Banshidhar Sahoo

Disease control in a food chain model supplying alternative food

Abstract Necessity to find a non-chemical method of disease control is being increasingly felt due to its eco-friendly nature. In this paper the role of alternative food as a disease controller in a disease induced predator–prey system is studied. Stability criteria and the persistence conditions for the system are derived. Bifurcation analysis is done with respect to rate of infection. The main goal of this study is to show the non-trivial consequences of providing alternative food in a disease induced predator–prey system. Numerical simulation results illustrate that there exists a critical infection rate above which disease free system cannot be reached in absence of alternative food whereas supply of suitable alternative food makes the system disease free up to certain infection level. We have computed the disease free regions in various parametric planes. This study is aimed to introduce a new non-chemical method for controlling disease in a predator–prey system.

Effects of supplying alternative food in a predator-prey model with harvesting

In this paper, we propose a tritrophic predator-prey model with harvesting where the top-predator population is partially supported with alternative food. We report the consequences of providing alternative food to the top-predator in a top-predator harvested model. The extinction criterion for top-predator population, local stability of equilibrium points and persistence conditions are discussed theoretically. Pontryagins maximum principle is used to characterize the optimal control of harvesting. We have derived the condition of Hopf bifurcation by varying harvesting effort. The bifurcation diagrams of the system with respect to harvesting effort in presence of alternative food are given. Our analysis show that alternative food can prevent top-predator extinction risk at higher harvesting effort and plays a vital role for biological conservation of species.

Effects of additional food in a delayed predator–prey model

We examine the effects of supplying additional food to predator in a gestation delay induced predator-prey system with habitat complexity. Additional food works in favor of predator growth in our model. Presence of additional food reduces the predatory attack rate to prey in the model. Supplying additional food we can control predator population. Taking time delay as bifurcation parameter the stability of the coexisting equilibrium point is analyzed. Hopf bifurcation analysis is done with respect to time delay in presence of additional food. The direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold theorem. The qualitative dynamical behavior of the model is simulated using experimental parameter values. It is observed that fluctuations of the population size can be controlled either by supplying additional food suitably or by increasing the degree of habitat complexity. It is pointed out that Hopf bifurcation occurs in the system when the delay crosses some critical value. This critical value of delay strongly depends on quality and quantity of supplied additional food. Therefore, the variation of predator population significantly effects the dynamics of the model. Model results are compared with experimental results and biological implications of the analytical findings are discussed in the conclusion section.

Diseased prey predator model with general Holling type interactions

Abstract Choice of interaction function is one of the most important parts for modelling a food chain. Many models have been proposed as a diseased-prey predator model with Holling type-I or type-II or type-III interactions, but there is no model with general Holling type interactions. In this paper, we study a diseased prey–predator model with general Holling type interactions. Local stability conditions of equilibrium points are derived. We obtain the permanence and impermanence conditions of the system. The conditions for global stability of the system are also derived. The system exhibits limit cycle, period-2, higher periodic oscillations and chaotic behaviour for different values of Holling parameters. One parameter bifurcation analysis is done with respect to general Holling parameters and infection rate. We utilize the MATCONT package to analyse the detailed bifurcation scenario as the two important interaction parameters are varied. It is interesting to note that a diseased system becomes a disease free system for proper choice of interaction functions. Our results give an idea for constructing a realistic food chain model through proper choice of general Holling parameters.

Generalized lag synchronization of delay coupled chaotic systems via linear transformations

In this paper, a scheme for generalized lag synchronization of delay coupled chaotic systems via linear transformation is proposed. The necessary and sufficient conditions for generalized lag synchronization of delay coupled systems are derived. Numerical simulation results are presented to show the effectiveness of the proposed synchronization scheme taking delay coupled chaotic Lorenz systems and Lorenz–Stenflo systems as examples.

Role of additional food in eco-epidemiological system with disease in the prey

An eco-epidemiological model supplying additional food is proposed.The dynamics of the system is studied both analytically and numerically.Optimal control problem is formulated and solved.Bifurcation is carried out with respect to additional food.Results show that disease free system can be obtained supplying additional food. An eco-epidemiological system with disease in the prey incorporating additional food to predator is proposed. The main objective of this study is to show the role of additional food in an eco-epidemiological system. We analyze the proposed system by calculating two reproduction numbers. The dynamical behavior of the system is investigated from the point of view of stability and persistence both analytically and numerically. Using Pontryagins Maximum Principle, an optimal control problem is formulated and solved in presence of additional food to achieve the control of disease. Numerical results illustrate that there exists a critical infection rate above which disease free system can not be reached in absence of additional food. On the other hand suitable additional food has the capability to obtain a disease free system up to certain infection level. The system becomes disease free also in presence of seasonally varying infection rate providing suitable additional food to predator. This study introduces a new non-chemical method for controlling disease in eco-epidemiological system.

Effects of additional food on an ecoepidemic model with time delay on infection

We propose a predator-prey ecoepidemic model with parasitic infection in the prey. We assume infection time delay as the time of transmission of disease from susceptible to infectious prey. We examine the effects of supplying additional food to predator in the proposed model. The essential theoretical properties of the model such as local and global stability and in addition bifurcation analysis is done. The parameter thresholds at which the system admits a Hopf bifurcation are investigated in presence of additional food with non-zero time lag. The conditions for permanence of the system are also determined in this paper. Theoretical analysis results are verified through numerical simulations. By supplying additional food we can control predator population in the model. Most important observation is that we can control parasitic infection of prey species by supplying additional food to predator. Eliminating the most infectious individuals from the prey population, predator quarantine the infected prey and prevent the spreading of disease.

Multistable behaviour of coupled Lorenz?Stenflo systems

In this paper, we propose three different schemes for designing multistable systems coupling Lorenz?Stenflo (LS) systems. In all of these three schemes the coupled LS-systems have been reduced to a single modified LS-system. Theoretically, pitchfork bifurcation and Hopf bifurcation conditions of the modified LS-system are derived. Phase diagrams are presented to show the multistable nature of the coupled LS systems for different initial conditions. One parameter bifurcation analysis is done with respect to difference in initial conditions of the two systems. Two parameter bifurcation analysis results are also presented. Our most important observation is that in coupling two m-dimensional dynamical systems multistable nature can be obtained if i number of variables of the two systems are completely synchronized and j number of variables keep a constant difference between them, where i?+?j?=?m and 1???i,j???m???1. Our observation may be applicable for designing physically or biologically useful multistable systems.

Effects of allochthonous inputs in the control of infectious disease of prey

Abstract Allochthonous inputs are important sources of productivity in many food webs and their influences on food chain model demand further investigations. In this paper, assuming the existence of allochthonous inputs for intermediate predator, a food chain model is formulated with disease in the prey. The stability and persistence conditions of the equilibrium points are determined. Extinction criterion for infected prey population is obtained. It is shown that suitable amount of allochthonous inputs to intermediate predator can control infectious disease of prey population, provided initial intermediate predator population is above a critical value. This critical intermediate population size increases monotonically with the increase of infection rate. It is also shown that control of infectious disease of prey is possible in some cases of seasonally varying contact rate. Dynamical behaviours of the model are investigated numerically through one and two parameter bifurcation analysis using MATCONT 2.5.1 package. The occurrence of Hopf and its continuation curves are noted with the variation of infection rate and allochthonous food availability. The continuation curves of limit point cycle and Neimark Sacker bifurcation are drawn by varying the rate of infection and allochthonous inputs. This study introduces a novel natural non-toxic method for controlling infectious disease of prey in a food chain model.

Dynamics of harvested-predator–prey model: role of alternative resources

In this paper, A predator–prey model with square root functional response for herd behaviour of prey incorporating predator harvesting is proposed and analysed. The predator population is provided with alternative resource. The proposed model is demonstrated in respect of theoretical as well as numerical results. The Pontryagin’s maximum principle is used to characterized the optimal harvesting strategy. Bifurcation study with the variation of harvesting effort and alternative resource are done in respect of numerical simulation. Simulation results show that suitable alternative resource has the capability to prevent predator extinction risk at higher harvesting level. The proposed model and obtaining results are usable in the field of conservation of biology.