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In this paper, we consider the dynamics of a prey–predator model with logistic growth incorporating refuge in the prey and cooperation among predators population. Furthermore, multiplicative Allee effect in the prey growth is added to account from biological and mathematical perspectives. First, the existence and stability of equilibria of the model are discussed. Next, the existence of several kinds of bifurcation is provided and also studied the direction and stability of Hopf bifurcation. In addition, we study the influence of hunting cooperation on the model analytically and numerically, and find that the hunting cooperation cannot only reduce the density of prey population, but also destabilize the system dynamics irrespective of Allee effect. We choose also the impact of refuge on the model numerically, and explore that refuge stabilize the system dynamics. Moreover, the comparison between the dynamics of strong and weak Allee effect is taken into consideration.

This article concentrates on the study of delay effect of a mangrove ecosystem of detritus, detritivores and predator of detritivores. Local stability criteria are derived in the absence of delays. Conditions are found out for which the system undergoes a Hopf bifurcation. Further conditions are derived for which there can be no change in stability.

A Holling I predator-prey model with mutual interference concerning pest control is proposed and analyzed. The prey and predator are considered to be a pest and a natural enemy, respectively. The model is forced by the addition of periodic impulsive terms representing predator import (biological control) and pesticide application (chemical control) at different fixed moments. By using Floquet theory and small amplitude perturbations, we show the existence and stability of pest-free periodic solutions. Further, we prove that when the stability of pest-free periodic solutions is lost, the system is permanent by using analytic methods of differential equation theory. Numerical solutions are also given, which show that when stability of pest-free periodic solutions is lost, more exotic behavior can occur, such as quasi-periodic oscillation or chaos. We investigate the effect of impulsive perturbations on the unforced continuous system, and find that the forced system has a different dynamical behavior with a different range of initial values which are inside or outside the unstable limit cycle of the unforced continuous system. Finally, we compare the validity of the combination of biological control and chemical control with classical methods and conclude that the synthetical strategy is more effective than classical methods if we take effective chemical control.

In most models of population dynamics, increases in population due to birth are assumed to be time dependent, but many species reproduce only a single period of the year. In this paper, we construct a stage-structured pest model with birth pulse and periodic spraying pesticide at fixed time in each birth period by using impulsive differential equation. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker function or Beverton-Holt function, and obtain the threshold conditions for their stability. Further, we show that the time of spraying pesticide has a strong impact on the number of the mature pest population. Our results imply that the best time of spraying pesticide is at the end of the season, that is before and near the time of birth. Finally, by numerical simulations we find that the dynamical behaviors of the stage-structured population models with birth pulse and impulsive spraying pesticide are very complex, including period-doubling cascade, period-halving cascade, chaotic bands with periodic windows and "period-adding" phenomena.

This paper deals with the analysis of the nonlinear glucose regulation system from its Piecewise Affine (PWA) equivalent Piecewise Affine (PWA) equivalent model. The use of PWA model was motivated by the fact that it has developed as an attractive tool for nonlinear system analysis in recent years, since it allows the use of well developed concepts in the field of linear systems and control theory. The analysis mainly involves the behavioral study of the system at steady state for different types of input and different sets of physiological parameters. The number of affine systems required to represent a nonlinear system is an important consideration in PWA modeling. In this paper, the optimal number of piecewise affine systems is obtained by a technique based on a fuzzy clustering. From the PWA representation, it is possible to determine the equilibrium point (basal condition) of the glucose regulation system quite easily without solving a complex transcendental equation. The presence of a compensating mechanism between insulin sensitivity and insulin secretion is utilized by the PWA model to determine a particular profile (characteristics) for insulin sensitivity for a modified insulin secretion characteristics that would maintain a normal basal glucose level. A condition for the existence of a stable limit cycle of plasma glucose concentration as a function of delay in hepatic glucose production is determined from the model. This allowed determining the point of hopf bifurcation without carrying out extensive simulations, as is done with the existing models. The response obtained from the numerical simulation of the model, is in line with the experimental results.

This paper deals with the eco-biological dynamics of a delayed diffusive autotroph-herbivore population ecosystem with nutrient recycling. The real situation is represented by a set of two-dimensional nonlinear ordinary differential equations involving autotroph-herbivore biomass. Plant populations undergo critical changes with different amplitude in plant ecology. We propose a description of plant communities as interesting systems which resembles to the behavior of real media. The delay and diffusion parameter have a great role to shape the dynamical features of the system. We have studied the growth of an autotroph and herbivore population depending on the limiting nutrient which is partially recycled through bacterial decomposition. We have analyzed the asymptotic stability and switching to instability with bifurcation of the model system with delay and diffusion where diffusion controls directly the delay parameters of our system. All the analytical results are interpreted ecologically and compared with the simulated computer results.

Some mathematical models are suggested to describe the tritrofic interactions among plants, herbivores and their carnivorous enemies attracted by defensive volatiles of plants. For the interactions of Volterra type, it is proved that the threshold value for the persistence of herbivore and carnivore populations is not affected by the chemical attractions. Furthermore, the attraction to carnivores is beneficial to reduce the density of herbivores and increase the density of plants. If the interaction of plants and herbivores takes the Leslie type, the model admits the fold bifurcation that induces bistable positive equilibria. Numerical computations indicate that the response time of carnivores to defensive volatiles of plants induces periodic cycles and irregular fluctuations.

A new deterministic model for the transmission dynamics of human papillomavirus (HPV) and related cancers, in the presence of the Gardasil vaccine (which targets four HPV types), is presented. In the absence of routine vaccination in the community, the model is shown to undergo the phenomenon of backward bifurcation. This phenomenon, which has important consequences on the feasibility of effective disease control in the community, arises due to the re-infection of recovered individuals. For the special case when backward bifurcation does not occur, the disease-free equilibrium (DFE) of the model is shown to be globally-asymptotically stable (GAS) if the associated reproduction number is less than unity. The model with vaccination is also rigorously analyzed. Numerical simulations of the model with vaccination show that, with the assumed 90% efficacy of the Gardasil vaccine, the effective community-wide control of the four Gardasil-preventable HPV types is feasible if the Gardasil coverage rate is high enough (in the range 78–88%).

This work deals with the dynamics of a bioeconomic continuous time model, where the combined action of the fishing effort exerted by men (as a predator) and multiple Allee effect or depensation on the growth rate of a self-regenerating resource (the prey) are considered.

It has been recently established that a depensation phenomenon appears by diverse causes and new functions have been proposed to describe multiple Allee effects. One of these formalizations is here incorporated in the well-known Smith's model, one of the simplest models to open access fisheries.

We prove that this new and complex expression is topologically equivalent to a simpler form. Then, we postulate that the parsimony principle must be used to describe this phenomenon.

It is also shown that in the phase plane of biomass-effort on the proposed model, the origin is an attractor equilibrium for all parameters values as a consequence of the Allee effect. Moreover, there is a subset of the parameter values, for which two limit cycles exist surrounding the unique positive equilibrium point of the system, one of them being asymptotically stable (the non damped oscillatory tragedy of the commons); hence, multiestability exists, particularly three-stability.

In this paper, we investigate the global dynamics of a system of delay differential equations which describes the interaction of hepatitis B virus (HBV) with both liver and blood cells. The model has two distributed time delays describing the time needed for infection of cell and virus replication. We also include the efficiency of drug therapy in inhibiting viral production and the efficiency of drug therapy in blocking new infection. We compute the basic reproduction number and find that increasing delays will decrease the value of the basic reproduction number. We study the sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Our analysis reveals that the model exhibits the phenomenon of backward bifurcation (where a stable disease-free equilibrium (DFE) co-exists with a stable endemic equilibrium when the basic reproduction number is less than unity). Numerical simulations are presented to evaluate the impact of time-delays on the prevalence of the disease.

A mathematical model of bacteria-phage interaction in the chemostat is formulated, which incorporates the host immune response with an aim to mimic phage therapy in vivo. It is shown that the host immune response induces the backward bifurcation. Thus, there exists the bistability of phage-free equilibrium with the phage-infection equilibrium. More importantly, it is found that the model exhibits the coexistence of a stable phage-infection equilibrium with a stable periodic solution. The crucial implication of these phenomena is that phage infection fails both from the smaller dose of initial injection and from the larger dose of initial injection. Thus, a proper design of phage dose is necessary for phage therapy. Further analysis indicate that the inhibition effects of bacteria and phages can induce periodic oscillations and modulated oscillation.

This paper concerns spatio-temporal pattern formation in a model for two competing prey populations with a common predator population whose movement is biased by direct prey-taxis mechanisms. By pattern formation, we mean the existence of stable, positive non-constant equilibrium states, or nontrivial stable time-periodic states. The taxis can be either repulsive or attractive and the population interaction dynamics is fairly general. Both types of pattern formation arise as one-parameter bifurcating solution branches from an unstable constant stationary state. In the absence of our taxis mechanism, the coexistence positive steady state, under suitable conditions, is locally asymptotically stable. In the presence of a sufficiently strong repulsive prey defense, pattern formation will develop. However, in the attractive taxis case, the attraction needs to be sufficiently weak for pattern formation to develop. Our method is an application of the Crandall–Rabinowitz and the Hopf bifurcation theories. We establish the existence of both types of branches and develop expressions for determining their stability.

The role of scavengers, which consume the carcasses of predators along with predation of the prey, has been ignored in comparisons to herbivores and predators. It has now become a topic of high interest among researchers working with food-web systems of prey–predator interactions. The food-web considered in these works contains prey, predators, and scavengers as the third species. In this work, we attempt to study a food-web model of these species in the presence of the multiplicative Allee effect and harvesting. It is observed that this makes the model more complex in the form of multiple co-existing steady states. The conditions for the existence and local stability of all possible steady states of the proposed system are analyzed. The global stability of the steady state lying on the x-axis and the interior steady state have been discussed by choosing suitable Lyapunov functions. The existence conditions for saddle-node and Hopf bifurcations are derived analytically. The stability of Hopf bifurcating periodic solutions with respect to both Allee and harvesting constants is examined. It is also observed that multiple Hopf bifurcation thresholds occur for harvesting parameters in the case of two co-existing steady states, which indicates that the system may regain its stability. The proposed model is also studied beyond Hopf bifurcation thresholds, where we have observed that the model is capable of exhibiting period-doubling routes to chaos, which can be controlled by a suitable choice of Allee and harvesting parameters. The largest Lyapunov exponents and sensitivity to initial conditions are examined to ensure the chaotic nature of the system.

In this paper, we have investigated global dynamics of a two-species food chain model with the Holling type III functional response that includes linear harvesting for the prey and nonlinear harvesting for the predator. The long-time continued existence of both species is discussed using uniform persistence theory. Stability of various equilibrium points is described in terms of model parameters. The local asymptotic stability of non-hyperbolic equilibrium points is determined with the help of center manifold theorem. Global behavior of solutions of the model system when both species are present is determined by considering the global properties of the coexistence equilibrium. Here, we have taken a comprehensive view by considering different bifurcations of co-dimension one and two and have discussed the importance of various model parameters on the system dynamics. The model system shows much more complex and realistic behavior compared to a model system without any harvesting, with constant harvesting or linear-yield harvesting of either or both of the species. Numerical simulations have been conducted to illustrate the theoretical findings.

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.

Understanding the Allee effect on endangered species is crucial for ecological conservation and management as it highly affects the extinction of a population. Due to several ecological mechanisms accounting for the Allee effect, it is necessary to study the dynamics of a predator–prey model incorporating this phenomenon. In 1999, Cosner et al. [Effects of spatial grouping on the functional response of predators, Theor Popul Biol 56:65–75, 1999] derived a new kind of functional response by considering spatially grouped predators. This paper deals with the dynamical behavior of a predator–prey system with functional response proposed by Cosner et al., and the growth of the prey population suffers a strong Allee effect. We find that the system undergoes various types of bifurcations such as Hopf bifurcation, saddle-node bifurcation, and Bogdanov–Takens bifurcation. We also observe that the model exhibits bistability and two different types of tristability phenomena. Our findings reveal that for such a kind of multistability in ecological systems, the initial population size plays a crucial role and also impacts the system’s state in the long term.

In this paper, a predator–prey model with prey refuge was developed to investigate how prey refuge affect the dynamics of predator–prey interaction. We studied the existence and stability of equilibria, and then derived the sufficient conditions for the bifurcation such as saddle-node, transcritical, Hopf and Bogdanov–Takens bifurcation. In addition, a series of numerical simulations were carried out to illustrate the theoretical analysis, and the numerical results are consistent with the analytical results. Our results demonstrate that prey refuge has a great impact on the predator–prey dynamics.

The harvesting of species occurs in terrestrial and aquatic habitats across the world. It not only causes alteration in the population structure of the species subjected to harvesting but also of the species in interaction with the harvested species. The present work investigates the effect of nonlinear prey harvesting on the dynamics of a ratio-dependent predator–prey system with a strong Allee effect in prey population. It is found that the system exhibits a rich spectrum of dynamics including saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation with respect to the parameters that shape the nonlinear harvesting rate, namely, the maximum harvesting rate and a half-saturation constant that represents the prey density at which half of the maximum harvesting rate is reached. It is found that the basin of attraction of the stable coexistence state shrinks as the harvesting rate increases and if the harvesting rate is above a threshold value at which saddle-node bifurcation occurs, the stable coexistence of predator and prey populations is not possible for any initial start. It is also found that the harvesting policies in which the harvesting rate increases less rapidly at low prey population size are more favorable for the stable coexistence of species. The presence of Allee effect in the prey population is found to increase the chances of extinction of both species by reducing the threshold value of the harvesting rate at which the unconditional extinction occurs. Numerical simulations are carried out to support the analytical findings.

p53 is a star protein in cancer biology, and its oscillatory dynamics have received much attention. However, most studies do not consider spatial effects. For this reason, we introduce the diffusion term in a classical p53-Mdm2 autoregulatory loop model. First, the equation is linearized at the positive equilibrium so that we can discuss the local asymptotic stability of this equilibrium. By taking the delay as a bifurcation parameter, the positive equilibrium transitions from stable to unstable and occurs a Hopf bifurcation. Second, we determine the stability of the bifurcating period solution by using a multiple time scales approach. Finally, the theoretical analysis is verified by the numerical simulations. Our study contributes to providing new insight in the controlling of p53 dynamics.

We analysed a model for the interaction of a macroparasite and a host population growing logistically. The model is obtained by approximating the parasite distribution with a negative binomial with a fixed clumping parameter.

By letting the contact rate k vary, we found a complex pattern of bifurcations, including subcritical bifurcations of the disease-free equilibrium, Hopf and homoclinic bifurcations. The specific pattern depends on the interaction of the various parameters; in particular, alternative stable equilibria may occur only when the carrying capacity K_N is sufficiently large, while periodic solutions may occur for all values of K_N, if k is large enough.