Narrow Results

A delayed diffusive predator–prey model with herd behavior subject to Neumann boundary conditions is studied both theoretically and numerically. Applying Hopf bifurcation analysis, we obtain the critical conditions under which the model generates spatially nonhomogeneous bifurcating periodic solutions. It is shown that the spatially homogeneous Hopf bifurcations always exist and that the spatially nonhomogeneous Hopf bifurcations will arise when the diffusion coefficients are suitably small. The explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorems for partial functional differential equations.

In this paper, we have investigated a diffusive predator–prey model with herd behavior. Also, we considered that the mortality of predators is linear as well as quadratic. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive numerical simulations, we obtain non-Turing spatiotemporal patterns in the model with linear mortality of predators, and Turing pattern formation, namely, spotted pattern and mixed pattern (spots-stripes) in model with quadratic mortality of predators. The results focus on the effect of the changing mortality rates of predator in pattern dynamics of a diffusive predator–prey model and help us in the better understanding of the dynamics of the predator–prey interaction in real environment.

In this paper, we consider a delayed diffusive predator–prey model with Leslie–Gower term and herd behavior subject to Neumann boundary conditions. We are mainly concerned with the impact of time delay on the stability of this model. First, for delayed differential equations and delayed-diffusive differential equations, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated respectively. It is observed that when time delay continues to increase and crosses through some critical values, a family of homogeneous and inhomogeneous periodic solutions emerge. Then, the explicit formula for determining the stability and direction of bifurcating periodic solutions are also derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are shown to support the analytical results.

In this paper, we propose a diffusive predator–prey model with herd behavior in prey and pack predation behavior in predator under the homogeneous Neumann boundary condition. Due to the pack predation-herd behavior, the predator–prey interaction occurs only at the outer edge of each group. First, we analyze the existence and local stability of the equilibria of temporal model using vector field analysis and characteristic equations. Second, we deduce the conditions under which Turing instability occurs with the help of linear stability analysis. Then, using standard multiple-scale analysis, we derive the amplitude equations for the excited modes. By numerical simulations, we find that the model exhibits complex pattern replication by varying the value of parameters $a$ and $d_{21}$, and study the relationships between the average density of both prey and predator and these two parameters.

In this paper, we study the dynamics of an ecoepidemic competition system where the individuals of one population gather together in herds with a defensive strategy, showing social behavior, while another predator population is subject to a transmissible disease and behaves individually. By analyzing the existence and stability of the equilibria of the system, we find that the relatively isolated population can be eradicated, or the population with group defense can live alone eventually under some constraints. Infected individuals end up in two possible situations. In the first case, the disease is eventually eliminated, meaning that only healthy and group-defense individuals in the system can survive. In the other case, the spread of the disease is controlled and eventually all three individuals can coexist. We also conduct a correlation analysis using competition parameter and recovery rate of disease as birfurcation parameters in order to study the transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation. The long-term dynamics of the boundary and interior equilibria are demonstrated by numerical simulations.

In this paper, we study a commensal model with Allee effect and herd behavior. Firstly, the stability of all possible equilibria are investigated. Secondly, using the Sotomayor’s theorem, we prove the existence of saddle-node bifurcation and pitchfork bifurcation. Finally, all the theoretical predictions on the bifurcation are verified by numerical simulations.

In prey–predator interaction, many factors, such as the fear effect, Allee effect, cooperative hunting, and group behavior, can influence the population dynamics. Hence, studying these factors in prey–predator makes the model more realistic. In this paper, we have proposed the prey–predator model having herd and Allee effect in prey population, where predators follow hunting cooperation. We have employed temporal analysis to examine the role of the Allee effect and hunting cooperation. Furthermore, we have extended the analysis to spatiotemporal analysis to examine the role of dispersal and the type of spatial structure formed by the population due to random movement. We first discuss the proposed model’s existence and positivity, then the stability of the existing equilibrium points through Routh–Hurwitz criteria. The temporal analysis is carried out through Hopf-bifurcation at the coexistence equilibrium point by considering the Allee threshold ($\alpha$), hunting cooperation ($\gamma$), and attack rate ($\beta$) as controlled parameters. With the addition of diffusion to the model, we examine the spatial model stability and derive the Turing instability condition, which will give rise to various Turing patterns. Finally, numerical simulations are performed to validate the analytical results. The theoretical study and numerical simulation results demonstrate that the Allee effect, hunting cooperation, and diffusion coefficient are sensitive parameters to the model’s stability.

The herd behavior of prey plays an important role in the predator–prey system. In this paper, we analyze the dynamical properties of a discrete predator–prey model with herd behavior. We found that due to the introduction of herd effects, this model exhibits complex dynamical behavior such as codimension-1 bifurcation (Flip and Neimark–Sacker bifurcations) and codimension-2 bifurcation (1:2, 1:3, and 1:4 strong resonance). Using the center manifold theorem and normal form theory, we obtained the critical conditions that may lead to changes in the stability of the system and even the emergence of new stable states or oscillating behaviors. Numerical simulations not only validate theoretical proofs but also reveal the complex and rich dynamical behavior of the discrete model.

The new idea of group defense as recently introduced by the author in the context of two interacting populations is in this paper applied to communities subject also to a disease. The system is formulated with the bare minimum of interactions among all the populations involved in order to highlight the effects of the nonlinearity describing the defense mechanism. A key parameter identified in the purely demographic model, which completely describes its outcomes, is seen here to have an important role also, in that it is dropping below a threshold prevents the disease from invading the environment and causes the healthy prey and predators to coexist via persistent oscillations.

This paper proposes a mathematical model describing the dynamics of both Antarctic krill and Blue whale population where Antarctic krill species (prey) has group defense when they aggregate in herds in order to provide a self-defense from Blue whale species (predator). Firstly, a sufficient condition for the global asymptotical stability of the positive equilibrium is given using Poincaré–Bendixson theorem excluding periodic solution. Furthermore, we discuss the existence of Hopf-bifurcation and give criteria for stability switches through the study of a first-order exponential polynomial characteristic equation. Finally, some numerical simulations are given to illustrate our results.

In this paper, we have considered a discrete prey–predator model with square-root functional response and optimal harvesting policy. This type of functional response is used to study the dynamics of the prey–predator model where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The considered population model has three fixed points; one is trivial, the second one is axial and the last one is an interior fixed point. The first two fixed points are always feasible but the last one depends on the parameter value. The interior fixed point experiences the flip and Neimark–Sacker bifurcations depending on the predator harvesting coefficient. Finally, an optimal harvesting policy has been introduced and the optimal value of the harvesting coefficient is determined.

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.

This paper presents a view of the economy as a complex system with heterogeneous interacting agents who collectively organize themselves to generate aggregate phenomena which cannot be regarded as the behavior of some average or representative individual. There is an essential difference between the aggregate and the individual and such phenomena as bubbles and crashes, herd behavior, the transmission of information and the organization of trade are better modeled in the sort of framework suggested here than in more standard economic models.

The clustering of agents in the market is a typical problem dealt with in recent approaches to macroeconomic modeling, describing macroscopic variables in terms of the behavior of a large collection of microeconomic entities. Clustering is subject to many economic interpretations, often described by the Ewens Sampling Formula (ESF). In contrast with the usual complex derivations, we suggest a finitary characterization of the ESF pointing to real economic processes. Our approach is finitary in the sense that we provide a probabilistic description of a system of n individuals considered as a closed system, a population, where individuals can change attributes over time. The probability is understood as the fraction of time the system spends in the considered partition. As the ESF represents an equilibrium distribution satisfying detailed balance, some properties difficult to prove are derived in a simple way. Besides the mean distribution of the cluster sizes, we study the probabilistic time behavior of clusters, in particular the mean survival as a function of the actual size and the correlation between size and age.

Large distributed systems, such as grid computing and cloud computing, promise to supply users with high performance. Consequently, scheduling is currently becoming a crucial problem. Herd behavior is a common phenomenon which causes severe performance decrease in the systems caused by bad scheduling behaviors. In this paper, based on the theoretical results of the homogeneous balls and bins model, it is proposed that a new and unique stochastic algorithm is used to avoid herd behavior. Experiments address that the multi-choice strategy can decrease herd behavior in large-scale sharing environment, at the same time providing increased scheduling performance and causing less scheduling burden than greedy algorithms. Distributed Hash Table (DHT) is used to organize grid computing resources. In the case of 1000 resources, the simulations show that for the heavy load (i.e., system utilization rate 0.5), the multi-choice algorithm reduces the number of incurred herds by a factor of 36, the average job waiting time by a factor of 8, and the average job turn-around time by 12% compared to the greedy algorithm. Moreover, in the cases of 2000 and 4000 nodes, two parameters (replica and d-group) are analyzed based on how they affect the performance of the algorithm. It is observed that there is an inflexion in the performance curve. Finally, a theoretic analysis of the algorithm performance is presented.

Herding has a great impact on stock market fluctuations, and it is possible for researchers to analyze the herding effect due to the recent popularity of mobile Internet and the development of big data analysis technology. In this paper, we propose both investor-based and stock-based sentiment propagation networks of Chinese stock markets based on the simple pairwise correlation of posts’ sentiment indexes. And the relationship between the herding effect and Chinese stock market fluctuations is studied by comparing the network indicators with the Shanghai Securities Composite Index (SSCI) and the Causeway International Value Index (CIVIX). Through the experimental results, we find that the indicators are indeed ahead of the Chinese stock market. This study is the first attempt to model stock market sentiment by using a complex network, and it proves that investor behavior has a great effect on the stock market.

In this work, we have introduced an eco-epidemiological model of an infected predator–prey system. Incorporation of prey refuge gives that a fraction of the infected prey is available to the predator for consumption. Moreover, to make the model more realistic to the environment, we have introduced strong Allee effect in the susceptible population. Boundedness and positivity of the solution have been established. Local stability conditions of the equilibrium points have been found with the help of Routh–Hurwitz criterion and it has been observed that if a prey population is infected with a lethal disease, then both the prey (susceptible and infected) and predator cannot survive simultaneously in the system for any parametric values. The disease transmission rate and the attack rate on the susceptible have an important role to control the system dynamics. For different values of these two key parameters, we have got only healthy or disease-free or predation-free or a fluctuating disease-free or even a fluctuating predator-free system with some certain parametric conditions.

The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior. The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction. This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models. For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devoted to study the effect of a non-selective hunting on the spread of the disease, where the local stability of the equilibria is investigated. Further the backward bifurcation is obtained concerning basic reproduction rate of the infection. The second case is for explaining the impact of selecting the weakest infected prey on the edge of the herd by a predator on the prevalence of the infection, where the local behavior is scrutinized. Moreover, for the graphical representation part, a numerical simulation scheme has been achieved using the Caputo fractional derivative operator.

In this paper, we introduce two classes of indices which can be used to measure the market perception concerning the degree of dependency that exists between a set of random variables, representing different stock prices at a fixed future date. The construction of these measures is based on the theory of comonotonicity. Both types of herd behavior indices are model-free and risk-neutral, derived from available option data. Depending on its particular definition, each index represents a particular aspect of the market sentiment concerning future co-movement of the underlying stock prices.

During smart anti-predator adaptations, preys use their size, fleeing capability and herd in their evolutionary tug-of-war with the predators. Predator–prey ecology is teeming with such examples where predators can be defeated or even killed by the group of victims, that is, preys, banding together. In this paper, we focus on a stage-structured predator–prey model with herd behavior and anti-predator phenomenon that results in the lesser ecological opportunities for predation in the presence of the smart anti-predatory constraints. It is our purpose in this paper to illustrate the combined effects of the herd behavior and anti-predator strategy on the predator–prey ecology. By constructing the model equations, we present an analytical study of the system and the occurrence of Hopf and transcritical bifurcation for the anti-predator rate as a bifurcation parameter. Conditions for co-existence under global stable interior equilibrium as well as permanence region for the survival of all species is adequately discussed. Also, the extinction of predators is explored via numerical simulations, that is, the prey population survives successfully in the habitat due to their defense mechanism and anti-predator behavior. In order to better understand the role of anti-predator traits in predator–prey system with square root functional response, we compare the theoretical and numerical results of the proposed model with a similar predator–prey system having Holling type-I functional response. Our findings show that the predator–prey dynamics under prey’s herd behavior and anti-predator tactics can have large influence on shifting the paradigm of predator–prey relationship by turning the predator–prey dynamics into the prey dominant dynamics, which is indeed, due to the smart synergy between both ecological actions. When underdog, sticking together, fights back, that would be a headache to deal with, no matter how menacing the predator is.