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An analysis is presented for a model of a two species prey-predator system subject to the combined effects of delay and harvesting. Our study shows that, both the delay and harvesting effort may play a significant role on the stability of the system. Computer simulations are carried out to explain some of the mathematical conclusions.

A gene regulatory network can be considered a dynamic cellular system which describes the behavior (development) of a living cell and depends completely on the current internal state plus any external inputs, if these exist. Although many details inside a cell are not precisely known, gene expression data on a genome scale provide useful insights into such a cellular system. With gene expression data, a wide variety of models, such as Boolean networks and differential/difference equations, have been proposed to model gene regulatory networks. In these previously proposed models, genes are viewed as the internal state variables of a cellular system. This viewpoint has suffered from the underestimation of the model parameters. In addition, these models ignore an important problem with a gene regulatory network — time delay. Instead, this paper proposes a state-space model with time delays for gene regulatory networks. The proposed model views genes as the observation variables, whose expression values depend on the current internal state variables and any external inputs. Bayesian information criterion (BIC) and probabilistic principal component analysis (PPCA) are used to estimate the number of internal state variables and their expression profiles from gene expression data. By constructing dynamic equations with time delays for the internal state variables and the relationships between the internal state variables and the observation variables (gene expression profiles), state-space models with time delays for gene regulatory networks are constructed. The parameters of the proposed model can be unambiguously identified from time-course gene expression data with a lower computational cost. The proposed model is applied to two time-course gene expression datasets, and two gene regulatory~networks are inferred, respectively. The analysis shows that the inferred gene regulatory networks have several features of the real gene regulatory networks, such as the stability, the robustness, and the periodicity. Further, compared to state-space models without time delays, the proposed model with time delays has better prediction accuracy.

In this paper, we have considered a prey-predator model with Holling type of predation and harvesting of mature predator species. Our study shows that, using the harvesting effort as control, it is possible to break the cyclic behavior of the system and drive it to a required steady state. It is also shown that the time delay can cause a stable equilibrium to become unstable and even switching of stabilities. We have also discussed the effects of bycatch of the immature species. Lastly, some numerical simulations are carried out.

A delayed dynamical system model incorporating modified Leslie–Gower and Holling-type II schemes with state-dependent impulses is investigated for the case of small delay. The boundedness of solutions is analyzed by applying the comparison argument. Existence and stability of the semi-trivial periodic solution and the positive period-1 solution are discussed by using the variational principle, constructing Poincaré map and applying Floquet multiplier theory. Flip bifurcation is also investigated and some numerical simulation is presented in detail.

In this paper, we have considered a nonautonomous stage-structured epidemic model having two stages of the period of infection according to the progressing process of some infectious diseases (e.g. Chagas' disease, hepatitis C, etc.) with varying total population size and distributed time delay to become infectious. The infected persons in the different stages have different ability of transmitting disease. We have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R₀ and R* and further obtained that the disease will be permanent when R₀ > 1 and the disease will be going to extinct when R* < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

A nonlinear mathematical model with delay to capture the dynamics of effect of awareness programs on the prevalence of any epidemic is proposed and analyzed. It is assumed that pathogens are transmitted via direct contact between susceptibles and infectives. It is assumed further that cumulative density of awareness programs increases at a rate proportional to the number of infectives. It is considered that awareness programs are capable of inducing behavioral changes in susceptibles, which result in the isolation of aware population. The model is analyzed using stability theory of differential equations and numerical simulations. The model analysis shows that, though awareness programs cannot eradicate infection, they help in controlling the prevalence of disease. It is also found that time delay in execution of awareness programs destabilizes the system and periodic solutions may arise through Hopf-bifurcation.

Two models for the interaction of susceptible and infected Tilapia population with Pelican population are studied. Here, we considered that Pelican interact with both susceptible and infected Tilapia in proportion to their abundance. Stability near nonzero equilibria is presented. In the second model, time delay is incorporated in the disease transmission term and Hopf bifurcation is analyzed by taking time delay as a bifurcation parameter. Numerical simulations are performed to support the analytical results.

The purpose of this paper is to study the effect of sex hormone binding globulin (SHBG) on the mathematical model of the hypothalamic-pituitary-gonadal (HPG) endocrine cycle which regulates the production of the male hormone testosterone. Large amounts of total circulating testosterone are bound to SHBG making them. Standard analytical techniques are used to analyze the modified mathematical model which includes a delay to account for the time required for luteinizing hormone emitted by the pituitary gland to reach the testis, to determine the steady state, its stability and the critical delay needed for the bifurcation. Numerical simulation of the solutions of the model is performed to illustrate the possible behaviors.

A novel $SIS$ epidemic spreading model with distributed delay on complex heterogeneous network is presented. The formula of the basic reproductive number is presented to the model. It is proven that the disease dies out if the basic reproductive number is less than unity, while if the basic reproductive number is more than unity, the disease is uniformly persistent. The results enrich the previous results.

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.

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.

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.

Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopf-bifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Time delay in the cooperation behavior generates irregular oscillations whereas delay in refuge stabilizes an otherwise unstable system. Seasonal variations in the level of fear and refuge generate higher periodic solutions and bursting patterns, respectively, which can be replaced by simple 1-periodic solution if the cooperation and fear are also allowed to vary with time in the former and latter situations. Higher periodicity and bursting patterns are also observed due to synergistic effects of delay and seasonality. Our results indicate that the combined effects of fear, refuge and hunting cooperation play a major role in maintaining a healthy ecological environment.

Media impact has significant effect on reducing the disease prevalence, meanwhile sanitation and awareness can control the epidemic by reducing the growth rate of bacteria and direct contacts with infected individuals. In this paper, we investigate the impacts of media and sanitation coverage on the dynamics of epidemic outbreak. We observe that the growth rate of social media advertisements carries out a destabilizing role, while the system regains stability if the baseline number of social media advertisements exceeds a certain threshold. The dissemination of awareness among susceptibles first destabilizes and then stabilizes the system. The disease can be wiped out if the baseline level of awareness or the rate of spreading global information about the disease and its preventive measures is too high. We obtain an explicit expression for the basic reproduction number ${{ℛ}}_0$ and show that ${{ℛ}}_0<1$ leads to the total eradication of infection from the region. To capture a more realistic scenario, we construct the forced delay model by seasonally varying the growth rate of social media advertisements and incorporating the time lag involved in reporting of total infective cases to the policy makers. Seasonal pattern in the growth rate of social media advertisements adds complexity to the system by inducing chaotic oscillations. For gradual increase in the delay in reported cases of infected individuals, the nonautonomous system switches finitely many times between periodic and chaotic states.

In this study, we investigate the effects of excessive inputs of bioavailable phosphorus into a lake from agricultural fields and households on algal bloom formation, and its potential management by using the lanthanum-modified clay Phoslock as a bioavailable phosphorus adsorbent. We also investigate the impact of time delay involved in the process of applying Phoslock after measuring the density of algal biomass in the lake. Moreover, the seasonal effects in the input of bioavailable phosphorus from the agricultural lands and the application rate of Phoslock have been investigated. Our simulation results show that the algal growth accelerates if the bioavailable phosphorus is excessively loaded through agricultural runoff and domestic discharges. However, algal biomass can be effectively controlled by employing Phoslock in a sufficiently large quantity. Further, we find that a delay in the application of Phoslock induces limit cycle oscillations. Furthermore, our findings show that the combined actions of delay and periodicity in the application of Phoslock bring forth dynamical complexity in the lake system by giving rise to higher periodic solutions and bursting patterns. Lastly, we investigate an optimal control problem to estimate the optimum dosage of Phoslock for the mitigation of algal biomass from the lake system.

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.

A mathematical model for the drug delivery to macrophages of the tissues by using a preassigned cohort of red blood cells loaded with a drug is presented. This model is a system of three nonlinear differential equations, with a discrete time delay and an input depending on the time. The input should be controlled in order to obtain the longest duration of the therapeutic effect.