This research presents a prey-predator model that integrates the ratio-dependent functional response, the carry-over fear effects (COF), and species diffusion behavior. We demonstrate the synergistic impacts of various processes on population dynamics and spatiotemporal patterns through theoretical analysis and numerical simulations. The findings indicate the presence of local asymptotic stability (LAS) and global asymptotic stability (GAS) conditions within the non-spatial system. When the predation rate exceeds the critical threshold, Hopf bifurcation occurs, leading to periodic oscillations. The Turing instability condition is established within the spatial system, and Hopf bifurcation is demonstrated to produce temporally periodic and spatially uniform oscillation patterns. Numerical simulations validate the theoretical findings and clarify the influence of various factors including spots and stripes on the stability and spatial pattern of the system. The interplay between the fear parameter f and the COE parameter c significantly influences stability: in a high-fear environment, the COE can mitigate stress and restore ecological equilibrium. This study provides a theoretical framework for examining population dynamics linked to multiple mechanisms and proposes innovative strategies for preserving ecosystem stability.

In this paper, we study a predator-prey model with additional food for predator. By using white noise to perturb the natural growth rates and introduce a jump process, we model the corresponding stochastic differential equations. The effect of fear and prey refuge on population dynamics is also considered. First, we use Itô’s formula to prove the existence and uniqueness of a global positive solution and its boundedness. Next, sufficient conditions for the extinction and persistence of both species have been given. Then the stochastic permanence of our system is investigated under some conditions. Our main results demonstrate that sufficiently large white noise could drive both species to extinction. However, Lévy noise enhances the survival of both prey and predator species. Our analytical derivations are justified through numerical simulations which show the reliability of the model from the ecological point of view. In addition, we have investigated the impact of fear effect, prey refuge and the additional food biomass on this model by numerical simulation.

In view of the importance of predator-dependent functional response and fear of prey induced by powerful predators, we construct a delayed prey–predator model with fear and Beddington–DeAngelis functional response. The existence, uniqueness, and global asymptotic stability of equilibrium points are investigated and some criteria are established. Next, Hopf bifurcation analysis is executed, and the critical values of such bifurcation parameters as fear and delay for the determinate system are obtained. Then we extend it to a random environment and study the boundedness of expectation of solutions and the global asymptotic stability. Finally, the main findings are validated by numerical examples. It is worth noting that the specific influences of fear by predator, time delay, and white noise are explored numerically. Simulation figures intuitively exhibit that fear, delay, and white noise bring serious influences on the stability of the system. Fear from predator leads to a lower equilibrium state of prey and predator, and it can change the system stability from unstable to stable after exceeding a certain critical value. The time delay has a significant impact on the system stability by producing Hopf bifurcations accompanied by limit cycles, and even lead to multiple stabilities. Larger white noise can change the system stability from stable to unstable.

The fear preoften leads to changes in the physiological characteristics of the prey. Different stages of prey exhibit different physiological behaviours, such as susceptibility to predator risk, which often leads to Allee effect. Taking into account the influence of these factors, a modified Leslie-Gower predator-prey model with Allee effect and stage structure is constructed in this paper. By use of variational technique and normal form theory, the criteria assuring the appearance of transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation and its direction are all established. Specially, such codimension 2 bifurcations as Bautin bifurcation and Bogdanov-Takens bifurcation are presented. The bubble phenomenon and bistability are detected. All theoretical findings are verified by numerical examples. The biological effects of fear level, Allee effect and stage structure on system stability are analysed.

In this paper, we study a population dynamics model containing one prey and two predators, combining the Smith growth model, the Holling Type II, and the Monod-Haldane functional response. We introduce time lags, nonlinear suppression terms, and Allee effects. We demonstrate the persistence of the system, showing that under specific parameter conditions, the system is able to maintain the population size within positive values and finite intervals. We also prove the global asymptotic stability of the system near the internal equilibrium point and investigate the effect of the time lag parameter on the stability of the system, which shows that the system will change from steady state to periodic oscillation when the time lag parameter exceeds a certain critical value. In the sensitivity analysis section, we develop the study using two approaches: firstly, the direct method reveals that the system shows high sensitivity to small changes in the time lag parameter in the early stage; and then, by combining the Latin Hypercubic Sampling (LHS) method and the Partial Correlation Coefficients (PRCC), we conduct global uncertainty and sensitivity analyses of the parameters in the system in order to assess the effect of different parameters on the model output. Numerical simulations validate our theoretical derivations and demonstrate the complex behavioral patterns of the system under different time lag conditions. This study provides an important theoretical basis for understanding predator-prey dynamics and suggests a strong methodological support for biodiversity conservation and ecosystem management.

The delayed fear of prey towards predator and delayed gestation of predator usually affect the system dynamical behaviors. On the other hand, additional food supplement is a commen measure for people to preserve species from extinction and keep natural balance for continuous development in system control. Thus in this paper, we construct a predator-prey model with predator induced fear, additional food, Holling-II type functional response and time delays, and then generalize it to the random environment. Next, we establish some criteria of the asymptotical stability of equilibrium state, and stationary distribution for stochastic case. Considering the impacts of fear, additional food and time delays, we analyze the Hopf bifurcation of them. Finally by some numerical examples and graphical analysis, we intuitively validate such properties as stability, instability and Hopf bifurcation. Our findings reveal the significant impacts of fear, additional food and time delays on the system stability.

The stability of predator–prey interactions in ecosystems is influenced by both inherent species interactions and external factors. For instance, the presence of additional food, as an external factor, may affect the system. To further explore this, a stage-structured predator–prey model is constructed, incorporating the influences of fear and delay on prey-population growth, which provides additional food for immature predators and facilitates cooperative behavior between mature and immature predators. The analysis evaluates the positivity, boundedness, equilibrium points, local stability around each equilibrium point, and certain bifurcations of the system. Additionally, numerical simulations are provided to correspond with the results of the theoretical analysis. It is observed that an appropriate level of fear contributes positively to system stability. While cooperation among predators can benefit immature predators, it also has the potential to harm the overall system. The introduction of additional food complicates the system dynamics, although it benefits predators, it places prey at a disadvantage. Furthermore, we observe a correlation between the level of fear and the effects of additional food, as well as the capacity of additional food to mitigate the influence of delay.

For food chain system with three populations, direct predation is the basic interaction between species. Different species often have different predation functional responses, so a food chain system with Holling-II response for middle predator and Beddinton–DeAngelis response for top predator is proposed. Apart from direct predation, predator population can significantly impact the survival of prey population by inducing the prey’s fear, but the impact often possesses a time delay. This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence. By use of Lyapunov functional method and bifurcation theory, the positiveness and boundedness of solutions, local and global behavior of species, the system stability around the equilibrium states and various kinds of bifurcation are investigated. Numerically, some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained. It is observed that fear and delay can not only stabilize, but also destabilize the system, which depends on the magnitude of the fear and delay. The system varies from unstable to stable due to the continuous increase of the prey’s fear by middle predator. Small fear induced by top predator or small delay of the prey’s fear can stabilize the system, while they are sufficiently large, the system stability is to be destroyed. Simultaneously, the conversion rate can also change the stability and even make the species to be extinct. Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited, which results in the convergence of the species from one stable equilibrium to another.

Considering the impact of fear levels, Allee effects and hunting cooperation factors on system stability, a Leslie-Gower predator-prey model was formulated. The existence, stability and bifurcation analysis of equilibrium points were studied by use of topological equivalence, characteristic equations, Sotomayor's theorem, and bifurcation theory. The sufficient conditions of saddle-node, Hopf, and Bogdanov-Takens bifurcations were established, respectively. Numerically, the theoretical findings were validated and some complicated dynamical behaviors as periodic fluctuation and multi-stability were revealed. The parameter critical values of saddle-node, Hopf bifurcation, and Bogdanov-Takens bifurcations were established. Biologically, how these factors of fear, Allee effect, and hunting cooperation affect the existence of equilibria and jointly affect the system dynamics were analyzed.

In this article, a stochastic predator-prey system with regime switching and Lévy jumps is formulated. First, the extinction and persistence in mean of each species are investigated by use of the stochastic differential comparison theorem and some stochastic inequalities. Second, the sufficient criteria for the stochastic permanence are established by constructing some Lyapunov functionals. In the end, simulation examples are given to illustrate the theoretical results and reveal the impacts of regime switching, distributed time delays and Lévy jumps on the dynamics, respectively.

Considering the impact of fear levels, Allee effects and hunting cooperation factors on system stability, a Leslie-Gower predator-prey model was formulated. The existence, stability and bifurcation analysis of equilibrium points were studied by use of topological equivalence, characteristic equations, Sotomayor's theorem, and bifurcation theory. The sufficient conditions of saddle-node, Hopf, and Bogdanov-Takens bifurcations were established, respectively. Numerically, the theoretical findings were validated and some complicated dynamical behaviors as periodic fluctuation and multi-stability were revealed. The parameter critical values of saddle-node, Hopf bifurcation, and Bogdanov-Takens bifurcations were established. Biologically, how these factors of fear, Allee effect, and hunting cooperation affect the existence of equilibria and jointly affect the system dynamics were analyzed.

In this paper, we investigate a stochastic prey-predator model with Holling-type IV functional responses, anti-predatory behavior (referring to prey resistance to predator), gestation time delay of prey and Lévy noise. We investigate the existence and uniqueness of global positive solutions through Itô's formulation and Lyapunov's method. We also provide sufficient conditions for the persistence and extinction of prey-predator populations. Additionally, we examine the stability of the system distribution and validate our analytical findings through detailed numerical simulations. Our paper concludes with the implications of our results.

In the predator–prey system, predators can affect the prey population by direct killing and predation fear. In the present study, we consider a delayed predator–prey model with fear and Beddington–DeAngelis functional response. The model incorporates not only the fear of predator on prey with an intraspecific competition relationship, but also fear delay and pregnancy delay. Apart from the local stability analysis of the equilibrium points of the model, we find that time delay can change the stability of the system and cause Hopf bifurcation. Taking time delay as the bifurcation parameter, the critical values of delays in several cases are derived. In addition, we extend it to the random environment and study the stochastic ultimate boundedness of the stochastic process. Finally, our theoretical results are validated by numerical simulation.

Taking into account the delayed fear induced by predators on the birth rate of prey, the counter-predation sensitiveness of prey, and the direct consumption by predators with stage structure and interference impacts, we proposed a prey-predator model with fear, Crowley-Martin functional response, stage structure and time delays. By use of the functional differential equation theory and Sotomayor's bifurcation theorem, we established some criteria of the local asymptotical stability and bifurcations of the system equilibrium points. Numerically, we validated the theoretical findings and explored the effects of fear, counter-predation sensitivity, direct predation rate and the transversion rate of the immature predator. We found that the functional response as well as the stage structure of predators affected the system stability. The fear and anti-predation sensitivity have positive and negative impacts to the system stability. Low fear level and high anti-predation sensitivity are beneficial to the system stability and the survival of prey. Meanwhile, low anti-predation sensitivity can make the system jump from one equilibrium point to another or make it oscillate between stability and instability frequently, leading to such phenomena as the bubble, or bistability. The fear and mature delays can make the system change from unstable to stable and cause chaos if they are too large. Finally, some ecological suggestions were given to overcome the negative effect induced by fear on the system stability.

In the predator-prey system, predators can affect the prey population by direct killing and inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. Therefore, it proposes a predator-prey model with anti-predation sensitivity induced by fear and Holling-Ⅱ functional response in the present paper. Through investigating the system dynamics of the model, we are interested in finding how the refuge and additional food supplement impact the system stability. With the changes of the anti-predation sensitivity (the refuge and additional food), the main result shows that the stability of the system will change accordingly, and it has accompanied with periodic fluctuations. Intuitively the bubble, bistability phenomena and bifurcations are found through numerical simulations. The bifurcation thresholds of crucial parameters are also established by the Matcont software. Finally, we analyze the positive and negative impacts of these control strategies on the system stability and give some suggestions to the maintaining of ecological balance, we perform extensive numerical simulations to illustrate our analytical findings.

In the predator-prey system, predators can affect the prey population by direct killing and inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. Therefore, it proposes a predator-prey model with anti-predation sensitivity induced by fear and Holling-Ⅱ functional response in the present paper. Through investigating the system dynamics of the model, we are interested in finding how the refuge and additional food supplement impact the system stability. With the changes of the anti-predation sensitivity (the refuge and additional food), the main result shows that the stability of the system will change accordingly, and it has accompanied with periodic fluctuations. Intuitively the bubble, bistability phenomena and bifurcations are found through numerical simulations. The bifurcation thresholds of crucial parameters are also established by the Matcont software. Finally, we analyze the positive and negative impacts of these control strategies on the system stability and give some suggestions to the maintaining of ecological balance, we perform extensive numerical simulations to illustrate our analytical findings.

In this paper, the finite-time control of nonlinear impulsive switched positive systems (ISPSs) is studied, where the impulses and bounded disturbance are both fully considered. By designing a novel event-triggered strategy, we present LMI-based conditions for the existence of a controller that guarantees finite-time boundedness (FTB) of the resulting closed-loop systems based on the average dwell time (ADT) method. Further, the finite-time weighted L_2 -gain performance of the considered systems is analyzed. Also, a lower bound of the minimum inter-event interval is obtained to preclude the Zeno behavior. Finally, a numerical example is given to compare the differences between different controllers and to verify the effectiveness of the presented conclusions.

Recent studies have shown that, in addition to direct predation, fear of predators alters the physiological behavior of prey. Based on this fact, this paper investigates a three-species food chain based on ratio-dependent and Beddington-DeAngelis type functional responses, which incorporates fear effects and two gestation delays. The positivity, boundedness and existence of equilibrium points of the system are investigated, and the local stability behavior of the equilibrium points and the occurrence of Hopf-bifurcation when the time lag parameters exceed the critical values are studied by analyzing the corresponding characteristic equations. The main results show that Hopf-bifurcation occurs when the time delay parameters attain the thresholds. Finally, numerical simulations are performed to verify our main results.

Taking into account the impacts of the fear by predator, anti-predation response, refuge for prey, additional food supplement for predator and the delayed fear induced by the predator, we establish a delayed predator-prey model in this paper. We analyze the persistence and extinction of species and the existence and uniqueness of a coexistence fixed point. Particularly, we investigate the local asymptotic stability of the equilibrium by use of the characteristic equation theory of a variational matrix. Applying the Hopf bifurcation theorem, we investigate and obtain the bifurcation thresholds of the parameters of fear, refuge coefficient, the quality and quantity of additional food and the anti-predation delayed response produced by prey. Finally we give some examples to verify our theoretical findings and clarify the detailed influences of these parameters on the system dynamics. The main conclusions reveal that these parameters play an important role in the long-term behaviors of species and should be applied correctly to preserve the continuous development of species.

This work is concerned with a stochastic predator-prey system with S-type distributed time delays, regime switching and Lévy jumps. By use of the stochastic differential comparison theory and some inequality techniques, we study the extinction and persistence in the mean for each species, asymptotic stability in distribution and the optimal harvesting effort of the model. Then we present some simulation examples to illustrate the theoretical results and explore the effects of regime switching, distributed time delays and Lévy jumps on the dynamical behaviors, respectively.