Gestation Delay Research Articles

Modelling and bifurcation analysis in a hybrid bioeconomic system with gestation delay and additive Allee effect

In this paper, we investigate a delayed differential algebraic prey–predator system, where commercial harvesting on predator and additive Allee effect on prey are considered. A discrete time delay is utilized to represent gestation delay of the predator population. Positivity of solutions and uniform persistence of system are discussed. In the absence of time delay, by taking economic interest as a bifurcation parameter, some sufficient conditions associated with additive Allee effect and economic interest are derived to show that the proposed system undergoes singularity-induced bifurcation around the interior equilibrium. In the presence of time delay, combined dynamic effects of time delay and additive Allee effect on population dynamics are discussed in the case of positive economic interest of commercial harvesting. Existence of Hopf bifurcation and local stability switch around the interior equilibrium are studied as gestation delay crosses the critical value. Furthermore, properties of Hopf bifurcation are investigated based on the center manifold theorem and the norm form of a delayed singular system. Existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem. Numerical simulations are provided to show consistency with theoretical analysis.

Global stability and Hopf-bifurcation of prey-predator system with two discrete delays including habitat complexity and prey refuge

In this paper, we consider a two-dimensional prey-predator system with two delays. One delay is for negative feedback of the prey population and another is for gestation delay of the predator population. The predator is partially dependent on the prey followed by Holling type-II functional response. Due to habitat complexity and prey refuge, the Holling type-II functional response is modified in this work. We discuss the boundedness, permanence, local and global asymptotic behavior of the non-delayed and delayed models. The existence of periodic solutions via Hopf-bifurcation with respect to both the delays is established. The stability and direction of Hopf-bifurcation is also analyzed by using Normal form theory and Centre manifold theory. Lastly, numerical simulations have been carried out to confirm the analytical findings. The main objective of this work is to balance the prey-predator relationship in the presence of habitat complexity, prey refuge and delays.

A STUDY OF BIFURCATION OF PREY–PREDATOR MODEL WITH TIME DELAY AND HARVESTING USING FUZZY PARAMETERS

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.

Analysis of a Delayed Predator–Prey System with Harvesting

Abstract This article is concerned with a Leslie–Gower predator–prey system with the predator being harvested and the prey having a delay due to the gestation of prey species. By regarding the gestation delay as a bifurcation parameter, we first derive some sufficient conditions on the stability of positive equilibrium point and the existence of Hopf bifurcations basing on the local parametrization method for differential-algebra system. In succession, we also investigate the direction of Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold by employing the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, several numerical simulations are given.

Dynamic relationship between the mutual interference and gestation delays of a hybrid tritrophic food chain model

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell– Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response. doi:10.1017/S144618111700044X

Stability and Hopf bifurcation dynamics of a food chain system: plant–pest–natural enemy with dual gestation delay as a biological control strategy

In this paper, a food chain system with gestation delay for both pest and the natural enemy is proposed. Here the boundedness of the system is studied. Stability analysis for all possible equilibrium points is carried out. The thresholds for Hopf bifurcation at interior and the natural enemy free equilibrium states are studied and analyzed. It is observed that the natural enemy free steady state is stable if the gestation delay for the pest is sufficiently low otherwise system observed oscillating behavior. Similar observations established for the interior equilibrium. The sensitivity analysis is performed to find the respective sensitive indices of the variables of the proposed system. Further, simulations have been carried out to support our analytic results.

Dynamical Behavior of a Stage Structured Prey–Predator Model with Two Delays and Monod–Haldane Functional Response

In this paper, a new stage structured prey–predator model with Monod–Haldane functional response is proposed and the stages for predator have been considered. The proposed mathematical model consists of three nonlinear delay differential equations to describe the interaction among prey, immature predator and mature predator populations. Two time delays viz. feedback delay and gestation delay have been used as the bifurcation parameter. A rigorous mathematical analysis has been carried out by considering all possible cases for both the delays. Conditions for local stability and Hopf bifurcation have been investigated in all cases. Furthermore, by using normal form method, Riesz representing theory and central manifold theorem, formulae are derived for the direction of Hopf bifurcation and stability of bifurcating periodic solutions. Global stability analysis is carried out and the numerical simulation to validate the theory is also executed at the end.

DYNAMIC RELATIONSHIP BETWEEN THE MUTUAL INTERFERENCE AND GESTATION DELAYS OF A HYBRID TRITROPHIC FOOD CHAIN MODEL

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell–Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response.

Bifurcation of a delayed Gause predator-prey model with Michaelis-Menten type harvesting

In this paper, a Gause predator-prey model with gestation delay and Michaelis–Menten type harvesting of prey is proposed and analyzed by considering Holling type III functional response. We first consider the local stability of the interior equilibrium by investigating the corresponding characteristic equation. In succession, we derive some sufficient conditions on the occurrence of the stability switches of the positive steady state by taking the gestation delay as a bifurcation parameter. It is shown that the delay can induce instability and small amplitude oscillations of population densities via Hopf bifurcations. Furthermore, the stability and direction of the Hopf bifurcations are determined by employing the center manifold argument. Finally, computer simulations are performed to illustrate our analytical findings, and the biological implications of our analytical findings are also discussed.

Effect of Multiple Delays in an Eco-Epidemiological Model with Strong Allee Effect

In the present article, we make an attempt to investigate the effect of two time delays, logistic delay and gestation delay, on an eco-epidemiological model. In the proposed model, strong Allee effect is considered in the growth term of the prey population. We incorporate two time lags and inspect elementary mathematical characteristic of the proposed model such as boundedness, uniform persistence, stability and Hopf-bifurcation for all possible combinations of both delays at the interior equilibrium point of the system. We observe that increase in gestation delay leads to chaotic solutions through the limit cycle. We also observe that the Allee effect play a major role in controlling the chaos. We execute several numerical simulations to illustrate the proposed mathematical model and our analytical findings.

Dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations

In this paper, we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton, where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered. Stochastic fluctuations are incorporated into the proposed system in form of Gaussian white noises to depict stochastic environmental factors in plankton system. For deterministic system without double time delays, existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting, and state feedback controllers are designed to eliminate singularity induced bifurcation. For deterministic system with double time delays, positivity and uniform persistence of solutions are studied, and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated. For stochastic system without double time delays, stochastic stability and existence of stochastic Hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory. For stochastic system with double time delays, existence and uniqueness of global positive solution are investigated, and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate Lyapunov functions. Numerical simulations are carried out to validate theoretical analysis.

New approach for stability and bifurcation analysis on predator–prey harvesting model for interval biological parameters with time delays

This paper presents a prey–predator harvesting model with time delay for bifurcation analysis. We consider the parameters of the proposed model with imprecise data as form of interval in nature, due to the lack of precise numerical information of the biological parameters such as prey population growth rate and predator population decay rate. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey under impreciseness of the biological parameters. Parametric functional form of interval number with two parameters is introduced. This study reveals that not only delay and harvesting effort play a significant role on the stability on the system but also interval parameters play a crucial role on the stability of the system. Computer simulations of numerical examples are given to explain our proposed imprecise model. We also address critically the biological implications of our analytical findings with proper numerical example.

Plant-pest-natural enemy dynamics with disease in pest and gestation delay for natural enemy

In this present study, a food chain system with the disease in pest species and gestation delay for the natural enemy is proposed. Here the boundedness and positivity of the system are studied. Stability analysis for all feasible equilibrium points is carried out. The Hopf bifurcation at interior equilibrium point is presented and analyzed. The sensitivity analysis is performed to find the respective sensitive indices of the variables of the proposed system. Further, simulations have been carried out to support our analytic results. https://doi.org/10.28919/jmcs/3416

A diffusive predator-prey system with prey refuge and gestation delay

The dynamics of a diffusive predator-prey system with prey refuge and gestation delay is investigated in this paper. For a non-delay system, global stability, Turing instability and Hopf bifurcation are studied. For a delay system, time delay induced instability and Hopf bifurcation are discussed. By the theory of normal form and the center manifold method, the direction and stability of the bifurcating periodic solution are discussed.

Complex dynamics and stability analysis in a discrete hybrid bioeconomic system with double time delays

In this paper, a discrete hybrid three-species food chain system is proposed, where commercial harvesting on top predator is considered. Two time delays are introduced to represent gestation delay for prey and predator population, respectively. In absence of time delay, sufficient conditions associated with economic interest and step size are derived to show system undergoes flip bifurcation. In presence of double time delays, existence of Neimark–Sacker bifurcation and local stability switch are discussed due to variations of time delays. Furthermore, by utilizing new normal form of delayed discrete hybrid system and center manifold theorem, direction and stability of Neimark–Sacker bifurcation are studied. Numerical simulations are performed not only to validate theoretical analysis, but also exhibit cascades of period-doubling bifurcation, chaotic behavior and stable closed invariant curve.

Dynamical analysis in a bioeconomic phytoplankton zooplankton system with double time delays and environmental stochasticity

This paper presents a double delayed bioeconomic phytoplankton zooplankton system with commercial harvesting on zooplankton and environmental stochasticity. Maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered. Environmental stochasticity is incorporated into the proposed system in form of Gaussian white noises. Some sufficient conditions are derived to show that the proposed system has a unique global positive solution. In absence of double time delays, stochastic stability and existence of stochastic Hopf bifurcation are studied based on invariant measure theory and singular boundary theory of diffusion process for the proposed system. In presence of double time delays, asymptotic behaviors of the interior equilibrium are discussed by constructing some appropriate Lyapunov functions.

Dynamical analysis in a hybrid bioeconomic system with multiple time delays and strong Allee effect

In this paper, a multiple delayed differential–algebraic prey–predator system is established, where commercial harvesting on predator and strong Allee effect in prey growth are considered. Three time delays are introduced to represent maturation delay for prey (τ1), reaction delay to changes of prey surviving environment due to strong Allee effect (τ2) and gestation delay for predator (τ3), respectively. Positivity of solutions and uniform persistence of system are discussed. In absence of time delay, existence of singularity induced bifurcation and local stability analysis are investigated due to variation of economic interest of commercial harvesting. In presence of multiple time delays, existence of Hopf bifurcation and local stability analysis are discussed by analyzing associated characteristic equation. By using new normal form of multiple delayed differential–algebraic system and center manifold theorem, properties of Hopf bifurcation are studied. Furthermore, existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem.

Plant-pest-natural enemy dynamics with disease in pest and gestation delay for natural enemy

In this present study, a food chain system with the disease in pest species and gestation delay for the natural enemy is proposed. Here the boundedness and positivity of the system are studied. Stability analysis for all feasible equilibrium points is carried out. The Hopf bifurcation at interior equilibrium point is presented and analyzed. The sensitivity analysis is performed to find the respective sensitive indices of the variables of the proposed system. Further, simulations have been carried out to support our analytic results.https://doi.org/10.28919/jmcs/3416

Optimal harvesting and complex dynamics in a delayed eco-epidemiological model with weak Allee effects

In this article, an eco-epidemiological system with weak Allee effect and harvesting in prey population is discussed by a system of delay differential equations. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system is analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section and maximum Lyapunov exponent. Our simulation results suggest that for increasing the delay parameter, the system undergoes chaotic oscillation via period doubling. We also observe a quasi-periodicity route to chaos and complex dynamics with respect to Allee parameter; such behavior can be subdued by the strength of the Allee effect and harvesting effort through period-halving bifurcation. To find out the optimal harvesting policy for the time delay model, we consider the profit earned by harvesting of both the prey populations. The effect of Allee and gestation delay on optimal harvesting policy is also discussed.

Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos

Food uptake ability of higher trophic level species are more complicated and interesting due to their choice and availability of food and consequently their growth and also the effect of top predator interference on the dynamics of a tritrophic food chain model. In this paper, we consider the general framework for calculating the stability of equilibria, Hopf and double Hopf-bifurcation of a prey–predator system with Holling type IV and Beddington–DeAngelis type functional responses of intermediate and top predator respectively. The top predator is of generalist type and its growth is considered due to sexual reproduction. Firstly, we have shown feasibility and boundedness of the solutions of the considered model system, behavior of equilibria and the existence of Hopf-bifurcation. Conditions are determined under which the coexistence equilibrium point remains globally asymptotically stable. We identify the critical values of delay parameter for which stability switches and nature of the Hopf-bifurcation by using normal form theory and center manifold theorem. We investigate the occurence of double Hopf bifurcation at positive equilibrium point when we choose appropriate measure of the predator’s immunity or tolerance of the prey. Furthermore, some dynamic behaviors, such as stability switches, chaos, bifurcation and double Hopf-bifurcation scenarios are observed using numerical simulations. The chaotic behavior of the system is clarified by standard numerical tests.