Narrow Results

Recent research suggests that the presence of predator may affect the physical characteristics of prey species to the level that it is more effective compared to direct predation in reducing the prey biomass. Furthermore, such nonlethal impacts might persist across seasons or generations. This research work centers on a predator–prey interaction considering the predator’s fear and its carryover effects (COEs) on prey as well as the Allee effect on predator species. This model incorporates both self- and cross-diffusions to explore the species distribution due to the movement of the species subject to the Neumann boundary condition. First, we study the positivity, boundedness, feasible equilibria and their local stability; different bifurcations appear in the nonspatial system. Global dynamics of the system are discussed, showing that coexistence equilibrium can never be globally stable. Both fear and its COEs destabilize the system, but the Allee effect in predators, once destabilized, again stabilizes the system. COEs of fear can influence the effect of Allee and destabilize the system even when the Allee effect is high, but fear has no such impact on the system dynamics, coupling with the Allee parameter. The system exhibits bistability, and the combined influence of fear and its COEs results in the “paradox of enrichment”. For the spatial model, Turing instability conditions, wave number range for instability and different instability regions are derived. The system dynamics permits mainly the spot stationary patterns. Diffusion causes an increase in gatherings with rising COEs and the Allee effect in predators, but an increase in the fear effect diminishes the prey gatherings in specific spots. Hopf instability leads to oscillation in the system. The delicate dynamics of fear and its COEs, the predator’s Allee effect along with diffusion reveal insights into the harmony in nature.

In this paper, we consider the dynamics of a prey–predator model with logistic growth incorporating refuge in the prey and cooperation among predators population. Furthermore, multiplicative Allee effect in the prey growth is added to account from biological and mathematical perspectives. First, the existence and stability of equilibria of the model are discussed. Next, the existence of several kinds of bifurcation is provided and also studied the direction and stability of Hopf bifurcation. In addition, we study the influence of hunting cooperation on the model analytically and numerically, and find that the hunting cooperation cannot only reduce the density of prey population, but also destabilize the system dynamics irrespective of Allee effect. We choose also the impact of refuge on the model numerically, and explore that refuge stabilize the system dynamics. Moreover, the comparison between the dynamics of strong and weak Allee effect is taken into consideration.

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.

Predator–prey interactions are considered to be key components in the structural framework of an ecosystem. Giving rise to rich dynamics, these engagements have always drawn attention of budding researchers. Biological processes like Allee effect have been investigated by several researchers in the field of eco-epidemiology. Regulation of chaotic dynamics under the influence of refugia cannot be neglected either. Hence, in our present investigation, we have demonstrated our model under the cumulative impact of both Allee effect and refugia. We shall also observe the role of nutritional value in regulating ailment propagation. The stability conditions for the biologically feasible equilibrium points are constructed using local stability analysis. Conditions for existence of Hopf bifurcation along with uniform persistence and global asymptotic stability have been established. Rigorous numerical investigation has been performed so as to scrutinize the impact of force of infection and nutritional value on the dynamics of the proposed eco-epidemic model.

In population biology, the Allee dynamics refer to negative growth rates below a critical population density. In this paper, we study a reaction-diffusion (RD) model of popoulation growth and dispersion in one dimension, which incorporates the Allee effect in both the growth and mortatility rates. In the absence of diffusion, the bifurcation diagram displays regions of both finite population density and zero population density, i.e. extinction. The early signatures of the transition to extinction at the bifurcation point are computed in the presence of additive noise. For the full RD model, the existence of traveling wave solutions of the population density is demonstrated. The parameter regimes in which the traveling wave advances (range expansion) and retreats are identified. In the weak Allee regime, the transition from the pushed to the pulled wave is shown as a function of the mortality rate constant. The results obtained are in agreement with the recent experimental observations on budding yeast populations.

We investigate a stochastic model for single species population growth with strong and weak Allee effects subjected to coupling between non-Gaussian and Gaussian colored noise as well as nonzero cross-correlation in between. Stationary probability distribution of population model is obtained depending on the Fokker–Planck equation. The mean first-passage time is also calculated in order to quantify the time of transition between survival state and extinction state with Allee effect in population. The intensity of non-Gaussian colored noise can induce phase transition, and population may be vulnerable to extinction due to the increase in the intensity of non-Gaussian colored noise. Whether Allee effect is strong or weak, the increase in Allee threshold will not contribute to the survival and stability of the population. Further, the phenomenon of resonant activation is firstly discovered in the study of population dynamics with Allee effect. These behaviors can be interpreted on the basis of a biological model of population evolution.

We consider a one-dimensional chain of identical sites, appropriate for colonization by a biological species. The dynamics at each site is subjected to the demographic Allee effect. We consider nonzero probability p of dispersal to the nearby sites and we prove, for small values of p, the existence of asymptotically stable time-periodic and space-localized solutions, such that the central site carries the vast majority of the metapopulation, while the populations at nearby sites attain very small values. We study numerically a chain of three sites, both for the case of open ends or periodic boundary conditions. We study the bifurcations leading to transition from chaotic to periodic behavior and vice-versa and note that the increase of the dispersal probability in both cases controls the chaotic behavior of the metapopulation.

We investigate a method of chaos control in which intervention is proportional to the difference between the current state and a fixed value. We prove that this method allows to stabilize the most usual one-dimensional maps used in discrete-time models of population dynamics about a globally stable positive equilibrium. From the point of view of targeting, this technique is very flexible, and we show how to choose the control parameter values to lead the system towards the desired target. Another important feature of this control scheme in the ecological context is that it can be designed to prevent the risk of extinction in models with the so-called Allee effect. We provide a useful geometrical interpretation, and give some examples to illustrate our theoretical results.

In this work we have considered a prey–predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation respectively.

The main purpose of this work is to study the dynamics and bifurcation properties of generic growth functions, which are defined by the population size functions of the generic growth equation. This family of unimodal maps naturally incorporates a principal focus of ecological and biological research: the Allee effect. The analysis of this kind of extinction phenomenon allows to identify a class of Allee’s functions and characterize the corresponding Allee’s effect region and Allee’s bifurcation curve. The bifurcation analysis is founded on the performance of fold and flip bifurcations. The dynamical behavior is rich with abundant complex bifurcation structures, the big bang bifurcations of the so-called “box-within-a-box” fractal type being the most outstanding. Moreover, these bifurcation cascades converge to different big bang bifurcation curves with distinct kinds of boxes, where for the corresponding parameter values several attractors are associated. To the best of our knowledge, these results represent an original contribution to clarify the big bang bifurcation analysis of continuous 1D maps.

In this paper, a Leslie-type predator–prey model with ratio-dependent functional response and Allee effect on prey is considered. We first study the existence of the multiple positive equilibria and their stability. Then we investigate the effect of delay on the distribution of the roots of characteristic equation and obtain the conditions for the occurrence of simple-zero, double-zero and triple-zero singularities. The formulations for calculating the normal form of the triple-zero bifurcation of the delay differential equations are derived. We show that, under certain conditions on the parameters, the system exhibits homoclinic orbit, heteroclinic orbit and periodic orbit.

In the present article, we make an attempt to investigate the effect of two time delays, logistic delay and gestation delay, on an eco-epidemiological model. In the proposed model, strong Allee effect is considered in the growth term of the prey population. We incorporate two time lags and inspect elementary mathematical characteristic of the proposed model such as boundedness, uniform persistence, stability and Hopf-bifurcation for all possible combinations of both delays at the interior equilibrium point of the system. We observe that increase in gestation delay leads to chaotic solutions through the limit cycle. We also observe that the Allee effect play a major role in controlling the chaos. We execute several numerical simulations to illustrate the proposed mathematical model and our analytical findings.

In this paper, we consider the cooperative system

Here, we have proposed a predator–prey model with Michaelis–Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh–Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson–Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.

The present paper mainly investigates the impact of hunting cooperation in a discrete-time predator–prey system through numerical simulations. We show that without hunting cooperation, an increase in the growth rate of prey population produces chaotic dynamics. We also show that hunting cooperation has the potential to modify the well-known period-doubling route to chaos by reverse period-halving bifurcations and makes the system stable. However, very high hunting cooperation can be detrimental and populations go to extinction. We observe that hunting cooperation induces strong demographic Allee effect in the system, where predator population persists due to hunting cooperation and would go to extinction without hunting cooperation. We perform extensive numerical simulations of the system and draw phase portraits, bifurcation diagrams, maximum Lyapunov exponents, two-parameter stability regions. We also observe the occurrence of flip and Neimark–Sacker bifurcations by taking the hunting cooperation rate as a bifurcation parameter.

In this paper, a predator–prey system with double Allee effects and time delay is considered. On the one hand, by using time delay as bifurcation parameter, the stability of the interior equilibrium point is studied. It is shown that under certain assumptions the equilibrium point of the delay model is locally asymptotically stable for all delay values, otherwise, there is a critical value, such that the equilibrium point is locally stable when time delay is less than the critical value, and Hopf bifurcation occurs when the delay crosses the critical value. Numerical simulations are carried out to validate the analytical results. On the other hand, we investigate the following four submodels respectively: the time-delayed system without Allee effect, the time-delayed system with Allee effect on predator, the time-delayed system with Allee effect on prey and the time-delayed system with double Allee effects. Compared with the situation without Allee effects, Allee effect on predator causes a decline of the equilibrium state of the predator, the equilibrium state of the prey and critical time delay are almost unaltered; Allee effect on prey and Allee effect on prey and predator all lead to the descent of the equilibrium state of the predator and prey, inversely, and critical time delay has an obvious rise. A comparison of the four cases indicates that Allee effect can alter the magnitudes of predator density and prey density at future time, the time that predator and prey cost to keeping steady state is also changed.

We complete the global bifurcation analysis of the Leslie–Gower system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations of limit cycles, we prove that such a system can have at most two limit cycles surrounding one singular point.

A scalar population model with delayed growth rate of Allee effect type is considered in this paper. The stability of equilibria and associated supercritical Hopf bifurcations are analyzed. The basins of attraction of the two locally stable equilibria are characterized in terms of parameter values. In particular, when the time delay is large, the basin of attraction of the persistence equilibrium and limit cycle shrinks to a single point, so a global extinction of population occurs as a combined result of Allee effect and time delay.

Predator foraging facilitation or cooperative hunting increases per predator consumption rate as predator density increases. This affects predator extinction in a prey–predator interaction model when the predator density is low. This is an indication of Allee effect in predator’s growth rate. Here, we take a Gause type model with a generalized type II functional response which depends on both prey and predator densities. We also assume that prey’s growth is subjected to Allee effect. Strong Allee effect in prey’s growth rate enhances the stability of the coexisting steady state. A region is found in a two-parameter plane where the coexisting steady state is a global attractor when the prey’s growth is subjected to weak Allee effect. In addition, codimension two bifurcation points (cusp and Bogdanov–Takens points) have also been found in the bifurcation diagram.

This paper deals with the positive solutions of a predator–prey model with additive Allee effect under Neumann boundary conditions. By applying the bifurcation theory, we provide a proof of the existence of local bifurcation solutions and describe the global behavior of these solutions. The result shows that the bifurcation curves can be extended infinitely along $d_2$ in the one-dimensional case. Moreover, the limiting behavior of the steady states is clarified using a shadow system approach. It appears that the shadow system exists with a positive solution and it can go into a terminal point when the parameter $d_2$ is sufficiently large.