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 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.

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.

This paper focuses on the construction along with investigation of a novel prey–predator model incorporating prey infertility and the integration of prey infertility with predator-incited fear. The model is built upon a basic prey–predator model, featuring a Michaelis–Menten response. Subsequently, fear effect is introduced to the model, followed by the integration of prey infertility into the concept of logistic growth. Finally, the prey infertility rate is included in the fear effect term. The system is formulated using fractional differential equations. This study covers the well-posedness, stability and Hopf bifurcation analysis of the model, conducting detailed numerical examinations together with biological interpretations. Pioneering and original results are achieved in terms of including the concept of prey infertility into the prey–predator model and considering it together with other biological facts, and incorporating the memory effect using fractional derivatives within the model.

In the present paper, we investigate the impact of fear in a tri-trophic food chain model. We propose a three-species food chain model, where the growth rate of middle predator is reduced due to the cost of fear of top predator, and the growth rate of prey is suppressed due to the cost of fear of middle predator. Mathematical properties such as equilibrium analysis, stability analysis, bifurcation analysis and persistence have been investigated. We also describe the global stability analysis of the equilibrium points. Our numerical simulations reveal that cost of fear in basal prey may exhibit bistability by producing unstable limit cycles, however, fear in middle predator can replace unstable limit cycles by a stable limit cycle or a stable interior equilibrium. We observe that fear can stabilize the system from chaos to stable focus through the period-halving phenomenon. We conclude that chaotic dynamics can be controlled by the fear factors. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to identify the chaotic behavior of the system.

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.

In the present paper, we investigate the impact of fear in an intraguild predation model. We consider that the growth rate of intraguild prey (IG prey) is reduced due to the cost of fear of intraguild predator (IG predator), and the growth rate of basal prey is suppressed due to the cost of fear of both the IG prey and the IG predator. The basic mathematical results such as positively invariant space, boundedness of the solutions, persistence of the system have been investigated. We further analyze the existence and local stability of the biologically feasible equilibrium points, and also study the Hopf-bifurcation analysis of the system with respect to the fear parameter. The direction of Hopf-bifurcation and the stability properties of the periodic solutions have also been investigated. We observe that in the absence of fear, omnivory produces chaos in a three-species food chain system. However, fear can stabilize the chaos thus obtained. We also observe that the system shows bistability behavior between IG prey free equilibrium and IG predator free equilibrium, and bistability between IG prey free equilibrium and interior equilibrium. Furthermore, we observe that for a suitable set of parameter values, the system may exhibit multiple stable limit cycles. We perform extensive numerical simulations to explore the rich dynamics of a simple intraguild predation model with fear effect.

In ecology, the predator’s impact goes beyond just killing the prey. In the present work, we explore the role of fear in the dynamics of a discrete-time predator-prey model where the predator-prey interaction obeys Holling type-II functional response. Owing to the increasing strength of fear, the system becomes stable from chaotic oscillations via inverse Neimark–Sacker bifurcation. Extensive numerical simulations are carried out to investigate the intricate dynamics for the organization of periodic structures in the bi-parameter space of the system. We observe fear induced multistability between different pairs of coexisting heterogeneous attractors due to the overlapping of multiple periodic domains in the bi-parameter space. The basin sets of the coexisting attractors are obtained and discussed at length. Multistability in the predator-prey system is important because the dynamics of the predator and prey populations in the critical parameter zone becomes uncertain.

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results.

Most biological systems have long-range temporal memory. Such systems can be modeled using fractional-order differential equations. The combination of fractional-order derivative and time delay provides the system more consistency with the reality of the interactions and higher degree of freedom. A fractional-order delayed prey–predator system with the fear effect has been proposed in this work. The time delay is considered in the cost of fear; therefore, there are no dynamical changes observed in the system due to time delay in the absence of fear. The existence and uniqueness of the solutions of the proposed system are studied along with non-negativity and boundedness. The existence of biologically relevant equilibria is discussed, and the conditions for local asymptotic stability are derived. Hopf bifurcation occurs in the system with respect to delay parameter. Further, a spatially extended system is proposed and analyzed. Hopf bifurcation also occurs in the extended system due to the delay parameter. Numerical examples are provided in support of analytical findings. Fractional-order derivative improves the stability and damps the oscillatory behaviors of the solutions of the system. Bistability behavior of the system admits stable dynamics by decreasing the fractional-order. Also, chaotic behavior is destroyed by decreasing fractional-order.

In this paper, we propose a Leslie–Gower predator–prey model with strong Allee effect on prey and fear effect on predator. We discuss the existence and local stability of equilibria by making full use of qualitative analytical theory. It is shown that the above system exhibits at most two positive equilibria and it can undergo a series of bifurcation phenomena. We indicate that the dynamical behavior of the model is closely related to the fear effect on predator. In detail, when the fear effect parameter $p=p\ast$, the system will undergo degenerate Hopf bifurcation. There exist two limit cycles (the inner is stable and the outer is unstable). However, when $p=p_{\ast }$, the system will undergo degenerate Bogdanov–Takens bifurcation. Also, by numerical simulation, we conclude that the stronger the fear effect, the bigger the density of prey species. The above shows that fear effect on predator is beneficial to the persistence of the prey species. Our results can be seen as a complement to previous works [González-Olivares et al., 2011; Pal & Mandal, 2014].

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.

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 one type of predator–prey model with simplified Holling type IV functional response by incorporating the fear effect into prey species. The existence and stability of all equilibria of the system are studied. And bifurcation behaviors including saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation of the system are completely explored. Numerical simulation is carried out to illustrate the theoretical analysis. It is shown that the fear effect does affect some dynamic behaviors of the system. Finally, we summarize the findings in a conclusion.

To reduce the chance of predation, many prey species adopt group defense mechanisms. While it is commonly believed that such defense mechanisms lead to positive feedback on prey density, a closer observation reveals that it may impact the growth rate of species. This is because individuals invest more time and effort in defense rather than reproductive activities. In this study, we delve into a predator–prey system where predator-induced fear influences the birth rate of prey, and the prey species exhibit group defense mechanism. We adopt a nonmonotonic functional response to govern the predator–prey interaction, which effectively captures the group defense mechanism. We present a detailed mathematical analysis, encompassing the determination of feasible equilibria and their stability conditions. Through the analytical approach, we demonstrate the occurrence of Hopf and Bogdanov–Takens (BT) bifurcations. We observe two distinct types of bistabilities in the system: one between interior and predator-free equilibria, and another between limit cycle and predator-free equilibrium. Our findings reveal that the parameters associated with group defense and predator-induced fear play significant roles in the survival and extinction of populations.

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results.

This study aims to investigate a diffusive predator–prey system incorporating additional food for predators, prey refuge, fear effect, and its carry-over effects. For the temporal model, the well-posedness and persistence of the system have been discussed. We investigated the existence and the stability behavior of the various equilibria. Furthermore, we explored the bifurcations of codimension-1 including transcritical, saddle-node, and Hopf, concerning the crucial parameters. The system also presents codimension-2 bifurcations such as Bogdanov–Takens and cusp bifurcation along with the global homoclinic bifurcation. We observed the bubbling phenomena, which illustrate the fluctuations in the amplitudes of the periodic oscillations. For the spatiotemporal system, we established the non-negativity and boundedness of the solutions. We derived the conditions for the diffusion-driven instabilities in a confined region with Neumann boundary conditions. Extensive numerical simulations have been conducted to depict the various stationary patterns in Turing space. It is observed that incorporating cross-diffusion divides the bi-parametric plane into various sub-regions and dynamic patterns are analyzed in these different regions. The intricate spatiotemporal dynamics exhibited by prey–predator interactions are crucial for unraveling the intricacies within ecological systems.

We explore a diffusive predator–prey system that incorporates the fear effect in advective environments. First, we analyze the eigenvalue problem and the adjoint operator, considering Constant-Flux and Dirichlet boundary conditions, as well as Free-Flow boundary conditions. Next, we use the principle of comparison to prove the non-negativity of the solution. Our investigation focuses on determining the direction and stability of spatial Hopf bifurcation, with the generation delay serving as the bifurcation parameter. Additionally, we examine the influence of both linear and Holling-II functional responses on the dynamics of the model. Through these analyses, we gain better understanding of the intricate relationship among advection, predation, and prey response in this system.