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In ecological systems, the predator-induced fear reduces the reproductive rate of prey species. In this study, we developed a predator–prey model incorporating both positive and negative effects of fear, refuge availability, and supplementary food using the Cosner functional response for prey consumption. Our mathematical and numerical analyses identify various bifurcations, including Hopf, transcritical, saddle-node, Bogdanov–Takens, and cusp bifurcations. Our numerical simulations highlight the critical role of prey birth rate, revealing scenarios ranging from species extinction to predator persistence to species coexistence. We also explore the impact of environmental white noise, investigating noise-induced transitions between different stable states. Using stochastic sensitivity functions (SSFs), we construct confidence ellipses and estimate noise intensity thresholds for state transitions. Our study provides significant insights into the complex dynamics of predator–prey interactions, emphasizing the importance of fear, refuge, and additional food sources.

By considering that the predators have some additional foods apart from the focal prey and the prey take refuge, a predator–prey model with the effect of seasonality in a fluctuating environment is investigated in this study. Gaussian white noises, created by random fluctuation of water level, are introduced on the prey’s growth rate and predator’s mortality rate. Both the deterministic and stochastic systems are analyzed mathematically as well as numerically. Our analytical findings show that the intensity of environmental noise, additional foods and prey refuge play significant roles in survival as well as extinction of both prey and predator populations in the aquatic system. Our numerical results show that the effectual level of additional foods, depending on water level, has positive as well as negative effects on the predator and prey populations. We find that both species strongly persist at certain water level and for the low intensity of noise. However, for an increased value of noise intensity, they persist weakly and then go to extinction. Overall, our findings show that the variations of water level together with additional food and white noise plays crucial roles in persistence and extinction of species in the ecosystem.

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.

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.

This paper considers a Leslie-Gower predator–prey system with Allee effect and prey refuge. By considering the prey refuge constant as a parameter, we analyze the stability of the equilibria in the system, and find that there are abundant dynamic behaviors. It is shown that the model can undergo a sequence of bifurcations including saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension two or three as the parameters vary. Moreover, the model underdoes a degenerate Hopf bifurcation of codimension two and has two limit cycles, where the inner one is stable and the outer one is unstable. Through some numerical simulations, the occurrence of Bogdanov–Takens bifurcation and Hopf bifurcation of codimension two are confirmed.

A fractional-order three-species food chain ecosystem with prey refuge and Holling-II type functional response for predation is proposed and studied. Several sufficient conditions for the existence and uniqueness of the solution of the fractional-order system are obtained. The boundedness of the solution of the system is proven. We investigate the asymptotic behavior of the model by using eigenvalue analysis, and some sufficient conditions on local asymptotic stability of the equilibrium points are given. Furthermore, the conditions for the occurrence of bifurcation at some equilibrium points are presented. We find that the order of the proposed fractional-order ecosystem is one of the parameters for its bifurcation. Several numerical simulations are provided to show the effectiveness of our findings in this paper. Lastly, some new numerical simulations are given to discuss the influence of the half-saturation constant, prey refuge coefficient and the order of fractional-order derivative on the stability of the discussed fractional-order system.

In this paper, a Beddington–DeAngelis prey–predator model with fear effect, refuge and harvesting is investigated. First, the positivity of solutions and boundedness of system are given. Then, the existence and local stability of equilibria of such system are obtained. Next, not only different codimension-one bifurcations, such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation take place, but also Bogdanov–Takens bifurcation of codimension-two occurs as predicted by the center manifold theorem and bifurcation theory. Finally, some numerical simulations are carried out to confirm our theoretical results.

In this paper, we consider a Holling type II predator–prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results.

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov–Takens bifurcation of a predator–prey model with alternative prey and prey refuges, which was proposed by Chen et al. [2023]. The results show that the predator–prey model can undergo a supercritical Hopf bifurcation or a Bogdanov–Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator–prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov–Takens bifurcation is at most two.

In population ecology, the interactions between predators and prey encompass more than just direct predation and killing. Various indirect effects also significantly influence the system’s dynamics. In this paper, we consider a refuge proportional to the prey population and assume that only unsuccessful refuge prey engage in group defense mechanisms. We introduce an exponential response function to model this behavior, taking into account the negative impact of defense measures on the prey growth rate. Subsequently, we analyze the non-negativity, boundedness, and persistence of the system. Furthermore, we study the existence and stability of different types of equilibrium points, and provide theoretical proofs for bifurcations in the system, such as transcritical, pitchfork, saddle-node, Hopf, and Bogdanov–Takens bifurcations. Finally, we verify our theoretical results through numerical simulations. Our study reveals that variations in refuge size and the intensity of group defense can alter the number and stability of equilibrium points, leading to phenomena such as bubbling and periodic oscillations. Additionally, group defense can induce bistability. Both prey refuge behavior and excessive group defense can lead to a reduction in predator numbers and eventually to their extinction. Moreover, compared to prey refuge, group defense as an anti-predator behavior has a greater effect on predators.

This paper, describes a prey–predator fishery model incorporating prey refuge. The proposed model reflecting the dynamic interaction between the net economic revenue and the fishing effort used to harvest the prey species in the presence of predation and a suitable tax. The steady states of the system are determined and the dynamic behavior of the model system is discussed. The occurrence of Hopf bifurcation of the proposed model system is examined through considering density-dependent mortality for the predator as bifurcation parameter. The optimal taxation policy is formulated and solved with the help of Pontryagin's maximal principle. The objective of the paper is to maximize the monetary social benefit as well as prevent the predator species from extinction, keeping the ecological balance. Results are illustrated with the help of numerical examples.

Considering that the ecological system is often deeply perturbed by human exploiting activities, this paper deals with a prey–predator model with prey refuge in which both species are independently harvested. First, some sufficient conditions for global stability of equilibria are obtained, and the existence and uniqueness of limit cycles are established. Our results indicate that over-exploitation would result in the extinction of the population and an appropriate harvesting strategy should ensure the sustainability of the population, which is in line with reality. Furthermore, the existence of bionomic equilibrium is discussed. Finally, the influences of prey refuge and harvesting efforts on equilibrium density values are considered and some numerical simulations are given to illustrate our results.

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

There is a global decline in marine fish abundance due to unsustainable harvesting. An effective harvesting policy can protect the overfished population from possible extinction. In this study, we used a mathematical model characterized by density-dependent refuge protection for herbivorous fish, exhibiting an anti-predator response in presence of a generalist invasive fish. The anti-predator behavior entails predator density-dependent reduced fecundity of the herbivorous fish. The model assumes a continuous threshold harvesting policy (CTHP) for the herbivorous fish and uses the catch-per-unit-effort (CPUE) hypothesis for harvesting the invasive fish. The CTHP allows harvesting of the herbivorous fish only when the density of the herbivorous fish exceeds a specified threshold value, thus ensuring the long-term sustainability of the herbivorous fish stock. The existence and stability of steady-state solutions and the bifurcations of the model are investigated. Our study reveals that the level of apprehension of the herbivorous fish and fishing efforts will play a significant role in the stability of the system. We examine the existence of the bionomic equilibrium and then study the dynamic optimization of the harvesting policy by employing Pontryagin’s maximum principle. We discuss different subsidies and tax policies for the effective management of a sustainable fishery. We use numerical simulations to compare the revenues corresponding to the harvest policies based on maximum sustainable yield (MSY), maximum economic yield (MEY), and optimal sustainable yield (OSY) for inferring an ecologically sustainable and economically viable harvesting policy.

In this paper, a predator–prey model with prey refuge was developed to investigate how prey refuge affect the dynamics of predator–prey interaction. We studied the existence and stability of equilibria, and then derived the sufficient conditions for the bifurcation such as saddle-node, transcritical, Hopf and Bogdanov–Takens bifurcation. In addition, a series of numerical simulations were carried out to illustrate the theoretical analysis, and the numerical results are consistent with the analytical results. Our results demonstrate that prey refuge has a great impact on the predator–prey dynamics.

The authors consider the combined harvesting of a prey-predator system in which both the prey and the predator species obey the law of logistic growth and some preys avoid predation by hiding. The dynamical behaviour of both the exploited and unexploited systems is examined. The possibility of existence of bionomic equilibria is discussed. The optimal harvest policy is studied and the solution is derived in the equilibrium case by using Pontryagin’s maximal principle. The biological as well as economic aspects of the optimal equilibrium solution are discussed. Finally, the model is recast into an S-system and its numerical solutions are derived in several cases using the ESSYNS algorithm.

It has been thought of as a class of models where a generalist predator feeds on two distinct prey species. We wish to examine how prey refuge factors affect prey–predator models when fear effects are present. In order to achieve this, a two-prey–one predator model with prey refuge has been taken into consideration in the presence of the predator’s fear effect on two prey species. Both prey species engage in intra-specific competition. We enhance our model by including a switching mechanism in predation. In order to account for the inherent imperfection of environmental conditions, some parameters have been taken as fuzzy numbers. Analytical and numerical results on the system have been examined in fuzzy sense. The system’s positivity, boundedness, and permanence are examined. The system’s local and global stability analyses have been investigated. Hopf bifurcation analysis around the positive interior equilibrium point has been explored. The stability of the limit cycle of our suggested fuzzy system has been discussed. The system’s numerical simulations have been investigated with pertinent tables and graphical illustrations. When the prey refuge parameter and fear parameter exceed the critical value, the system experiences Hopf bifurcation at the positive interior equilibrium point.

Prey–predator interactions are perhaps the most ordinarily noticed phenomena in the environment. In this article, we have proposed a three-species prey–predator model incorporating three important factors, namely, prey refuge, group defense, and the growth rate of two prey species which is reduced for the amount of fear of the predator species. All the biological parameters of our system have been presented as fuzzy numbers to make them more realistic. The system has been studied analytically and numerically in the fuzzy sense. Model analyses such as positivity, boundedness, and permanence of the system are investigated. Stability analysis at all equilibrium points of the system has been studied. Hopf bifurcation analysis around the positive interior equilibrium point has been discussed. All the numerical simulations of the system are presented with suitable tables and graphical diagrams by using MATHEMATICA and MATLAB. Numerically, we have seen that the fear effect and prey refuge parameter can stabilize the system from chaos to a stable region. The effect of fear and prey refuge on stability has been analyzed in the numerical section. Stable focus and limit cycle analysis are investigated in crisp as well as fuzzy environment. The system undergoes Hopf bifurcation at the positive equilibrium point when the fear parameter $k$ and refuge parameter $m$ cross the threshold value in crisp as well as fuzzy environment.

This paper describes a prey–predator type ecological model with infection in the prey populations. We also consider here a nonlinear functional response for disease transmission and a constant amount of refuge for the sound prey populations. The dynamical behavior of the mathematical model is described from the point of view of stability and bifurcation. A geometric method is also applied to establish the global asymptotic stability at the co-existence equilibrium point. Some computer simulation works have been presented to illustrate the theoretical results.

A predator–prey model with Holling type II functional response incorporating a constant prey refuge and independent harvesting in either species is investigated. Some sufficient conditions of the instability and stability properties to the equilibria and the existence and uniqueness to limit cycles for the model are obtained. We also show that influence of prey refuge and harvesting efforts on equilibrium density values. One of the surprising finding is that for fixed prey refuge, harvesting has no influence on the final density of the prey species, while the density of predator species is decreasing with the increasing of harvesting effort on prey species and the fixation of harvesting effort on predator species. Numerical simulations are carried out to illustrate the obtained results and the dependence of the dynamic behavior on the harvesting efforts or prey refuge.