Analysis of a Delay Prey-Predator Model with Disease in the Prey Species Only (original) (raw)
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Dynamical Properties of a Delay Prey-Predator Model with Disease in the Prey Species Only
Discrete Dynamics in Nature and Society, 2010
A three-dimensional ecoepidemiological model with delay is considered. We first investigate the existence and stability of the equilibria. We then study the effect of the time delay on the stability of the positive equilibrium. The existence of a Hopf bifurcation at the positive equilibrium is obtained through the study of an exponential polynomial equation with delay-dependent coefficients. Numerical simulation with a hypothetical set of data has been carried out to support the analytical findings.
A mathematical study on the dynamics of an eco-epidemiological model in the presence of delay
In the present work a mathematical model of the prey-predator system with disease in the prey is proposed. The basic model is then modified by the introduction of time delay. The stability of the boundary and endemic equilibria are discussed. The stability and bifurcation analysis of the resulting delay differential equation model is studied and ranges of the delay inducing stability as well as the instability for the system are found. Using the normal form theory and center manifold argument, we derive the methodical formulae for determining the bifurcation direction and the stability of the bifurcating periodic solution. Some numerical simulations are carried out to explain our theoretical analysis.
Mathematical Biosciences, 2011
We consider a system of delay differential equations modeling the predator–prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.► An eco-epidemic model. ► Effect of delay. ► Investigate possible unforeseen consequences in the human intervention. ► A threshold is identified for the disease to die out. ► Numerical simulations support our theoretical analysis.
Stability and Bifurcation Analysis of an Eco-Epidemiological Model with Multiple Delays
2016
We propose and analyze an eco-epidemic model with disease in pr dator. The model dynamics is studied with gestation delay in predator and incub ation delay in disease transmission along with four different incidence functions. Our findings re-es tablish the claim of de Jong et al. that the mass action and standard incidence functions behave in a sim ilar fashion. In the absence of timedelay, the stability conditions of the equilibrium points a re derived in terms of basic reproduction numbers. We observe that disease has a stabilization effect . Further, we study the stability dynamics of the interior equilibrium for various combinations of the d lay factors and observe that the delay 2010 Mathematics Subject Classification: 92Bxx, 65Lxx, 37Mxx
Stability and bifurcation in a generalized delay prey–predator model
Nonlinear Dynamics, 2017
The present paper considers a generalized prey-predator model with time delay. It studies the stability of the nontrivial positive equilibrium and the existence of Hopf bifurcation for this system by choosing delay as a bifurcation parameter and analyzes the associated characteristic equation. The researcher investigates the direction of this bifurcation by using an explicit algorithm. Eventually, some numerical simulations are carried out to support the analytical results.
Stability and Hopf Bifurcation of a Delay Eco‐Epidemiological Model with Nonlinear Incidence Rate
Mathematical Modelling and Analysis, 2010
In this paper, a three‐dimensional eco‐epidemiological model with delay is considered. The stability of the two equilibria, the existence of Hopf bifurcation and the permanence are investigated. It is found that Hopf bifurcation occurs when the delay τ passes a sequence of critical values. Moreover, by applying Nyquist criterion, the length of delay is estimated for which the stability continues to hold. Numerical simulation with a hypothetical set of data has been done to support the analytical results. This work is supported by the National Natural Science Foundation of China (No. 10771104 and No.10471117), Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No. 2010IRTSTHN006) and Program for Key Laboratory of Simulation and Control for Population Ecology in Xinyang Normal University (No. 201004) and Natural Science Foundation of the Education Department of Henan Province (No. 2009B1100200 and No. 2010A110017)
International Journal of Differential Equations, 2011
A SI-type ecoepidemiological model that incorporates reproduction delay of predator is studied. Considering delay as parameter, we investigate the effect of delay on the stability of the coexisting equilibrium. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. Computer simulations have been carried out to illustrate different analytical findings. Results indicate that the Hopf bifurcation is supercritical and the bifurcating periodic solution is stable for the considered parameter values. It is also observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the reproduction period of predator exceeds the critical value.
Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system
Applied Mathematics and Computation, 2008
In this paper, a delayed predator-prey epidemiological system with disease spreading in predator population is considered. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the local asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcation of periodic solutions are investigated. Moreover, we also study the direction of Hopf bifurcations and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.
Stability and Hopf bifurcation analysis in ecological system with two delays
International Journal of Engineering, Science and …, 2011
This paper aims to study the effect of time-delay on a food chain model. Two delays ) and ( 2 1 τ τ are considered in the model to describe the time that juveniles of prey and predator take to mature. The stability analysis of the proposed model is carried out. The Hopf bifurcation conditions of the interior equilibrium point are established. Finally, numerical simulations are done to support the analytical findings. In addition, critical value of time delays are determined and it is found that maturation delay always acts as a destabilizing factor.
Dynamics of a prey-generalized predator system with disease in prey and gestation delay for predator
Modeling Earth Systems and Environment, 2016
In the present study, a prey-generalized predator model is proposed with disease in the prey and gestation delay for predator. The asymptotic behavior of the model is studied for all the feasible equilibrium states. The criterion for local stability of the system are established around steady states and thresholds for Hopf bifurcation are determined at the endemic as well as disease-free state. The respective sensitive indices of the variables are identified at the endemic state by performing the sensitivity analysis. Further numerical simulations have been carried out to justify our analytic findings.