Fear Of Predators Research Articles

Stability and Hopf-bifurcation analysis of diffusive Leslie–Gower prey–predator model with the Allee effect and carry-over effects

Identification of the various dynamical kinetics relating to prey–predator interconnection is the main goal of theoretical ecologists. In addition to direct killing (the lethal effect) in prey–predator interactions, a number of studies have suggested that there are some indirect effects (non-lethal), such as fear of predation. This non-lethal effect slows down the growth rate of prey and this phenomenon can be carried over to upcoming seasons or generations. The Allee effect also has an impact not only on prey populations but also on predator populations. Considering these, we first propose a modified Leslie–Gower predator–prey model that takes into account the fear-induced carry-over effect (COE), the predator’s Allee effect, and intraspecific competition among predators. The ecologically viable equilibrium points and corresponding local and global stability criteria are derived for the model. Moreover, we establish the conditions for the occurrence of Hopf-bifurcation around coexistence equilibrium and compute the first Lyapunov coefficient to verify the stability of the limit cycle. Secondly, we extend the proposed model to include the spatial variable by considering the spatial movements of the species. For the spatial model with zero flux boundary conditions, the criteria for Turing instability and Hopf-bifurcation are derived. Numerical simulations aim to verify the theoretical results. Our study suggests that the parameters related to fear-induced COE, the predator’s Allee effect, and intraspecific competition have a greater impact on system dynamics.

Hopf bifurcation and patterns formation in a diffusive two prey-one predator system with fear in preys and help

Recognizing the relationship between the spatial patterns in species concentrations and ecological heterogeneity is crucial for understanding demographics and species governance in a given domain, as ecological patterning processes are believed to be imitated in real ecosystems. In this present article, we have considered a two-prey-one-predator system with Holling type-II functional response where both the preys have fear of predator and compete in the absence of predator. Additionally, we have assumed that both the prey help each other when the predator attacks them. We have incorporated self-diffusion in this system under Neumann boundary conditions. The existence and stability conditions of different equilibria are discussed. We have studied the Hopf bifurcation around positive offset and derived the stability and direction of periodic solution. We have done a series of numerical simulations to ensure our theoretical finding. Diffusion-driven instability conditions are obtained. Different instability regions and fascinating patterns, such as spots, mixtures, and stripes, are depicted. Spatiotemporal dynamics also reflects that prey populations are located in isolated areas of low population concentration with increasing level of fear, whereas an increase in handling time forms lower-density stripes. But with increasing help, prey becomes more concentrated in some stripe regions. In the Hopf-Turing region, it is observed that the diffusion coefficient of predators can stabilize the system. Overall, pattern creation in predator-prey systems can help anticipate long-term system behaviour and analyze the influence of ecological factors on system dynamics.

IMPACT OF HUNTING COOPERATION AND FEAR EFFECT IN A GENERALIST PREDATOR–PREY MODEL

Predator foraging facilitation or hunting cooperation and the antipredator behavior of prey are essential mechanisms in evolutionary biology and ecology and may strongly influence the predator–prey dynamics. In a real-world scenario, this behavioral tendency is well documented, but less is known about how it could affect the dynamics between predator and prey. Here, we investigate the impact of the fear of predator on prey and the hunting cooperation in predator on the predator–prey dynamics, where the predator is assumed to be of generalist type. We observe that without fear, even with the high level of hunting cooperation, both populations may coexist, though the increasing level of hunting cooperation reduces the prey density at coexistence equilibrium. Moreover, increasing level of fear also destabilizes the system with and without hunting cooperation. Further, in the presence of hunting cooperation and fear effect, the model shows three different types of bistability phenomena: bistability between two coexisting equilibria, bistability between coexisting equilibria and prey-free equilibrium, and bistability between stable limit cycle and coexisting equilibria. In addition, saddle-node, Hopf, transcritical bifurcation of codimension one, Bautin (generalized Hopf), Bogdanov–Takens, and cusp bifurcation of codimension two are observed.

Effects of predation risk on parasite-host interactions and wildlife diseases.

Landscapes of fear can determine the dynamics of entire ecosystems. In response to perceived predation risk, prey can show physiological, behavioral, or morphological trait changes to avoid predation. This in turn can indirectly affect other species by modifying species interactions (e.g., altered feeding), with knock-on effects, such as trophic cascades, on the wider ecosystem. While such indirect effects stemming from the fear of predation have received extensive attention for herbivore-plant and predator-prey interactions, much less is known about how they alter parasite-host interactions and wildlife diseases. In this synthesis, we present a conceptual framework for how predation risk-as perceived by organisms that serve as hosts-can affect parasite-host interactions, with implications for infectious disease dynamics. By basing our approach on recent conceptual advances with respect to predation risk effects, we aim to expand this general framework to include parasite-host interactions and diseases. We further identify pathways through which parasite-host interactions can be affected, for example, through altered parasite avoidance behavior or tolerance of hosts to infections, and discuss the wider relevance of predation risk for parasite and host populations, including heuristic projections to population-level dynamics. Finally, we highlight the current unknowns, specifically the quantitative links from individual-level processes to population dynamics and community structure, and emphasize approaches to address these knowledge gaps.

Fear of predators reduces body and physiological condition affecting offspring survival and the ‘quality’ of the survivors

Abstract Predators affect the survival of developing young by eating them. More recently, the fear of being eaten has been uncovered as a powerful driver of offspring survival reducing recruited offspring numbers by 53%, but the mechanisms driving these effects are not well understood. We exposed song sparrows to predator or non‐predator playbacks throughout three breeding seasons. We followed offspring as eggs through development and into maturity. We quantified the repercussions of being reared in a fearful environment on body (fat, mass, size) and physiological condition using 12 different measures categorized into six systems (oxygen carrying capacity, energy reserves, immune function, antioxidant protection, oxidative stress/damage, glucocorticoids). We further tracked offspring survival in the nest to maturity and determined which measures of condition significantly predicted survival. We report that fear of predators affected offspring condition at all stages of life, with survival consequences. Predator‐playback offspring hatched from heavier eggs (8%) but quickly lost this advantage and never regained it. Nestlings reared by ‘frightened’ parents were as heavy and much leaner than non‐predator playback nestlings. This pattern extended throughout development into maturity where offspring reared by frightened parents were up to 31% leaner and 5% lighter. Predator‐playback nestlings also were evidently in poorer physiological condition, having shorter telomeres, while late dependency stage fledglings carried relatively less uric acid and more haptoglobin perhaps indicative of chronic susceptibility to infection. Only body condition (mass and fat) predicted survival during peak mortality periods (nestling and early fledging), with the leanest lightest offspring from the predator‐playback treatment being least likely to survive. Thereafter, mass predicted survival irrespective of playback treatment. Similarly, at this later stage, offspring with less telomere loss and high uric acid levels survived best regardless of playback group. Our findings reveal that fear of predators is a potent form of early life adversity which persistently alters the condition of offspring affecting early stage survival and the ‘quality’ of the survivors throughout their life. Development is fuelled by food and fear‐induced reductions in parental provisioning at the earliest stages likely instigated these effects, with consequences on the population growth rate. Read the free Plain Language Summary for this article on the Journal blog.

Dangerous neighbours: Birds and bird-eating bats sharing tree cavities.

Mounting evidence indicates the non-consumptive effects of predators significantly impact prey physiology, ecology and behaviour. Passerine birds experience adverse effects on nesting and reproductive success when in proximity to predators. Fear of predators is context-dependent and influenced by hunting habitats and foraging strategies. While some bat species prey on birds, the greater noctule (Nyctalus lasiopterus) stands out by specialising in avian prey, especially during peak bird migration. N. lasiopterus is thought to seize avian prey in flight, but direct evidence is lacking. If birds were taken from nests, they would likely avoid nesting near these bats. However, no observations support this view. This study documents the successful reproduction of Eurasian blue tits (Cyanistes caeruleus) nesting alongside a colony of approximately 25 greater noctules. This bird species is a prey species for greater noctules in Italy. Over about 1 month (April-May 2023), we observed parent birds provisioning food to chicks, with at least two chicks alive and fed outside the tree cavity by the end of the period. While acknowledging the limitations of a single observation, we propose that this previously unknown behaviour indirectly supports the idea that greater noctules only capture avian prey in flight, not within confined spaces. This observation challenges the perception that these bats pose a threat when sharing roosting spaces in trees, as evidenced in our observed case. We hope this novel observation inspires future research on variations in bird nesting behaviour and reproductive success in the presence of bird-eating bats, as well as an assessment of the long-term impact on population dynamics and reproductive success of nesting birds sharing such roosting spaces.

Bifurcation analysis of autonomous and nonautonomous modified Leslie-Gower models.

In ecological systems, the predator-induced fear dampens the prey's birth rate; yet, it fails to extinguish their population, as they endure and survive even under significant fear-induced costs. In this study, we unveil a modified Leslie-Gower predator-prey model by incorporating the fear of predators, cooperative hunting, and predator-taxis sensitivity. We embark upon an exploration of the positivity and boundedness of solutions, unearthing ecologically viable equilibrium points and their stability conditions governed by the model parameters. Delving deeper, we unravel the scenario of transcritical, saddle-node, Hopf, Bogdanov-Takens, and generalized-Hopf bifurcations within the system's intricate dynamics. Additionally, we observe the bistable nature of the system under some parametric conditions. Further, the nonautonomous extension of our model introduces the intriguing interplay of seasonality in some crucial parameters. We establish a set of sufficient conditions that guarantee the permanence of the seasonally driven system. By conducting a numerical study on the seasonally forced model, we observe a myriad of phenomena manifesting the predator-prey dynamics. Notably, periodic solutions, higher periodic solutions, and bursting patterns emerge, alongside intriguing chaotic dynamics. Specifically, seasonal variations of the predator-taxis sensitivity and hunting cooperation can lead to the extinction of prey species and even the control of chaotic (higher periodic) solutions through the generation of a simple periodic solution. Remarkably, the seasonal forcing has the capacity to govern the chaotic behavior, leading to an exceptionally quasi-periodic arrangement in both prey and predator populations.

A generalist predator–prey system with the effects of fear and refuge in deterministic and stochastic environments

In the present study, we investigate the dynamics of a generalist predator–prey system by considering that the fear of predators suppresses the birth rate and enhances the intraspecies competition of the prey species. Also, we assume that a constant proportion of the prey population is taking refuge, which results in a reduction in the predation pressure. We consider Cosner functional response to describe the interactions between prey and predator species. We rigorously analyze the proposed model analytically as well as numerically. We see that the system exhibits multi-stable configurations under identical ecological conditions by allowing different bifurcation scenarios, including saddle–node, Hopf and transcritical bifurcations with varying degrees of fear, on the prey’s reproduction and intraspecies competition, induced by the predator species. The emergence of multistability scenarios in the system is also stimulated by the birth rate of prey species and the growth in predator species due to the additional food sources. We modify our proposed model to its stochastic counterpart by considering the impact of environmental white noises on the natural death rates of both prey and predator species. The simulation results for the stochastic system demonstrate that the fluctuations in the densities of prey and predator populations due to stochastic perturbations are highly sensitive to the intensities of the environmental noises. Additionally, we find that the stochastic system showcases noise-induced transitions whenever the deterministic system exhibits bistability.

Predation risk by largemouth bass modulates feeding functional responses of native and non-native crayfish

Context-dependency is prevalent in nature, challenging our understanding and prediction of the potential ecological impacts of non-native species (NNS). The presence of a top predator, for example, can modify the foraging behaviour of an intermediate consumer, by means of non-consumptive effects. This raises the question of whether the fear of predation might modulate consumption rates of NNS, thus shaping the magnitude of ecological impacts. Here, we quantified the functional feeding responses of three non-native crayfish species – red swamp crayfish Procambarus clarkii, rusty crayfish Faxonius rusticus and virile crayfish Faxonius virilis – compared to the native analogue signal crayfish Pacifastacus leniusculus, considering the predation risk imposed by a top fish predator, the globally invasive largemouth bass Micropterus salmoides. We applied the comparative functional response (FR) approach using snails as prey and exposing crayfish to water containing predator and dietary chemical cues or not. All crayfish species presented a destabilising Type II FR, regardless of the presence of chemical cues. Predation risk resulted in significantly longer handling times or lower attack rates in non-native crayfish; however, no significant differences were observed in signal crayfish. We estimated per capita impacts for each species using the functional response ratio (FRR; attack rate divided by handling time). The FRR metric was lower for all crayfish species when exposed to predation risk. Rusty crayfish demonstrated the highest FRR in the absence of chemical cues, followed by signal crayfish, virile crayfish and red swamp crayfish. By contrast, the FRR of signal crayfish was nearly twice that of rusty crayfish and virile crayfish and ten times greater than red swamp crayfish when chemical cues were present. The latter result agrees with the well-recognised ecological impacts of signal crayfish throughout its globally-introduced range. This study demonstrates the importance of considering the non-consumptive effects of predators when quantifying the ecological impacts of intermediate non-native consumers on prey. The direction and magnitude of the modulating effects of predators have clear implications for our understanding of NNS impacts and the prioritisation of management actions.

Fear, concealment, and time of day interactively predict group size of a common ungulate

AbstractAnimals commonly form groups with conspecifics. Hypotheses on the drivers of group size often (but not always) pertain to fear of predation and food availability or quality. However, for a more complete understanding of group forming, multiple, possibly interacting mechanisms should be considered. We used camera traps, detectability estimates from distance sampling, and hierarchical Bayesian modeling to index group size and test multiple additive and interacting group‐formation hypotheses in white‐tailed deer (Odocoileus virginianus), within Indiana, USA. We found a strong relationship between our index of group size and (1) a three‐way interaction between spatiotemporal pulses in fear of natural predation risk and anthropogenic development and (2) a three‐way interaction between the local cover (defined as the amount of vegetative cover in the immediate area of the camera), the area of concealment (defined as forests or wetlands) within the larger landscape, and time of day. Specifically, we documented larger groups in areas near anthropogenic development, in areas with high predator use intensity, and when predators were active. Additionally, we found larger groups during dawn and dusk when deer were most active in locations with sparse local cover when the area of concealment in the surrounding landscape was small, and within dense local cover when the area of concealment in the surrounding landscape was large. Our methods can aid future behavioral and ecological studies, as we present a clear and easily replicable strategy that infers group membership from spatiotemporal proximity, corrects for differences in detectability, and enables sampling across large spatial and long temporal scales. Our findings indicate that fear of natural predators and anthropogenic disturbance interactively explain variation in group size.

Detection of Hopf bifurcations induced by pregnancy and maturation delays in a spatial predator–prey model via crossing curves method

We investigate the stability switches behaviors due to Hopf bifurcations triggered by maturation and pregnancy delays for a diffusive predator–prey model under the joint impacts of fear and Allee effect of predators. We firstly carry out the linear analysis of the proposed model by applying the recently developed crossing curves method to cope with the corresponding characteristic equation incorporating wave number as well as delay-dependent parameters. We then procure the Hopf bifurcation curves through which delays pass, the stability of the associated constant equilibrium can change accordingly. Thereupon, we deduce the normal form regarding Hopf bifurcation and thus clarify its properties. Finally, we present a numerical example and perform some simulations to verify the accuracy of the obtained results. It is shown that both the maturation and pregnancy periods can respectively induce stability switches.

Appearance of chaos and bi-stability in a fear induced delayed predator–prey system: A mathematical modeling study

Since many years ago, it has been only a priori assumption that the diversity of ecological demography is largely influenced by the direct interactions of predator and prey species. But over the years, it has also been recognized that an indirect effect can also affect the system more strongly. The effect of fear on prey species is that it restricts physical activities and feeding times of prey thereby reducing interactions with their predators and leading to an intraspecific competition among predators. By these motives, we investigate a delay-induced model inducing fear of predators on the prey population. Existence of feasible steady points, well posedness of the solutions, local & global stability, subcritical & supercritical bifurcation are well presented. The results show that predator fear reduces both prey and predator populations, but does not affect the steady state of the system. As delay increases within a specific range, it changes the dynamics of the system in a stable–unstable–stable sequences. A small delay shifts an unstable interior equilibrium to a predator-free stable equilibrium. The main result of this study is that a large delay produces chaotic dynamics when cost of fear is high and prey birth is low, whereas a small delay creates bi-stable dynamics when cost of fear is low and prey birth is high. Furthermore, by increasing the competitiveness among the predators, the chaos can be completely controlled. We examine that small fear has a greater effect while relatively large fear has less effect. Delay parameter is responsible for appearing stability switch more than once. Extensive numerical simulations are also carried out for various parameters.

Role reversal in a stage-structured prey-predator model with fear, delay, and carry-over effects.

The present work highlights the reverse side of the same ecological coin by considering the counter-attack of prey on immature predators. We assume that the birth rate of prey is affected by the fear of adult predators and its carry-over effects (COEs). Next, we introduce two discrete delays to show time lag due to COEs and fear-response. We observe that the existence of a positive equilibrium point and the stability of the prey-only state is independent of fear and COEs. Furthermore, the necessary condition for the co-existence of all three species is determined. Our system experiences several local and global bifurcations, like, Hopf, saddle-node, transcritical, and homoclinic bifurcation. The simultaneous variation in the attack rate of prey and predator results in the Bogdanov-Takens bifurcation. Our numerical results explain the paradox of enrichment, chaos, and bi-stability of node-focus and node-cycle types. The system, with and without delay, is analyzed theoretically and numerically. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also derived. The cascade of predator attacks, prey counter-attacks, and predator defense exhibit intricate dynamics, which sheds light on ecological harmony.

Uncertain parameter based dynamical model among the interaction of immature prey, mature prey and predator

ABSTRACT In this paper, a mathematical model for the interaction of immature prey, mature prey, and predators has been developed. Here, most of the model parameters are considered fuzzy numbers. It is considered that the growth of immature prey depends on mature prey as well as the fear of predators. It is also considered that after remaining in the immature stage for a while, some immature prey transforms into mature prey. It is assumed that predators consume immature prey as well as mature prey. The intra-species competition among the mature prey has been considered. The boundedness of the solution of the proposed model is studied. Different equilibrium points have been determined, and the stability of the system has been investigated. The global stability of the system around the interior equilibrium point has been discussed through a geometric approach. From the analysis of the model, it is found that uncertain values of the model parameters have greatly impacted the solution of the model. It is observed that the increasing rate of transfer of immature prey to mature prey may destabilize the system. It is found that the system may become stable due to the increase in intra-species competition among mature prey.

Easier said than done! Organic farmers consider free-ranging important for laying hen welfare but outdoor areas need more shelter – important gaps between research and practice

ABSTRACT 1. The aim of the present study was to investigate the design and management of free-range areas and their use by birds on commercial organic laying hen farms in Sweden and to document farmers’ perspectives on outdoor access for poultry. 2. Eleven Swedish organic laying hen farms were visited. The farmers were interviewed about general farm management, bird health and behaviour and outdoor access. The free-range areas were assessed in terms of proportion covered by protective (high) vegetation and any artificial shelters provided. The numbers of hens ranging at different distances from the house were recorded twice during the day. 3. The outdoor area within 250m from the house contained 0–5% vegetation cover on six of the farms and at least 80% pasture on seven farms. On 10 farms, no more than 13% of the flock was observed outdoors. Of the hens observed in the free-range area, the median proportion ranging within 20m from the house or veranda per observation event was 99% (IQR=55–100%), confirming reports by the farmers. 4. Free-range access was considered important by all farmers, primarily for welfare reasons and most agreed that protective vegetation cover and/or artificial shelters were important in encouraging free-ranging. However, there was marked variation among the farmers in their suggestions on how to attract hens outside.

Fear of predators alters herbivore regulation of soil microbial community function.

Fear of predation can affect important ecosystem processes by altering the prey traits expression that, in turn, regulates the quantity and quality of nutritional inputs to soil. Here, we aimed to assist in bridging a knowledge gap in this cascading chain of events by exploring how risk of spider predation may affect grasshopper prey performances, and the activity of various microbial extracellular enzymes in the soil. Using a mesocosms field-experiment, we found that grasshoppers threatened by spider predation ate less, grew slower, and had a higher body carbon to nitrogen ratio. Herbivory increased activity of all microbial extracellular enzymes examined, likely due to higher availability of root exudates. Predation risk had no effect on C-acquiring enzymes but decreased activity of P-acquiring enzymes. We found contrasting results regarding the effect of predation on the activity of N-acetyl-glucosaminidase and leucine arylamidase N-acquiring enzymes, suggesting that predation risk may alter the composition of N-inputs to soil. Our work highlighted the importance of soil microbial enzymatic activity as a way to predict how changes in the aboveground food-web dynamics may alter key ecosystem processes like nutritional-cycling.

Hopf bifurcation in a delayed prey–predator model with prey refuge involving fear effect

This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points.

Spatiotemporal dynamics of a predator-prey system with fear effect

Recent field experiments on vertebrates show that though mere presence of a predator causes a dramatic change in prey demography, the fear of predators increases the survival probability of prey leading to a cost of prey production. Based on the experimental findings, we proposed and analyzed a mathematical model that incorporates the fear-induced birth reduction in the prey population due to presence of predator. A modified and more realistic fear function is proposed in this study. Qualitative behavior of the model is performed including positivity and boundedness of solutions, existence of critical points and their local stability analysis, existence of transcritical and Hopf bifurcation. We analyzed Hopf bifurcation with respect to the prey growth rate and the level of fear. Transcritical bifurcation is analyzed by varying the prey growth rate. Distribution of the population of interacting species in a large scale natural system is heterogeneous and subject to alter for different reasons. Thus, we investigate how behavioral modification in prey population due to fear for predators and mutual interference among predator species can create various spatiotemporal pattern formation in population distribution. In the spatially extended system, we provide a detailed stability analysis and obtain the conditions for Turing instability. Numerical simulations are performed to validate analytical results for both non-spatial and spatial models. Warm spot patterns are obtained by considering three different types of initial data and discussed the biological significance of these patterns for the two-dimensional spatial model. Our numerical simulation demonstrates that the fear effect in a diffusive predator-prey system with mutual interference between predators may exhibit more complicated dynamics.

Stability and bifurcation analysis of delayed tri‐trophic food chain model in poisoned environment with fear effect and additional food

One of the most important goals of theoretical ecologists is to find a strategy for controlling the chaos in ecological models to maintain healthy ecosystems. We investigate the influence of fear and the supply of additional food to predators in a delayed tri‐trophic food chain model, where interference among the species is determined by Holling type‐II functional response. The proposed model incorporates the fact that the growth rates of prey and middle predators are decreased due to the fear of middle and special predators, respectively. The additional food is supplied to both predators by either nature or external agencies. Furthermore, the poisoned prey in the predator's environment is taken into account. In the proposed model, we use two time delays: one for the growth term of the middle predator and another for the gestation delay of the special predator. We determine the conditions for the existence of ecologically feasible equilibrium points and their local and global stability. In addition, we establish the conditions for the existence of Hopf bifurcation around interior equilibrium to seek periodic behavior in nondelayed and delayed models. Numerical investigations are performed to justify the proposed theoretical findings through phase portraits, time series of solutions, and bifurcation diagrams. We observe that the chaotic dynamics of a tri‐trophic food chain model can be controlled by the proper choice of the fear effect and by supplying additional food parameters. As a result, our findings provide more ecological insights into the dynamics of the delayed tri‐trophic food chain model with the fear effect and additional food supply.

Dynamic analysis and bifurcation control of a delayed fractional-order eco-epidemiological migratory bird model with fear effect

In this paper, a new delayed fractional-order model including susceptible migratory birds, infected migratory birds and predators is proposed to discuss the spread of diseases among migratory birds. Fear of predators is considered in the model, as fear can reduce the reproduction rate and disease transmission rate among prey. First, some basic mathematical results of the proposed model are discussed. Then, time delay is regarded as a bifurcation parameter, and the delay-induced bifurcation conditions for such an uncontrolled system are established. A novel periodic pulse feedback controller is proposed to suppress the bifurcation phenomenon. It is found that the control scheme can successfully suppress the bifurcation behavior of the system, and the pulse width can be arbitrarily selected on the premise of ensuring the control effect. Compared with the traditional time-delay feedback controller, the control scheme proposed in this paper has more advantages in practical application, which not only embodies the advantages of low control cost and easy operation but also caters to the periodic changes of the environment. The proposed control scheme, in particular, remains effective even after the system has been disrupted by a constant. Numerical simulation verifies the correctness of the theoretical results.