Strong Allee Effect Research Articles

On the Allee effect and directed movement on the whole space.

It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.

Dynamics of a Leslie–Gower Model with Weak Allee Effect on Prey and Fear Effect on Predator

In this paper, a Leslie–Gower model with weak Allee effect on prey and fear effect on predator is proposed. Compared with the case without fear effect on predator, the new model undergoes richer dynamic behaviors such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Also, different from the strong Allee effect on prey, the system with weak Allee effect has bistable attractors which are a largely stable limit cycle and a stable positive equilibrium, two stable equilibria, or a stable limit cycle and a stable trivial equilibrium. When the Allee effect coefficient is intermediate, fear effect on the predator can protect the prey and the predator from being extinguished. The results in this paper can be seen as a complement to those in the literatures about the Leslie–Gower model with Allee effect and fear effect.

Mathematical Modeling and Dynamical Analysis for Tumor Cells and Tumor Propagating Cells Controlled by G9a Inhibitors

It is of critical importance to comprehend the biological environment and core tumor populations when trying to design successful therapeutic solutions for fighting cancers. In several diseases, G9a has been recognized as a novel epigenetic therapeutic target, and its blockage can shift tumor cells (TCs) toward tumor propagating cells (TPCs). This study combines mathematical modeling based on ordinary differential equations and dynamical analysis to quantitatively and qualitatively understand the interactions among G9a, TCs, and TPCs, denoted as G9a-TC-TPC. We propose four different dynamical systems with the impact of the strong Allee effect, named the Hill–Hill system, Logistic–Logistic system, Hill–Logistic system and Logistic–Hill system, to simulate different biological processes through the Hill functions and the Logistic functions that are often used in the models of biological systems. Based on theoretical analysis of these models, including the positivity, boundedness and stability of equilibria, we find that the Hill–Logistic system can display bistable states that correspond to the wild-type tumors and the aggressive tumors. Consequently, we use bifurcation analysis and numerical simulations to illustrate the complicated dynamical behavior of this system. It has been shown that under a specific therapy that changes the relative apoptotic rate of TCs (G9a suppresses the apoptosis of TCs), which can affect the bistability and instability of the system, the wide-type state can be obtained. We also discover that the relative handling time of TCs and TPCs can cooperatively enhance bistability, whereas the cooperative coefficient of feedback can contribute to all tumor cells moving from high-level monostability to bistable states in a restricted region, then to low-level stable states. These results offer new insights for more precisely understanding epigenetic therapy treatments with G9a.

Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model.

In the present study, the effects of the strong Allee effect on the dynamics of the modified Leslie-Gower predator-prey model, in the presence of nonlinear prey-harvesting, have been investigated. In our findings, it is seen that the behaviors of the described mathematical model are positive and bounded for all future times. The conditions for the local stability and existence for various distinct equilibrium points have been determined. The present research concludes that system dynamics are vulnerable to initial conditions. In addition, the presence of several types of bifurcations (e.g., saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic bifurcation) has been investigated. The first Lyapunov coefficient has been evaluated to study the stability of the limit cycle that results from Hopf bifurcation. The presence of a homoclinic loop has been demonstrated by numerical simulation. Finally, possible phase drawings and parametric figures have been depicted to validate the outcomes.

Unidirectional Migration of Populations with Allee Effect

In this note we consider two populations living on identical patches, connected by unidirectional migration, and subject to strong Allee effect. We show that by increasing the migration rate, there are more bifurcation sequences than previous works showed. In particular, the number of steady states can change from 9 (small migration) to 3 (large migration) at a single bifurcation point, or via a sequence of bifurcations with the system having 9, 7, 5, 3 steady states or 9, 7, 9, 3 steady states, depending on the Allee threshold. This is in contrast with the case of bidirectional migration, where the number of steady states always goes through the same bifurcation sequence of 9, 5, 3 steady states as we increase the migration rate, regardless of the value of the Allee threshold. These results have practical implications as well in spatial ecology.

Modeling the Dynamics of a Ratio-Dependent Leslie–Gower Predator–Prey System with Strong Allee Effect

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.

Güçlü Allee Etkili Av-Avcı Modelinin Kararlılığı, Neimark-Sacker Çatallanma Analizi ve Kaos Kontrol

In this study, the dynamical behaviors of a prey–predator model with multiple strong Allee effect are investigated. The fixed points of the model are examined for existence and topological classification. By selecting as the bifurcation parameter $\beta$, it is demonstrated that the model can experience a Neimark-Sacker bifurcation at the unique positive fixed point. Bifurcation theory is used to present the Neimark-Sacker bifurcation conditions of existence and the direction of the bifurcation. Additionally, some numerical simulations are provided to back up the analytical result. Following that, the model's bifurcation diagram and the triangle-shaped stability zone are provided.

Application of Non-singular Kernel in a Tumor Model with Strong Allee Effect

We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plotted.

Complex dynamics and bifurcation analysis for a Beverton–Holt population model with Allee effect

In this paper, we have derived a discrete evolutionary Beverton–Holt population model. The model is built using evolutionary game theory methodology and takes into consideration the strong Allee effect related to predation saturation. We have discussed the existence of the positive fixed point and examined its asymptotic stability. Analytically, we demonstrated that the derived model exhibits Neimark–Sacker bifurcation when the maximal predator intensity is at lower values. All chaotic behaviors are justified numerically. Finally, to avoid these chaotic features and achieve asymptotic stability, we implement two chaos control methods.

Analysis of a Prey-predator Model with Prey Refuge in Infected Prey and Strong Allee Effect in Susceptible Prey Population

Analysis of a Prey-predator Model with Prey Refuge in Infected Prey and Strong Allee Effect in Susceptible Prey Population

Dynamics of a spatially explicit eco-epidemic model with double Allee effect

This article deals with the dynamics of a reaction–diffusion eco-epidemic model that assimilates a double Allee effect. Allee effects are strongly related to extinction vulnerability of populations and two or more Allee effects can emerge simultaneously for the same population. Similarly, diseases can generate high mortality, decrease in reproduction rate, persistence, change in population size, etc. Surprisingly, relatively few studies capture the effect of both factors on a single population. Here we assume that susceptible prey populations suffer from both infectious and double Allee thresholds. We provide precise analysis of the proposed model, including (1) global existence of solutions; (2) the stability of positive constant solution; (3) sufficient conditions leading to Turing instabilities. We also conduct a thorough analysis of the associate local (ODE) system to enhance understanding of the effects of Allee and diffusion on the stability of the steady-state solution. The direction and stability of Hopf-bifurcation are analyzed for the local system concerning the Allee parameter. We have derived conditions for the occurrence of the overexploitation phenomenon, also demonstrated this through simulation. We emanate the pseudo-disease-free subsystem numerically with various dynamics related to the Allee threshold. Also this study shows that both Allee thresholds can generate chaotic dynamics. A brief simulation is performed to explain the developments of the pattern and its mechanism. Furthermore, we see that both weak and strong Allee effects can show enriched pattern diagrams.

Invasion reversal by front transitions and their implications for ecosystem management

Alien species often become invasive by triggering the growth of pathogens that exert a strong negative effect on the native species. Using an extended Lotka–Volterra plant competition model that includes pathogen dynamics with a strong Allee effect we identify a bistability range of counter-propagating fronts representing invasion and recovery dynamics. The fronts differ in the levels of the pathogen in the front zone; high, beyond the Allee threshold for the invasion front and low, below that threshold for the recovery front. Invasion reversal is then studied as induced transitions from invasion fronts to recovery fronts. The study suggests managing invasion by local manipulations in the front zone that increase the Allee threshold and thereby reverse the front from invasion to recovery. We demonstrate numerically direct and indirect manipulations of this kind.

Wave speed and critical patch size for integro-difference equations with a strong Allee effect.

Simplified conditions are given for the existence and positivity of wave speed for an integro-difference equation with a strong Allee effect and an unbounded habitat. The results are used to obtain the existence of a critical patch size for an equation with a bounded habitat. It is shown that if the wave speed is positive there exists a critical patch size such that for a habitat size above the critical patch size solutions can persist in space, and if the wave speed is negative solutions always approach zero. An analytical integral formula is developed to determine the critical patch size when the Laplace dispersal kernel is used, and this formula shows existence of multiple equilibrium solutions. Numerical simulations are provided to demonstrate connections among the wave speed, critical patch size, and Allee threshold.

Nonspreading solutions and patch formation in an integro-difference model with a strong Allee effect and overcompensation

Previous work involving integro-difference equations of a single species in a homogenous environment has emphasized spreading behavior in unbounded habitats. We show that, under suitable conditions, a simple scalar integro-difference equation incorporating a strong Allee effect and overcompensation can produce solutions where the population persists in an essentially bounded domain without spread despite the homogeneity of the environment. These solutions are robust in that they occupy a region of full measure in parameter space. We develop orbit diagrams showing various patterns of nonspreading solutions from stable equilibria, period-two, to higher periodicity. We show that from a relatively uniform initial density with small stochastic perturbations, a population consisting of multiple isolated patches can emerge. In ecological terms, this work suggests a novel endogenous mechanism for the creation of patch boundaries.

Phenomenological bifurcation in a generally stochastic population model with Allee effect.

A general population model with Allee effect driven by correlated additive and multiplicative white noises is considered. This paper aims to investigate noise-induced phenomenological bifurcation (P-bifurcation) and the influence of noises on the population model. With the help of Fokker-Planck equation, we obtain the stationary probability distribution (SPD) of the model, and find that the shape of SPD experiences a transition from one structure to another when the noise intensity passes through a critical value, i.e., the P-bifurcation occurs. Moreover, detailed analysis and simulations for stochastic logistic-like models with weak and strong Allee effects show that the correlated noises have complex effects on the eventual distribution of population size. This paper aims to investigate noise-induced phenomenological bifurcation (P-bifurcation) for a general population model with Allee effect driven by correlated additive and multiplicative white noises. We find that the shape of SPD experiences a transition from one structure to another when the noise intensity passes through a critical value, i.e., the P-bifurcation occurs.

The role of host refuge and strong Allee effects in a host–parasitoid system

We propose and study a discrete host–parasitoid model of difference equations with a spatial host refuge where hosts in the refuge patch are free from parasitism but undergo a demographic strong Allee effect. If the growth rate of hosts in the non-refuge patch is less than one, a host Allee threshold is derived below which both populations become extinct. It is proven that both populations can persist indefinitely if the host growth rate in the non-refuge patch exceeds one and the maximum reproductive number of parasitoids is greater than one. Numerical simulations reveal that the host refuge can either stabilize or destabilize the host–parasitoid interactions, depending on other model parameters. A large number of parasitoid turnover from a parasitized host may be detrimental to the parasitoids due to Allee effects in the hosts within the refuge patch. Moreover, it is demonstrated numerically that if the host growth rate is not small, the population level of parasitoids may suddenly drop to a small value as some parameters are varied.

Non-constant steady states and Hopf bifurcation of a species interaction model

In this paper, we are concerned with the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. To explore nonhomogeneous solutions of the model, we consider the existence and non-existence of non-constant steady states and temporal bifurcation. Then we show the boundedness of the global positive solutions for the parabolic system and present the upper and lower bounds of positive solutions for the associated elliptic system. The non-existence and existence of the non-constant steady states of the elliptic system with the homogeneous Neumann boundary conditions are obtained by using the priori estimates, maximum principle and index theory. Furthermore, the existence and the direction of Hopf bifurcation are investigated via the stability analysis, center manifold theory and normal form reduction. Numerical simulations are carried out to verify our theoretical analysis and to illustrate that the ratio-dependent Holling III functional response and strong Allee effect have strong impact on dynamical behaviors of the species interaction systems.

Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation

In this paper, we consider a Leslie–Gower cross diffusion predator–prey model with a strong Allee effect and hunting cooperation. We mainly investigate the effects of self diffusion and cross diffusion on the stability of the homogeneous state point and processes of pattern formation. Using eigenvalue theory and Routh–Hurwitz criterion, we analyze the local stability of positive equilibrium solutions. We give the conditions of Turing instability caused by self diffusion and cross diffusion in detail. In order to discuss the influence of self diffusion and cross diffusion, we choose self diffusion coefficient and cross diffusion coefficient as the main control parameters. Through a series of numerical simulations, rich Turing structures in the parameter space were obtained, including hole pattern, strip pattern and dot pattern. Furthermore, We illustrate the spatial pattern through numerical simulation. The results show that the dynamics of the model exhibits that the self diffusion and cross diffusion control not only form the growth of dots, stripes, and holes, but also self replicating spiral pattern growth. These results indicate that self diffusion and cross diffusion have important effects on the formation of spatial patterns.

Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect

Abstract We propose and analyze a Lotka-Volterra commensal model with an additive Allee effect in this article. First, we study the existence and local stability of possible equilibria. Second, the conditions for the existence of saddle-node bifurcations and transcritical bifurcations are derived by using Sotomayor’s theorem. Third, we give sufficient conditions for the global stability of the boundary equilibrium and positive equilibrium. Finally, we use numerical simulations to verify the above theoretical results. This study shows that for the weak Allee effect case, the additive Allee effect has a negative effect on the final density of both species, with increasing Allee effect, the densities of both species are decreasing. For the strong Allee effect case, the additive Allee effect is one of the most important factors that leads to the extinction of the second species. The additive Allee effect leads to the complex dynamic behaviors of the system.

Optimal management of stochastic invasion in a metapopulation with Allee effects

Invasive species account for incalculable damages worldwide, in both ecological and bioeconomic terms. The question of how a network of invasive populations can be optimally managed is one that deserves further exploration. A study accounting for partial observability and imperfect detection, in particular, could yield useful insights into species eradication efforts. Here, we generalized a simple model system that we developed in previous work. This model consists of three interacting populations with underlying strong Allee effects and stochastic dynamics, inhabiting distinct locations connected by dispersal, which can generate bistability. To explore the stochastic dynamics, we formulated an individual-based modeling approach. Next, using the theory of continuous-time Markov chains, we approximated the original high-dimensional model by a Markov chain with eight states, with each state corresponding to a combination of population thresholds. We then used the reduced model as the core for a powerful decision-making tool, referred to as a Partially Observable Markov Decision Process (POMDP). Analysis of this POMDP indicates when the system results in optimal management outcomes.