Analysis of an Eco-Epidemiological Model under Optimal Control Measures for Infected Prey

This paper examines the analysis of an eco-epidemiological model with optimal control strategies for infected prey. A model is proposed and analyzed qualitatively using the stability theory of the differential equations. A local and global study of the model is performed around the disease-free equilibrium and the endemic equilibrium to analyze the global stability using the Lyapunov function. The time-dependent control is introduced into the system to determine the best strategy for controlling the disease. The results obtained suggested the separation of the infected population plays a vital role in disease elimination

Modelling of a seasonally perturbed competitive three species impulsive system

The population of biological species in the ecosystem is known sensitive to the periodic fluctuations of seasonal change, food resources and climatic conditions. Research in the ecological management discipline conventionally models the behavior of such dynamic systems through specific impulsive response functions, but the results of such research are applicable only when the environments conform exactly to the conditions as defined by the specific response functions that have been implemented for specific scenarios. This means that the application of previous work may be somewhat limited. Moreover, the intra and inter competitions among species have been seldom studied for modelling the prey-predator ecosystem. To fill in the gaps this paper models the delicate balance of two-prey and one-predator system by addressing three main areas of: ⅰ) instead of using the specific impulse response this work models the ecosystem through a more general response function; ⅱ) to include the effects due to the competition between species and ⅲ) the system is subjected to the influences of seasonal factors. The seasonal factor has been implemented here in terms of periodic functions to represent the growth rates of predators. The sufficient condition for the local and global asymptotic stability of the prey-free periodic solution and the permanence of the system have been subsequently obtained by using the Comparison techniques and the Floquet theorems. Finally, the correctness of developed theories is verified by numerical simulation, and the corresponding biological explanation is given.2017005,2017019: Shanxi Agricultural University of Science and Technology Innovation Fund Projects

Dynamics of Bacterial white spot disease spreads in Litopenaeus Vannamei with time-varying delay

In this paper, we mainly consider a eco-epidemiological predator-prey system where delay is time-varying to study the transmission dynamics of Bacterial white spot disease in Litopenaeus Vannamei, which will contribute to the sustainable development of shrimp. First, the permanence and the positiveness of solutions are given. Then, the conditions for the local asymptotic stability of the equilibriums are established. Next, the global asymptotic stability for the system around the positive equilibrium is gained by applying the functional differential equation theory and constructing a proper Lyapunov function. Last, some numerical examples verify the validity and feasibility of previous theoretical results

Stability and Hopf bifurcation of a delay eco‐epidemiological model with nonlinear incidence rate

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) First published online: 10 Feb 201

Influence of impreciseness in designing tritrophic level complex food chain modeling in interval environment

Abstract In this paper, we construct a tritrophic level food chain model considering the model parameters as fuzzy interval numbers. We check the positivity and boundedness of solutions of the model system and find out all the equilibrium points of the model system along with its existence criteria. We perform stability analysis at all equilibrium points of the model system and discuss in the imprecise environment. We also perform meticulous numerical simulations to study the dynamical behavior of the model system in detail. Finally, we incorporate different harvesting scenarios in the model system and deploy maximum sustainable yield (MSY) policies to determine optimum level of harvesting in the imprecise environment without putting any unnecessary extra risk on the species toward its possible extinction

Mathematical study of eco-epidemiological system

In this paper, a mathematical model consisting of the prey- predator involving infectious disease in prey population, is proposed and analyzed. And this disease passed from a prey to predator through attacking of predator to prey. The model represented mathematically by the set of nonlinear differential equations. The existence, uniqueness and boundedness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The occurrence of local bifurcation (such as saddle-node, transcritical and pitchfork) a long with Hopf bifurcation near each of the equilibrium points are discussed. Finally, numerical simulation is used to study the global dynamics of this model. Keywords: eco-epidemiological model, SI epidemics disease, prey-predator model, stability analysis, Hopf bifurcation

Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response

In this paper, we analyze a recently proposed predator-prey model with ratio dependence and Holling type III functional response, with particular emphasis on the dynamics close to extinction. By using Briot-Bouquet transformation we transform the model into a system, where the extinction steady state is represented by up to three distinct steady states, whose existence is determined by the values of appropriate Lambert W functions. We investigate how stability of extinction and coexistence steady states is affected by the rate of predation, predator fecundity, and the parameter characterizing the strength of functional response. The results suggest that the extinction steady state can be stable for sufficiently high predation rate and for sufficiently small predator fecundity. Moreover, in certain parameter regimes, a stable extinction steady state can coexist with a stable prey-only equilibrium or with a stable coexistence equilibrium, and it is rather the initial conditions that determine whether prey and predator populations will be maintained at some steady level, or both of them will become extinct. Another possibility is for coexistence steady state to be unstable, in which case sustained periodic oscillations around it are observed. Numerical simulations are performed to illustrate the behavior for all dynamical regimes, and in each case a corresponding phase plane of the transformed system is presented to show a correspondence with stable and unstable extinction steady state