Search Results (45)

This paper investigates a model of amensalism, in which the first species is influenced by the combined effects of refuge and fear, while the second species exhibits an additive Allee effect. The paper analyzes the existence and stability of the equilibria of the system and derives the conditions for various bifurcations. In the global structure analysis, the stability at infinity is examined, and the phenomena of global stability and bistability in the system are analyzed. Additionally, a sensitivity analysis is employed to evaluate the impact of system parameters on populations. The study reveals that refuge has a significant positive effect on the first population, and refuge’s effect becomes more pronounced as the fear level increases. Under the strong Allee effect, when the initial density of the second species is low, the second species may eventually become extinct; when the initial density is high, if the refuge parameter is below a certain threshold, increasing the refuge parameter slows down the extinction of the first species, whereas, when the refuge parameter exceeds this threshold, the two species can coexist. Under the weak Allee effect, when the refuge parameter surpasses a certain threshold, the two species can achieve long-term, stable coexistence, and the threshold for the weak Allee effect is higher than that for the strong Allee effect. Full article

In this paper, an eco-epidemiological model with a weak Allee effect and prey disease dynamics is discussed. Mathematical features such as non-negativity, boundedness of solutions, and local stability of the feasible equilibria are discussed. Additionally, the transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation are proven using Sotomayor’s theorem and Poincare–Andronov–Hopf theorems. In addition, the correctness of the theoretical analysis is verified by numerical simulation. The numerical simulation results show that the eco-epidemiological model with a weak Allee effect has complex dynamics. If the prey population is not affected by disease, the predator becomes extinct due to a lack of food. Under low infection rates, all populations are maintained in a coexistent state. The Allee effect does not influence this coexistence. At high infection rates, if the prey population is not affected by the Allee effect, the infected prey is found to coexist in an oscillatory state. The predator population and the susceptible prey population will be extinct. If the prey population is affected by the Allee effect, all species will be extinct. Full article

Depensation (or an Allee effect) has recently been detected in stock–recruitment relationships (SRRs) in four Atlantic herring stocks and one Atlantic cod stock using a Bayesian statistical approach. In the present study, we define the Allee effect threshold (B_AET) for these five stocks and propose B_AET as a candidate limit reference point (LRP). We compare B_AET to traditional LRPs based on proportions of equilibrium unfished biomass (B₀) and biomass at maximum sustainable yield (B_MSY) assuming a Beverton–Holt or Ricker SRR with and without depensation, and to the change point from a hockey stick SRR (B_CP). The B_AET for the case studies exceeded 0.2 B₀ and 0.4 B_MSY for three of the case study stocks and exceedances of 0.2 B₀ were more common when the Ricker form of the SRR was assumed. The B_AET estimates for all case studies were less than B_CP. When there is depensation in the SRR, multiple equilibrium states can exist when fishing at a fixed fishing mortality rate (F) because the equilibrium recruits-per-spawner line at a given F can intersect the SRR more than once. The equilibrium biomass is determined by whether there is excess recruitment at the initial projected stock biomass. Estimates of equilibrium F_MSY in the case studies were generally higher for SRRs that included the depensation parameter; however, the long-term F that would lead the stock to crash (F_crash) in the depensation SRRs was often about half the F_crash for SRRs without depensation. When warranted, this study recommends exploration of candidate LRPs from depensatory SRRs, especially if Allee effect thresholds exceed commonly used limits, and simulation testing of management strategies for robustness to depensatory effects. Full article

This study investigates a modified Lotka–Volterra commensalism system that incorporates the weak Allee effect in prey and symmetric (equal harvesting effort for both species) non-selective harvesting, addressing a critical gap in ecological modeling. Unlike previous work, we rigorously examine how the interaction between the Allee effect and harvesting disrupts system stability, giving rise to novel bifurcation phenomena and population collapse thresholds. Using eigenvalue analysis and the Dulac–Bendixson criterion, we derive sufficient conditions for the existence and stability of equilibria. We find that harvesting destabilizes the system by inducing two saddle-node bifurcations. Notably, prey abundance can increase with greater Allee intensity under controlled harvesting—a rare and counterintuitive ecological outcome. Moreover, exceeding a critical harvesting threshold drives both species to extinction, while controlled harvesting allows sustainable coexistence. Numerical simulations support these analytical findings and identify critical parameter ranges for species coexistence. These results contribute to theoretical ecology and offer insights for designing sustainable harvesting strategies that balance exploitation with conservation. Full article

This study examines abrupt changes in system dynamics, focusing on a Hassell-type density-dependent model with an Allee effect. It aims to analyze tipping points leading to extinction and bistability, including chaotic dynamics. Key methods include computing the topological entropy and Lyapunov exponents when varying the carrying capacity, the intrinsic growth rate and the initial conditions, providing a detailed characterization of chaotic regimes. Meanwhile, we derive an inverse square-root scaling law near a saddle-node bifurcation using a complex analysis. This study uniquely integrates chaos theory, a bifurcation analysis and scaling laws into a density-dependent ecological model with an Allee effect, revealing how chaotic regimes, bistability and an analytically derived inverse square-root scaling law near extinction shape the tipping point dynamics and critical transitions in ecological systems. Full article

In this study, two discrete mosquito population-control models incorporating the Allee effect are developed to investigate the impact of different sterile mosquito release strategies. By applying the theory of difference equations, a comprehensive analysis is conducted on the existence and stability of fixed points in scenarios with and without sterile mosquito releases. Conditions for the existence and stability of positive fixed points are rigorously derived. The findings reveal that in the absence of a positive fixed point, the wild mosquito population inevitably declines to extinction. When a single positive fixed point exists, the population dynamics exhibit dependence on the initial population size, potentially leading to either extinction or stabilization. In cases where two positive fixed points are present, a bistable dynamic emerges, indicating the coexistence of two mosquito populations. Full article

This article is concerned with the mathematical modeling of cancer virotherapy, emphasizing the impact of Allee effects on tumor cell growth. We propose a modeling framework that describes the complex interaction between tumor cells and oncolytic viruses. The efficacy of this therapy against cancer is mathematically investigated. The analysis involves linear and logistic growth scenarios coupled with different Allee effects, including weak, strong, and hyper Allee forms. Critical points are identified, and their existence and stability are analyzed using dynamical system theories and bifurcation techniques. Also, bifurcation diagrams and numerical simulations are utilized to verify and extend analytical results. It is observed that Allee effects significantly influence the stability of the system and the conditions necessary for tumor control and eradication. Full article

The Allee effect and group defense are two naturally occurring phenomena in the prey species of a predator–prey system. This research paper examines the impact of integrating the Allee effect on the dynamics of a predator–prey model, including a density-dependent functional response that reflects the defensive strategies of the prey population. Initially, the positivity and boundedness of the solutions are examined to ascertain the biological validity of the model. The presence of ecologically significant equilibrium points are established, followed by examining parametric restrictions for the local stability to comprehend the system dynamics in response to minor perturbations. A detailed computation encompasses diverse bifurcations, both of codimension one and two, which provide distinct dynamic behaviors of the model, such as oscillations, stable coexistence, and potential extinction scenarios. Numerical simulation has been provided to showcase complex dynamical behavior resulting from the Allee effect and prey group defense. Full article

In this paper, I extend the Simple Equations Method (SEsM) and adapt it to obtain exact solutions of systems of fractional nonlinear partial differential equations (FNPDEs). The novelty in the extended SEsM algorithm is that, in addition to introducing more simple equations in the construction of the solutions of the studied FNPDEs, it is assumed that the selected simple equations have different independent variables (i.e., different coordinates moving with the wave). As a consequence, nonlinear waves propagating with different wave velocities will be observed. Several scenarios of the extended SEsM are applied to the time-fractional predator–prey model under the Allee effect. Based on this, new analytical solutions are derived. Numerical simulations of some of these solutions are presented, adequately capturing the expected diverse wave dynamics of predator–prey interactions. Full article

This contribution concerns studying a realistic predator–prey interaction, which was achieved by virtue of formulating a modified Leslie–Gower predator–prey model under the influence of the double Allee effect and fear effect in the prey species. The initial theoretical work sheds light on the relevant properties of the solution, presence, and local stability of the equilibria. Both analytic and numerical approaches were used to address the emergence of diverse bifurcations, like saddle-node, Hopf, and Bogdanov–Takens bifurcations. It is noteworthy that while making the assumption that the characteristic equation of the Jacobian matrix J has a pair of imaginary roots $C\left(\rho \right)\pm \iota D\left(\rho \right)$, it is sufficient to consider only $C\left(\rho \right)+\iota D\left(\rho \right)$ due to symmetry. The impact of the fear effect on the proposed model is discussed. Numerical simulation results are provided to back up all the theoretical analysis. From the findings, it was established that the initial condition of the population, as well as the phenomena (fear effect) introduced, played a crucial role in determining the stability of the proposed model. Full article

Allées used to be the essential artistic tools and indispensable parts of the strictly architectural, formal Baroque gardens. Beyond the practical purposes of edging paths and garden ways for walking, hunting, or horse and carriage riding, allées played a vital role in marking visual and landscape connections and thus the spatial projection of the noble estate, its wealth, and social rank. In Historical Hungary, Baroque architecture and garden art appeared in German-Austrian and French examples in the 18th century. The Széchenyi Linden Allée is an outstanding linear garden space of Baroque Garden art at Nagycenk, West Hungary. The generous composition, created by the prominent Count Széchenyi family in the mid-18th century, has remained a magnificent entity in the landscape ever since. Despite barely two hundred years of detected or unknown environmental or habitat changes, as early as 1942, the allée received a nature conservation nomination. More than a half-century later, in 2002, the allée became a historical and landscape aesthetical heritage within the Fertő-Hanság Cultural Landscape World Heritage site. Unfortunately, the once magnificent tree lines have severely eroded in recent decades due to mature trees’ subsequent death, inadequate replacement, lack of regular maintenance and tree care, and effects of climate change. In recent years (2011, 2018), landscape and horticultural analyses and visual and instrumental tree assessments were performed to help the conservation and rebirth of the allée, maintain the mature trees, and restore the landscape within a long-term renewal plan. Along with the 2018 survey and plan, the short-term maintenance works were completed in 2019–2020. This study, based on site surveys in 2022 and 2024, aims to identify the results of the primary management, analyses the vitality of mature trees after crown reductions, and then proposes a resilient and sustainable regeneration method with the habitat, cultural, natural, and genetic heritage, and the feasible maintenance contexts in focus. As proposed in the 2018 plan, the reproduction of mature trees started in 2020 and resulted in well-developing grafts for a later allée restoration. Due to the challenges of climate change, the regeneration project requires a special, long-term restoration management plan with a special focus on the still vital and possible remaining mature trees, the well-growing individuals from previous replanting, and the nursery school seedlings conserving the genetic heritage of the Széchenyi lime trees with long-viability capacity. Full article

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. Full article

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 $\gamma$-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 $\gamma$ is even or odd. In addition, the analysis of the bifurcation structure of this $\gamma$-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. Full article

We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker–Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. One is the unstable state, which lies between the two stable equilibria. Our primary focus is on elucidating the transition pathways from extinction to the upper stable state in this single-species model, particularly under the influence of jump-diffusion noise. This helps us to study the biological behavior of species. The identification of the most probable path relies on solving the nonlocal FPE tailored to the population dynamics of the single-species model. This enables us to pinpoint the corresponding maximum possible stable equilibrium state. Additionally, we derive the Onsager–Machlup function for the stochastic model and employ it to determine the corresponding most probable paths. Numerical simulations manifest three key insights: (i) when non-Gaussian noise is present in the system, the peak of the stationary density function aligns with the most probable stable equilibrium state; (ii) if the initial value rises from extinction to the upper stable state, then the most probable trajectory converges towards the maximally probable equilibrium state, situated approximately between 9 and 10; and (iii) the most probable paths exhibit a rapid ascent towards the stable state, then maintain a sustained near-constant level, gradually approaching the upper stable equilibrium as time goes on. These numerical findings pave the way for further experimental investigations aiming to deepen our comprehension of dynamical systems within the context of biological modeling. Full article

This paper investigates the complex dynamics of a ratio-dependent predator–prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation; (c) The derivation of the normal form near the Turing–Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results. Full article