic food chain model incorporating an Allee effect, sexually reproductive generalist top predators, and Holling type IV and Beddington–DeAngelis functional responses for interactions across different trophic levels. Analytically, we explore the feasible equilibria, their local stability, and various bifurcations, including Hopf, saddle-node, transcritical, and Bogdanov–Takens bifurcations. Numerical findings suggest that higher Allee intensity in prey growth leads to the inability of species coexistence, resulting in a decline in species density. Likewise, a lower reproduction rate and a higher strength of intraspecific competition among top predators also prevent the coexistence of species. Conversely, a rapid increase in the reproduction rate and a decrease in the strength of intraspecific competition among top predators enhance the densities of prey and top predators while decreasing intermediate predator density. We also reveal the presence of bistability and tristability phenomena within the system. Furthermore, we extend our autonomous model to its nonautonomous counterpart by introducing seasonally perturbed parameters. Numerical analysis of the nonautonomous model reveals that higher seasonal strength in the reproduction rate and intraspecific competition of top predators induce chaotic behavior, which is also confirmed by the maximum Lyapunov exponent. Additionally, we observe that seasonality may lead to the extinction of species from the ecosystem. Factors such as the Allee effect and growth rate of prey can cause periodicity in population densities. Understanding these trends is critical for controlling changes in population density within the ecosystem. Ecologists, environmentalists, and policymakers stand to benefit significantly from the invaluable insights garnered from this study. Specifically, our findings offer pivotal guidance for shaping future strategies aimed at safeguarding biodiversity and maintaining ecological stability amidst changing environmental conditions. By contributing to the existing body of knowledge, our study advances the field of ecological science, enhancing the comprehension of predator–prey dynamics across diverse ecological conditions. [ABSTRACT FROM AUTHOR]