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In this paper, we investigate the behavior of a predator–prey model with cooperation and Allee effect considering both deterministic and stochastic approaches. The main aim of this paper is to investigate the effect of environmental fluctuation in a deterministic predator–prey model. During the analysis of the deterministic model, it is shown that the system has saddle-node point of co-dimension 1, Hopf bifurcation and Bogdanov–Takens bifurcation of co-dimension 2. To study the effect of environmental fluctuation, we use perturbation to the birth rate of prey and death rate of predator density by Gaussian white noise. The persistence of the model and the stationary distribution is shown by forming a suitable Lyapunov function. Finally, numerical simulations are performed to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

The SARS-CoV-2 leads to a worldwide COVID-19 pandemic, which has caused tremendous damage to the world. In this paper, we develop a dynamic model in vivo, fitting and estimating parameters for T lymphocytes and pro-inflammatory cytokines IL-6 in patients with mild and severe COVID-19 at Yale New Haven Hospital through the GWMCMC algorithm. Meanwhile, we also analyze the structural identifiability and practical identifiability of the model. Further, we add time-varying parameters to the model, using the least squares method to perform data fitting and parameter estimation on survivors and non-survivors of the Italian infectious disease hospital. Then analyze the similarities and differences in immune response mechanisms between the two countries. Finally, we demonstrate the existence and stability of the equilibrium state of the model and analyze the Hopf bifurcation at the positive equilibrium state by using the central manifold theory and normal form theory. This result may explain the recurrence of infection in some COVID-19 patients. [ABSTRACT FROM AUTHOR]

This paper excogitates a bifurcation control strategy for a delayed fractional-order population dynamics model with incommensurate orders. First and foremost, by using stability theory of fractional differential equations, the sufficient conditions for the stability of the positive equilibrium are established. It is not difficult to find that the fractional-order system has a wider stability region than the traditional integer-order system. Second, taking time delay as bifurcation parameter, the sufficient criteria for Hopf bifurcation are obtained. In the next place, it is interesting to introduce a delayed feedback controller to guide Hopf bifurcation. The results reveal that the bifurcation dynamics of the model could be effectively controlled as long as the delay or fractional order is carefully adjusted. In conclusion, numerical simulations are carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of tumor–immune systems perturbed by the environmental noise and focus on the longtime behaviors. The existence and uniqueness of the globally positive solution to the tumor–immune system are proved using stochastic Lyapunov analysis and Itô's formula. We study the boundedness of moments for tumor cells and effector cells. By considering the dynamics on the boundary, applying the comparison theorem and the strong ergodic theorem, we obtain a threshold λ which is used to characterize the stochastic permanence in the sense that there is a unique invariant measure and extinction of the stochastic tumor–immune system. We also give biological interpretations about our analytical results of stochastic system. In addition, we present numerical examples and discussions to illustrate our analysis results. We find that the small noises preserve Hopf bifurcation of the deterministic system in stochastic setting and study numerically how the stochastic Hopf bifurcation with parameters occurs. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey system with multiple anti-predator behaviors is developed and studied, where not only the prey may spread between patches but also the fear effect and counter-attack behavior of the prey are taken into account. First, the stability and existence of coexistence equilibria are presented. The unique positive equilibrium may be a saddle-node or a cusp of codimension 2. Then, various transversality conditions of bifurcations such as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation are obtained. Moreover, compared with a single strategy, the multiple anti-predator strategies are more beneficial to the persistence and the population density of prey. [ABSTRACT FROM AUTHOR]

In this paper, we designed a population model that shows how a prey species defends itself against a generalist predator by exhibiting group defence. A non-monotonic functional response is used to represent the group defence functionality. We have demonstrated the model's local stability in the vicinity of the coexisting equilibrium solution employing a local Lyapunov function. Condition for existence of Hopf bifurcation is obtained along with its normal form. The suggested model has been validated by numerical simulations, which have also been used to verify the acquired analytical results. The parameters are subjected to sensitivity analysis by utilizing partial rank correlation coefficient (PRCC) and Latin hypercube sampling (LHS). The Z-type dynamic method is used to prevent population blow-up. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Age structure and delay play a significant role in determining the dynamics of the diseases. In this paper, an SEIRS epidemic model with age structure and time delay is investigated, where the loss of the acquired immunity and delay are incorporated. Through some rigorous analyses, an explicit formula for the basic reproduction number of the model is calculated, and some results about stability and instability of equilibria for the model are established. The findings show that the age structure and delay can produce Hopf bifurcation for an SEIRS model. The numerical examples are executed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Introduction of biopesticides and predators in the crop fields has a crucial impact on the pest management as the biopesticides induce infection among the pest population and the infected pests are easy to be captured by the predators. In this paper, we investigate a mathematical model for a pest management system with farming awareness-based inclusion of predators and biopesticides. The level of awareness among the farmers is assumed to grow at a constant rate besides its growth due to an increase in the pest population in the crop field. The proposed model is analyzed mathematically for the feasibility and stability of the equilibrium points; existence of Hopf bifurcation together with direction and stability of the bifurcating periodic solutions are discussed; sufficient conditions are derived for which the system does not exhibit any periodic solution. Our simulation results show that the activeness of farmers for employing biopesticides in the crop field potentially suppresses the oscillations and brings forth stability in the system. Further, we observe that the system showcases the appearances of Generalized Hopf bifurcation, Zero-Hopf bifurcation, transcritical bifurcation and bistability. Overall, our findings manifest that by imparting knowledge among the farmers regarding the use of biopesticides and predators in the crop fields, the pest population can be effectively controlled. [ABSTRACT FROM AUTHOR]

This paper considers predator–prey systems in which the predator adapts the search speed to prey's density. The prey admits the cost of fear because of the predator and moves between source–sink patches to escape from predation. Using dynamical systems theory, we demonstrate equilibrium stability, uniform persistence and bifurcations in the system. It is shown that enhancing the search speed of predator and/or decreasing prey's dispersal from the source to sink could promote persistence of the system, while increasing the fear cost would stabilize the system. Moreover, dispersal could make the prey persist in both source–sink patches, even drive the predator into extinction. The dispersal could also make the prey reach a total population abundance larger than that without dispersal, even larger than the carrying capacity. Asymmetry in dispersal plays a role in the persistence and abundance. A novel finding of this work is that dispersal may lead to results reversing those without dispersal. Numerical simulations illustrate and extend our results. This work is important in understanding complexity in predation systems. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical behavior of the modified May–Holling–Tanner model in the presence of dynamic alternative resources. We study the role of dynamic alternative resources on the survival of the species when there is prey rarity. Detailed mathematical analysis and numerical evaluations, including the situation of ecosystem collapsing, have been presented to discuss the coexistence of species', stability, occurrence of different bifurcations (saddle-node, transcritical, and Hopf) in three cases in the presence of prey and alternative resources, in the absence of prey and in the absence of alternative resources. It has been obtained that the multiple coexisting states and their stability are outcomes of variations in predation rate for alternative resources. Also, the occurrence of Hopf bifurcation, saddle-node bifurcation, and transcritical bifurcation are due to variations in the parameters of dynamic alternative resources. The impact of dynamic alternative resources on species' density reveals the fact that if the predation rate for alternative resources increases, then the prey biomass increases (under some restrictions), and variations in the predator's biomass widely depend upon the quality of food items. This study also points out that the survival of predators is possible in the absence of prey. In the theme of ecological balance, this study suggests some theoretical points of view for the eco-managers. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic size-dependent phytoplankton–zooplankton (PZ) model with additive Allee effect as well as its stochastic differential equation version is formulated to explore the growth dynamic of phytoplankton. A stability and Hopf bifurcation analysis is performed towards deterministic PZ model, and stochastic dynamic analysis is carried out on the stochastic PZ model, including the stochastic extinction, persistence in mean and the unique ergodic stationary distribution, which in turn provides a theoretical basis for numerical simulation analysis. Numerical simulations reveal that the synergistic effect of plankton body size and Allee effect can have a complex impact on the growth of phytoplankton in the deterministic and stochastic environments. One of the most interesting findings suggests that the reduced phytoplankton cell size can trigger bistable phenomenon, while the increased phytoplankton cell size or the weakened Allee effect has the ability to destroy such bistable phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, rich dynamics and complex bifurcations of an SIR epidemic model with saturated incidence rate and strong Allee effect are investigated. First, the existence of disease-free and endemic equilibria is explored, and we prove that the system has at most three positive equilibria, which exhibit different types such as hyperbolic saddle and node, degenerate unstable saddle (node) of codimension 2, degenerate saddle-node of codimension 3 at disease-free equilibria, and cusp, focus, and elliptic types Bogdanov–Takens singularities of codimension 3 at endemic equilibria. Second, bifurcation analysis at these equilibria are investigated, and it is found that the system undergoes a series of bifurcations, including transcritical, saddle node, Hopf, degenerate Hopf, homoclinic, cusp type Bogdanov–Takens of codimensional 2, and focus and elliptic type Bogdanov–Takens bifurcation of codimension 3 which are composed of some bifurcations with lower codimension. The system shows very rich dynamics such as the coexistence of multiple periodic orbits and homoclinic loops. Finally, numerical simulations are conducted on the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an HIV infection model with eclipse phase, humoral immune response and immunological delay has been discussed. By studying the characteristic equations of the model, the local stability analysis of various equilibrium points has been explored. Treating the delay as the bifurcation parameter, it has been shown that the delay can destabilize the stability of the infected steady-state leading to Hopf bifurcation and periodic solutions. By using Lyapunov functionals and LaSalle's invariance principle, the global stability analysis of the boundary equilibrium points has also been explored. It has also been shown numerically that the inclusion of the drug therapy in the model generates sporadic outbursts of virus called viral blips. In the end, numerical simulations have been employed to justify the analytical results proved in the paper. Biologically, the proposed model can explain the presence of viral blips in the system, during the introduction of HAART, as observed in the HIV-infected patient. These blips could mark the viral advancement in the system, thus resulting in complete immunological failure. One of the reasons for these viral blips can be the presence of delay in the activation of immunological response. But the development of drug-resistive virus could also be the reason for this sudden rise in viral loads. Moreover, the incorporation of the delay in the model generates oscillations and periodicity in the model, thus validating the long latency period seen in most HIV-infected patients. Also, a longer delay in the activation of immune response marks a viral advancement in the viral timeline resulting in viral blips, which signify the evolution of the virus. [ABSTRACT FROM AUTHOR]

In ecological environment, Allee effect is one of the important factors which cause significant changes to the system dynamics. In this paper, using the theory of dynamical systems, we analyze a variation of a standard cannibalistic two-dimensional prey–predator model with Holling type-II functional response in the presence of both weak and strong Allee effects. We have analyzed the impact of strong and weak Allee effects on the dynamics of a cannibalistic system, knowing the dynamics of the cannibalistic model without Allee effect. We have deduced that in the presence of cannibalism, both strong and weak Allee effects generate bistability between equilibrium points. For strong Allee effect, bistability occurs between trivial equilibrium point and predator-free equilibrium point as well as between trivial and coexistence equilibrium points. But for weak Allee effect, bistability occurs only between coexistence equilibrium points. We also pointed out that the cannibalistic system without Allee effect exhibits tristability among the trivial equilibrium point, coexistence equilibrium point having low prey concentration and coexistence equilibrium point having comparatively high prey concentration. But in the presence of strong Allee effect, cannibalistic system experiences tristability among trivial and two other stable coexistence equilibrium points. By a comprehensive bifurcation analysis, we have observed that Allee effect enriches both the local and global dynamics of the system. Here, we have reported all possible codimension-one and codimension-two bifurcations extensively by choosing cannibalism, Allee effect and predator natural death rate as the bifurcation parameters. In the analysis of bifurcations, we have explored the existence of transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation. Our analytical findings are validated through exhaustive numerical simulations. Finally, we have reported a comparative study between the impacts of strong and weak Allee effects on the dynamics of the cannibalistic system. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator–prey system in which the prey population is infected from a parasite and the growth of susceptible prey is suppressed due to fear of predation. We consider that the predators have the ability to distinguish between the susceptible and infected prey items, and they avoid the infected ones to reduce fitness cost. The predators are assumed to die naturally and also due to intraspecific competition. The proposed model is analyzed mathematically for the feasibility and stability of the system's equilibria. We also discuss the existence of Hopf bifurcation by taking the feeding preference of predators as a bifurcation parameter. We perform global sensitivity analysis to identify model parameters having significant impact on the density of predator population in the ecosystem. Our simulation results show the stabilizing role of selective feeding of predators whereas fear factor and disease prevalence induce limit cycle oscillations. Feeding more the predators with additional foods bring stability in the system by evacuating the persistent oscillations. To model the situation more realistically, we consider that the parameters representing the cost of fear and the feeding preference of predators vary with time. For the seasonally forced system, conditions are obtained for which the system has at least one positive periodic solution; global attractivity of the positive periodic solution is also discussed. Our seasonally forced model demonstrates the appearance of a unique periodic solution, higher periodic solutions and complex bursting patterns. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a system in which both prey and predator are harvested with constant rate. Our main objective is to find the effects of harvesting on equilibria, stability, and bifurcations in the system, which may be useful for biological management. The existence and stability of equilibrium points of the model are further investigated. A thorough qualitative analysis has been carried out based on bifurcation theory in dynamical systems and to validate our analytical findings, a large scale numerical simulation has been performed by using plausible values of parameters involved. It is shown that the model can exhibit Hopf bifurcation. The first Lyapunov coefficient is calculated to determine the direction of limit cycle of Hopf bifurcation. Also, it has been proven analytically that the system exhibits Bogdanov–Takens bifurcation of codimension 2. Moreover, discrete-time delay effect has been included due to gestation of the predator species on the same system and observed Hopf bifurcation with respect to the delay parameter. This study renders important tools for investigations of the dynamics of biotic organisms for the management and control of over harvesting. Some phase plane analysis has been carried out to support our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a three-species food chain model has been developed by considering the interaction between prey, predator and super-predator species. It is assumed that in the absence of predator and super-predator species, the prey species grow logistically. It is also assumed that predator and super-predator consume prey and predator, respectively. It is assumed that the predator shows refuge behavior to the super-predator. Again, harvesting of super-predator population has been considered. It is assumed that the consumption of prey and predator follows Crowley–Martin-type functional form. Boundedness of the solution of the system has been studied and different equilibrium points are determined and the stability of the system around these equilibrium points has been investigated. Existence conditions of Hopf bifurcation with respect to γ 1 of the system have been studied. It is found that the system shows some complex and critical dynamics due to increase of handling time of prey. It is also found that the system moves towards stable steady state due to increase of predator interference. It is observed that predator refuge may be responsible for the stability of the system. The chaotic dynamics of the system have been found due to the increase of the harvesting rate of super-predator. [ABSTRACT FROM AUTHOR]

In this paper, we study a within-host malaria infection model recently proposed by Schneider et al. in 2018. The stability and Hopf bifurcation analysis at the interior equilibrium are carried out, finding that the basic reproduction number plays a key role in the dynamics of the model, and incrementing the time delay will induce Hopf bifurcation at this equilibrium. The global extension of the local Hopf branch is further tracked numerically by the MatLab package DDE-BIFTOOL. Neimark-Sacker bifurcation of Poincaré map and period-doubling bifurcation of the bifurcated periodic solution are also detected, resulting in the existence of quasi-periodic and multiple periodic solutions, respectively. These results reveal that Hopf bifurcation will indeed bring about the rich dynamics of the model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two predator–prey models which take into consideration hunting cooperation (i.e., mutualism) between two different predators and within one predator species, respectively. Local and global dynamics are obtained for the model systems. By a detailed bifurcation analysis, we investigate the dependence of predation dynamics on mutualism (cooperative predation). From our study, we prove that mutualism may enhance the survival of mutualist predators in a severe condition and break the competitive exclusion principle. We further provide quantitative information about how the cooperative predation (mutualism) may (i) establish multiple stability switches on the positive equilibrium; (ii) generate backward bifurcation on equilibria; (iii) induce supercritical or subcritical Hopf bifurcations; and (iv) establish bi-stability phenomenon between the predator-free equilibrium and a positive equilibrium (or a limit cycle). [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the dynamical behavior of a two-species predator–prey model with Holling type IV functional response and nonlinear predator harvesting. The positivity and boundedness of the solutions of the model have been established. The considered system contains three kinds of equilibrium points. Those are the trivial equilibrium point, axial equilibrium point and the interior equilibrium points. The trivial equilibrium point is always saddle and stability of the axial equilibrium point depends on critical value of the conversion efficiency. The interior equilibrium point changes its stability through various parametric conditions. The considered system experiences different types of bifurcations such as Saddle-node bifurcation, Hopf bifurcation, Transcritical bifurcation and Bogdanov–Taken bifurcation. It is clear from the numerical analysis that the predator harvesting rate and the conversion efficiency play an important role in stability of the system. [ABSTRACT FROM AUTHOR]

This paper presents a generalized predator-prey system and considers the effect of habitat complexity on the dynamical consequences. The results show that habitat complexity has a major impact on the dynamical consequences of the considered system. On the one hand, habitat complexity has a stabilizing impact under certain conditions. A numerical simulation in our study and in experiments conducted in the published studies elaborate on this stabilizing effect. On the other hand, the most interesting and open issue is that a destabilizing effect of habitat complexity is found theoretically. All results are explained and illustrated from the ecological viewpoint. [ABSTRACT FROM AUTHOR]

In this paper, we study the spatiotemporal dynamics of a diffusive Leslie-type predator–prey system with Beddington–DeAngelis functional response under homogeneous Neumann boundary conditions. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the diffusive model, firstly, it is shown that Turing (diffusion-driven) instability occurs which induces spatial inhomogeneous patterns. Next, it is proved that the diffusive model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the diffusive model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Numerical simulations are also presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The progressive and increasing invasion of an opportunistic predator, the lionfish (Pterois volitans) has become a major threat for the delicate coral-reef ecosystem. The herbivore fish populations, in particular of Parrotfish, are taking the consequences of the lionfish invasion and then their control function on macro-algae growth is threatened. In this paper, we developed and analyzed a stage-structured mathematical model including P. volitans (lionfish), a cannibalistic predator, and a Parrotfish, its potential prey. As control upon the over predation, a rational harvest term has been considered. Further, to make the system more realistic, a delay in the growth rate of juvenile P. volitans population has been incorporated. We performed a global sensitivity analysis to identify important parameters of the system having significant correlations with the fishes. We observed that the system generates transcritical bifurcation, which takes the P. volitans-free equilibrium to the coexistence equilibrium on increasing the values of predation rate of adult P. volitans on Parrotfish. Further increase in the values of the predation rate of adult P. volitans on Parrotfish drives the system into Hopf bifurcation, which induces oscillation around the coexistence equilibrium. Moreover, the conversion efficiency due to cannibalism also has the property to alter the stability behavior of the system through Hopf bifurcation. The effect of time delay on the dynamics of the system is extensively studied and it is observed that the system develops chaotic dynamics through period-doubling oscillations for large values of time delay. However, if the system is already oscillatory, then the large values of time delay causes extinction of P. volitans from the system. To illustrate the occurrence of chaotic dynamics in the system, we drew the Poincaré map and also computed the Lyapunov exponents. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper studies a differential-algebraic predator-prey system with prey harvesting, which consists of two differential equations and an algebraic equation. By using the differential-algebraic system theory, bifurcation theory and formal series expansions, we investigate the Hopf bifurcation and center stability of the differential-algebraic predator-prey system. Some sufficient conditions on these issues are obtained. In addition, numerical simulations illustrate the effectiveness of our results and their biological implications are discussed. [ABSTRACT FROM AUTHOR]

Understanding the spatio-temporal dynamics of cholera outbreaks may help in devising more effective control procedures. In this paper, we have considered a reaction-diffusion system for biological control of cholera epidemic. Firstly, we have focused on temporal evolution of cholera in a region and its control using lytic bacteriophage in the aquatic reservoirs. Then, we have explored the effect of spatial dispersion of populations on the disease dynamics. We have observed the onset of sustained oscillations via Hopf-bifurcation for the endemic state of temporal system. This onset of fluctuations in populations depends upon the phage adsorption rate. But in the spatially extended setting, all the populations stabilize i.e., the spatio-temporal distribution of all the populations becomes uniform. Some numerical computations have been done to verify analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of multi-group epidemic models allowing heterogeneity of the host population and that has taken into consideration with general relapse distribution and nonlinear incidence rate. We establish that the global dynamics are completely determined by the basic reproduction number R0. The proofs of the main results utilize the persistence theory in dynamical systems, Lyapunov functionals and a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach. Biologically, the disease (with any initial inoculation) will persist in all groups of the population and will eventually settle at a constant level in each group. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behaviors of the basic SIR model. On the other hand, the global dynamics exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions. [ABSTRACT FROM AUTHOR]

In this paper, we extend the famous Ross's model to establish a new model for malaria to incorporate the impact of blood meal resources for mosquitoes and temperature on the transmission of malaria. A dynamical analysis for the new model is provided and it is shown that with the new growth rate for mosquitoes, the transmission dynamics of malaria becomes more complex and the Hopf bifurcation may occur which induces sustained oscillations not only in the mosquito population but also in the infected human population. Our results suggest that the abundance of blood meal resource for mosquitoes can be a factor which is important to characterize the transmission dynamics of malaria in a region. The impact of maturation time delay related to temperature changes is also analyzed which suggests that increasing the temperature exacerbates the transmission of malaria. [ABSTRACT FROM AUTHOR]

Our current research is based on the investigation of an eco-epidemiological model that solely includes illness in predators. Predators, both healthy and diseased, are thought to consume prey and breed; however, the offsprings are expected to be vulnerable. To achieve a more realistic and explicit outcome of the existing phenomena correlated with our model system, we consider that the process of disease transmission is mediated by some time lag and the intensity of disease prevalence is seasonally forced. Our simulation results show that the disease dies out for lower intensity of disease prevalence or higher rate of consumption of prey by susceptible predator. The system undergoes subcritical/supercritical Hopf bifurcation as the parameters representing the intensity of disease prevalence, consumption rate of prey by susceptible/infected predator vary. The system exhibits two types of bistabilities: the first one between stable coexistence and oscillating coexistence, and the second one between two oscillatory coexistence cycles. Moreover, we see that with gradual increase in the incubation delay, the system shows transitions from stable focus to limit cycle oscillations to period doubling oscillations to chaotic dynamics. Chaotic dynamics is also observed for the periodic changes in the intensity of disease prevalence if it takes much time for the pathogens to develop sufficiently inside body of the susceptible predators. [ABSTRACT FROM AUTHOR]

In recent years, the stability of the predator–prey model subject to the Allee effect has become an interesting issue. This study investigates the effect of Allee effect on the stability of a discrete-time predator–prey model with Holling type-II functional response. Using equilibrium analysis, stability analysis and bifurcation theory, the mathematical characteristics of the proposed model are examined. Model experiences flip bifurcation and Neimark–Sacker bifurcation based on the center manifold theorem and bifurcation theory. Our analytical results are demonstrated by numerical simulations. [ABSTRACT FROM AUTHOR]

Predator–prey interactions are the ubiquitous and natural phenomenon in an ecological system. Predators reduce the prey population's density by direct killing, which is an essential part of any ecological system. Based on the experimental works, for overcoming predation pressure, prey uses a variety of mechanisms. With Holling type-II functional response, we examined a prey–predator system incorporating anti-predator behavior and the cost of fear into prey. Prey anti-predator activity is a counterattacking strategy in which adult prey targets adolescent predators in order to counteract the potential predation pressure. Fear of predation may disrupt the physiological state of prey species and lead to long loss of prey species. In this study, we investigated this aspect to use a dynamical modeling approach. This research finds a plethora of fascinating phenomena. The studied system exhibits a wide range of dynamics and bifurcations, including saddle-node, Hopf, homoclinic, and a Bogdanov–Takens bifurcation in co-dimension two are among the dynamics and bifurcations observed in the analyzed system. We performed some numerical simulations to investigate the effects of anti-predator behavior and fear on prey and found both affect the prey–predator dynamics significantly. Our numerical examples clearly show that as prey carrying capacity increases, so does the prey's ability to perceive the risk of predation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Some infectious diseases produce lifelong immunity while others only produce temporary immunity. In the case of short-lived immunity, the level of protection wanes over time and may be boosted upon re-exposure, via infection or vaccination. Previous work developed a simple model capturing waning and boosting immunity, known as the Susceptible-Infectious-Recovered-Waned-Susceptible (SIRWS) model, which exhibits rich dynamical behavior including supercritical and subcritical Hopf bifurcations among other structures. Here, we extend the bifurcation analyses of the SIRWS model to examine the influence of all parameters on these bifurcation structures. We show that the bistable region, involving both a stable fixed point and a stable limit cycle, exists only for a small region of biologically realistic parameter space. Furthermore, we contrast the SIRWS model with a modified version, where immune boosting may involve the occurrence of a secondary infection. Analysis of this extended model shows that oscillations and bistability, as found in the SIRWS model, depend on strong assumptions about infectivity and recovery rate from secondary infection. Understanding the dynamics of models of waning and boosting immunity is important for accurately assessing epidemiological data. [ABSTRACT FROM AUTHOR]

A within-human-host malaria parasite model, integrating key variables that influence parasite evolution-progression-advancement, under innate and adaptive immune responses, is analyzed. The implicit role of immunity on the steady state parasite loads and parasitemia reproduction number ( ℛ 0 ), a threshold parameter measuring the parasite's annexing ability of healthy red blood cells (HRBCs), eventually rendering a human infectious to mosquitoes, is investigated. The impact of the type of recruitment function used to model HRBC growth is also investigated. The model steady states and ℛ 0 , both obtained as functions of immune system variables, are analyzed at snapshots of immune sizes. Model results indicate that the more the immune cells, innate and adaptive, the more efficient they are at inhibiting parasite development and progression; consequently, the less severe the malaria disease in a patient. Our analysis also illustrates the existence of a Hopf bifurcation leading to a limit cycle, observable only for the nonlinear recruitment functions, at reasonably large ℛ 0 . [ABSTRACT FROM AUTHOR]

Recently, pest control has become a very interesting research topic because it is closely associated with agricultural and economic loss. Empirical evidence shows that pest insects are responsible for lower crop production and many other adverse effects on the farming sector. There are several biological, physical and chemical control mechanisms. However, the biological control of pest populations by using natural enemies is one of the most important ecosystem services adopted in agriculture around the world. In the present study, we consider an ecological model consisting of prey (pest) and its natural enemy as the predator. Different system equilibria are obtained, their stability is analyzed, and Hopf bifurcation of the system around the interior equilibrium is discussed. The sufficient permanence criteria of the system are also derived. Moreover, we perform bifurcation analysis to explore the existence of limit cycle. We also investigate the stability property of the positive periodic solution when the interior equilibrium loses its stability. Our analytical results are further verified through numerical simulations. Our findings suggest that, in the absence of a super predator, pest and natural enemy show stable coexistence. However, in the presence of super predator, if the natural enemy is killed at a lower rate, both pest and natural enemy coexist. Finally, above a threshold value, the natural enemy is eradicated from the system and pest outbreak occurs. [ABSTRACT FROM AUTHOR]

In this work, a fuzzy prey–predator system with time delay is proposed. The model consists of two preys and one predator. The biological coefficients/parameters are considered as imprecise in nature and quantified by triangular fuzzy numbers. We have studied the effect of gestation delay on the stability of the system in fuzzy environment. The signed distance method for the defuzzification of the proposed fuzzy prey–predator system is adopted. For the underlying fuzzy model, we have provided a solution procedure to find all possible equilibrium points and studied their stabilities in the fuzzy sense. It is observed that there are stability switches, and Hopf-bifurcation occurs when the delay crosses some critical value in fuzzy sense. Numerical illustrations are provided in crisp as well as fuzzy environment with the help of graphical presentations to support our proposed approach. [ABSTRACT FROM AUTHOR]

We consider a two predator and one prey model with Holling type II functional response incorporating a constant prey refuge. Depending upon the constant prey refuge m, which provides a criterion for protecting m of prey from predation, sufficient conditions for stability and global stability of equilibria are obtained. We find the critical value of this refuge parameter m for which the dynamical system undergoes a Hopf bifurcation and then makes use of center manifold theorem and normal form methods to find the direction of the Hopf bifurcation as well as the stability of the resulting limit cycle. The influence of the prey refuge parameter is also investigated at the interior equilibrium. Numerical simulations were carried out to illustrate and support the analytical results. [ABSTRACT FROM AUTHOR]