BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN ALLEE EFFECT AND A DELAYED MECHANISM

Regarding delay-induced predator-prey models, much research has been done on delayed destabilization, but whether delays are stabilizing or destabilizing is a subtle issue. In this study, we investigate predator-prey dynamics affected by both delays and the Allee effect. We analyze the consequences of delays in different feedback mechanisms. The existence of a Hopf bifurcation is studied, and we calculate the value of the delay that leads to the Hopf bifurcation. Furthermore, applying the normal form theory and a center manifold theorem, we consider the direction and stability of the Hopf bifurcation. Finally, we present numerical experiments that validate our theoretical analysis. Interestingly, depending on the chosen delay mechanism, we find that delays are not necessarily destabilizing. The Allee effect generally increases the stability of the equilibrium, and when the Allee effect involves a delay term, the stabilization effect is more pronounced.

Dynamic Analysis of an Algae-Fish Harvested Model with Allee Effect

In this paper, an algae-fish harvested model with Allee effect was established to further explore the dynamic evolution mechanism under the influence of key factors. Mathematical theoretical work not only investigated the existence and stability of all possible equilibrium points, but also probed into the occurrence of transcritical and Hopf bifurcation. The numerical simulation works verified the effectiveness of the theoretical derivation results and displayed rich bifurcation dynamical behaviors, which showed that Allee effect and harvest have played a vital role in the dynamic relationship between algae and fish. In summary, it was expected that these research results would be beneficial for promoting the study of bifurcation dynamics in aquatic ecosystems.

Existence and nonexistence of positive solutions of semilinear elliptic equation with inhomogeneous strong Allee effect

In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.

Dynamics Analysis of an Aquatic Ecological Model with Allee Effect

In this paper, based on the dynamic relationship between algae and protozoa, an aquatic ecological model with Allee effect was established to investigate how some ecological environment factors affect coexistence mode of algae and protozoa. Mathematical derivation works mainly gave some key conditions to ensure the existence and stability of all possible equilibrium points, and to induce the occurrence of transcritical bifurcation and Hopf bifurcation. The numerical simulation works mainly revealed ecological relationship change characteristics of algae and protozoa with the help of bifurcation dynamics evolution process. Furthermore, it was also worth emphasizing that Allee effect had a strong influence on the dynamic relationship between algae and protozoa. In a word, it was hoped that the research results could provide some theoretical support for algal bloom control, and also be conducive to the rapid development of aquatic ecological models.

Switching effect on a two prey-one predator system with strong Allee effect incorporating prey refuge

In an environment,the food chains are balanced by the prey-predator interactions.When a predator species is provided with more than one prey population,it avails the option of prey switching between prey species according to their availability.So,prey switching of predators mainly helps to increase the overall growth rate of a predator species.In this work,we have proposed a two prey-one predator system where the predator population adopts switching behavior between two prey species at the time of consumption.Both the prey population exhibit a strong Allee effect and the predator population is considered to be a generalist one.The proposed system is biologically well-defined as the system variables are positive and do not increase abruptly with time.The local stability analysis reveals that all the predator-free equilibria are saddle points whereas the prey-free equilibrium is always stable.The intrinsic growth rates of prey,the strong Allee parameters,and the prey refuge parameters are chosen to be the controlling parameters here.The numerical simulation reveals that in absence of one prey,the other prey refuge parameter can change the system dynamics by forming a stable or unstable limit cycle.Moreover,a situation of bi-stability,tri-stability,or even multi-stability of equilibrium points occurs in this system.As in presence of the switching effect,the predator chooses prey according to their abundance,so,increasing refuge in one prey population decreases the count of the second prey population.It is also observed that the count of predator population reaches a comparatively higher value even if they get one prey population at its fullest quantity and only a portion of other prey species.So,in the scarcity of one prey species,switching to the other prey is beneficial for the growth of the predator population.

Dynamic analysis of a predator-prey model of Gause type with Allee effect and non-Lipschitzian hyperbolic-type functional response

In this work,we study a predator-prey model of Gause type,in which the prey growth rate is subject to an Allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response,which is neither differentiable nor locally Lipschitz at the predator axis.This kind of functional response is an extension of the so-called square root functional response,used to model systems in which the prey have a strong herd structure.We study the behavior of the solutions in the first quadrant and the existence of limit cycles.We prove that,for a wide choice of parameters,the solutions arrive at the predator axis in finite time.We also characterize the existence of an equilibrium point and,when it exists,we provide necessary and sufficient conditions for it to be a center-type equilibrium.In fact,we show that the set of parameters that yield a center-type equilibrium,is the graph of a function with an open domain.We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation.In particular,we guarantee the existence of a unique limit cycle,for small perturbations of the system.

Global Dynamics in a Predator-Prey Model with Allee Effect

Since the last century, various predator-prey systems have garnered widespread attention. In particular, the predator-prey systems have sparked significant interest among applied mathematicians and ecologists. From the perspectives of both mathematics and biology, a predator-prey system with the Allee effect and featuring the Bazykin functional response has been established. For this model, analyses have been conducted on its boundedness, the properties of its solutions, the existence of equilibrium points, as well as its local stability and Hopf bifurcations.

Stability and Turing Patterns of a Predator-prey Model with Holling Type Ⅱ Functional Response and Allee Effect in Predator

In this paper, we are concerned with a predator-prey model with Holling type Ⅱ functional response and Allee effect in predator. We first mathematically explore how the Allee effect affects the existence and stability of the positive equilibrium for the system without diffusion. The explicit dependent condition of the existence of the positive equilibrium on the strength of Allee effect is determined. It has been shown that there exist two positive equilibria for some modulate strength of Allee effect. The influence of the strength of the Allee effect on the stability of the coexistence equilibrium corresponding to high predator biomass is completely investigated and the analytically critical values of Hopf bifurcations are theoretically determined.We have shown that there exists stability switches induced by Allee effect. Finally, the diffusion-driven Turing instability, which can not occur for the original system without Allee effect in predator, is explored, and it has been shown that there exists diffusion-driven Turing instability for the case when predator spread slower than prey because of the existence of Allee effect in predator.

Hopf bifurcation of a delayed predator-prey model with Allee effect and anti-predator behavior

This paper proposes a diffusive predator-prey model with Allee effect,time delay and anti-predator behavior.First,the existence and stability of all equilibria are analyzed and the conditions for the appearance of the Hopf bifurcation are studied.Using the normal form and center manifold theory,the formulas which can determine the direction,period and stability of Hopf bifurcation are obtained.Numerical simulations show that the Allee effect can determine the survival abundance of the prey and predator populations,and anti-predator behavior can greatly improve the stability of the coexisting equilibrium.

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.

Analysis of an avian influenza model with Allee effect and stochasticity

In this paper,we investigate a two-dimensional avian influenza model with Allee effect and stochasticity.We first show that a unique global positive solution always exists to the stochastic system for any positive initial value.Then,under certain conditions,this solution is proved to be stochastically ultimately bounded.Furthermore,by constructing a suitable Lyapunov function,we obtain sufficient conditions for the existence of stationary distribution with ergodicity.The conditions for the extinction of infected avian population are also analytically studied.These theoretical results are conformed by computational simulations.We numerically show that the environmental noise can bring different dynamical outcomes to the stochastic model.By scanning different noise intensities,we observe that large noise can cause extinction of infected avian population,which suggests the repression of noise on the spread of avian virus.

Dynamics in a Discrete-time Predator-prey System with Allee Effect

In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases （within a limited value）. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.

QUALITATIVE ANALYSIS OF A PREDATOR-PREY SYSTEM WITH ALLEE EFFECT FOR PREY

In this paper, we study some predator-prey system with Allee effect for prey. In addition, we discuss the properties of equilibrium points and the existence and uniqueness of limit cycle.

STABILITY ANALYSIS OF A LOTKA-VOLTERRA COMMENSAL SYMBIOSIS MODEL INVOLVING ALLEE EFFECT

In this paper, we present a stability analysis of a Lotka-Volterra commensal symbiosis model subject to Allee effect on the unaffected population which occurs at low population density. By analyzing the Jacobian matrix about the positive equilibrium, we show that the positive equilibrium is locally asymptotically stable. By applying the differential inequality theory, we show that the system is permanent, consequently, the boundary equilibria of the system is unstable. Finally, by using the Dulac criterion, we show that the positive equilibrium is globally stable. Although Allee effect has no influence on the final densities of the predator and prey species, numeric simulations show that the system subject to an Allee effect takes much longer time to reach its stable steady-state solution, in this sense that Allee effect has unstable effect on the system, however, such an effect is controllable. Such an finding is greatly different to that of the predator-prey model.

A Markov-switching predator-prey model with Allee effect for preys

This paper concerns with a Markov-switching predator-prey model with Allee effect for preys.The conditions under which extinction of predator and prey populations occur have been established.Sufficient conditions are also given for persistence and global attractivity in mean.In addition,stability in the distribution of the system under con-sideration is derived under some assumptions.Finally,numerical simulations are carried out to illustrate theoretical results.

Dynamical behaviors of a constant prey refuge ratio-dependent prey-predator model with Allee and fear effects

In this paper,we consider a nonlinear ratio-dependent prey-predator model with constant prey refuge in the prey population.Both Allee and fear phenomena are incorporated explicitly in the growth rate of the prey population.The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail.Hopf bifurcation including its direction and stability for the model is also studied.We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system.It is also observed that for a fixed strength of fear,an increase in the Allee parameter makes the system unstable,whereas an increase in prey refuge drives the system toward stability.However,higher values of both the Allee and prey refuge parameters have negative impacts and the populations go to extinction.Further,we explore the variation of densities of the populations in different bi-parameter spaces,where the coexistence equilibrium point remains stable.Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software.

Habitat Loss,Uneven Distribution of Resources and Fragmented Landscapes—A Resource Based Model of the Patch Size Effect

The problem of habitat fragmentation is recently an important issue in ecological research as well as in the practical approach of nature conservation. According to the most popular approaches, habitats are considered as the homogenous parts of the landscape. Also the metapopulation concept problem of the inert habitat heterogenity is considered quite seldom. These approaches have some weak points resulting from the assumption that the border between habitat patches and the metapopulation matrix is fairly sharp. This paper presents a resource-based concept of habitats, based on mathematical theory of point processes, which can be easily applied to analysing the problem of uneven distribution of resources. The basic assumption is that the random distribution of resources may be mathematically described as the realisation of a certain point process. According to our method, it is possible to calculate the expected quantities of available resources as well as the minimum area of habitat that includes the expected abundance of the resource. This approach may be very useful to understand some crucial phenomena in landscape ecology, such as the patch size effect and its connection to habitat loss and fragmentation.

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

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

Qualitative Analysis of a Fractional Pandemic Spread Model of the Novel Coronavirus (COVID-19)

In this study,we classify the genera of COVID-19 and provide brief information about the root of the spread and the transmission from animal(natural host)to humans.We establish a model of fractional-order differential equations to discuss the spread of the infection from the natural host to the intermediate one,and from the intermediate one to the human host.At the same time,we focus on the potential spillover of bat-borne coronaviruses.We consider the local stability of the co-existing critical point of the model by using the Routh–Hurwitz Criteria.Moreover,we analyze the existence and uniqueness of the constructed initial value problem.We focus on the control parameters to decrease the outbreak from pandemic form to the epidemic by using both strong and weak Allee Effect at time t.Furthermore,the discretization process shows that the system undergoes Neimark–Sacker Bifurcation under specific conditions.Finally,we conduct a series of numerical simulations to enhance the theoretical findings.

Global Dynamics of a Predator-prey Model

In this paper, we consider a predator-prey model. A sufficient conditionis presented for the stability of the equilibrium, which is different from the one for themodel with Hassell-Varley type functional response. Furthermore, by constructing aLyapunov function, we prove that the positive equilibrium is asymptotically stable.