Narrow Results

This piece of work circles around the investigation of a predator–prey model that incorporates nonlinear harvesting and group defense, both of which are important phenomena occurring in the ecosystem. To analyze the proposed model, both analytical and numerical methods are employed. The model experiences a series of bifurcations, including saddle-node, Hopf and Bogdanov–Takens. The system exhibits bifurcations of codimension one and two but no bifurcation of codimension three emerged. Phase portrait diagrams are also presented to confirm the validity of both the analytical and numerical findings.

In this paper, we establish a predator–prey model with focus on the Allee effect and prey group defense. The positivity and boundedness of the model, existence of equilibrium point, and stability change caused by Allee effect are studied. Bifurcation (transcritical bifurcation, Hopf bifurcation) analysis is discussed, and the direction of Hopf bifurcation is determined by calculating the first Lyapunov number. Then we introduce delay into the original model and consider the influence of delay on the stability of the model. By selecting delay as the bifurcation parameter, we obtain the existence conditions of Hopf bifurcation and the direction of Hopf bifurcation. Finally, we verify the theoretical analysis by numerical simulation. Considering both the Allee effect and the prey group defense, the dynamic behavior near the origin becomes more complex than only considering Allee effect or prey group defense in the model. Allee effect can bring the risk of extinction and the change of stability, and the delay effect can make the stable coexistence equilibrium unstable and lead to periodic oscillation.

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.

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.

In this paper, we focus on some important aspects of model building. The discussion specifically concerns the case of predator–prey interactions. We introduce here two models whose slight difference lies just in the way predators survive. In the former, they are taken to feed only on the modeled prey, i.e., to be specialists; in the second one, they are generalists, i.e., they can survive on other not explicitly modeled food resources.

But our main focus is on the prey, that may disappear due to the Allee effect, if reduced to very low numbers. On the other hand, they also exhibit herd behavior. Our main aim is the discussion of the issues of mathematical modeling of such situation. We show on this example that modeling requires much more than equations patching from different systems.

The analysis of the models indicates that the ecosystem may collapse in the case of specialist predators. If the predators have other feeding resources, they instead can thrive. Both types of models exhibit bistability between the prey-free state and coexistence.

Ecological systems often exhibit complex dynamics due to interactions between predator and prey species, which can be influenced by both direct predation and indirect effects like fear and population thresholds. We introduce a predator–prey model that incorporates non-consumptive effects such as the Allee effect, group defense, and prey refuge. These non-consumptive effects are triggered by predator fear, along with a predator-taxis sensitivity parameter. We explore interactions between fear and defense mechanisms. Our ODE model’s comprehensive mathematical analysis addresses the positivity and boundedness of solutions, species extinction, and the existence and stability of equilibria, including various local bifurcations. We present a one-parameter bifurcation analysis focusing on the Allee parameter and predator-taxis sensitivity. This analysis reveals multiple interior equilibrium points with varying stability, occurrences of saddle-node, transcritical, and Hopf bifurcations, as well as instances of bistability and tristability. We find that a strong Allee effect initially supports species coexistence through tristability. However, beyond a certain threshold, the Allee parameter leads to unsustainable coexistence. Similarly, initial predator sensitivity may encourage coexistence through multiple stable states, but excessive sensitivity makes coexistence infeasible. The outcomes — ranging from complete extinction to predator extinction — depend critically on initial population densities. Further, we delve into the spatial dynamics of the associated reaction–diffusion prey–predator interaction model that incorporates cross-diffusion effects. We establish conditions for stability and Turing instability, validate through numerical simulations. These simulations reveal diverse spatial patterns such as irregular mixtures of spots, and cluster of spots like patterns that underscore the influence of time iteration, cross-diffusion coefficients and the Allee threshold on spatial density distribution of species.

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.

In this paper, an integrated pest management Filippov model with group defense behavior is established, which takes the population density of pests as the control index of integrated pest management. First, under the condition that both subsystems have a globally asymptotically stable equilibrium, the dynamics of the established model are systematically analyzed, including the sliding mode dynamics, the existence and global stability of the real, virtual and pseudo equilibrium, as well as boundary-node and boundary-focus bifurcation. Next, we study the complex dynamics in the Filippov model when an unstable node (focus) or a stable limit cycle occurs in the subsystem by using numerical simulations. The results show that although there are no closed orbits in subsystem, a stable periodic solution may exist for Filippov system after switching perturbation. Finally, we conclude that the group defense behavior of pest makes it harder to control.