The dynamics of prey-predator model with disease in prey

Mathematical theory and modeling, 2015

In this paper, a mathematical model consisting of a prey-predator involving a prey refuge and infectious disease in the predator has been proposed and analyzed. Two types of functional responses are used to describe the feeding of the predator on the available prey. The existence, uniqueness and boundedness of the solution of the system are discussed. The dynamical behavior of the system has been investigated locally as well as globally using suitable Lyapunov function. The persistence conditions of the system are established. Local bifurcation near the equilibrium points has been investigated. The Hopf bifurcation conditions around the positive equilibrium point are derived. Finally, numerical simulations are carried out to specify the control parameters and confirm the obtained results Keywords: Prey-Predator, Disease, Refuge, Stability, Bifurcation.

Ecological Genetics and Genomics, 2019

In this paper, a delayed predator-prey model has been developed. Here, on the basis of infectious disease, the prey population has been divided into two sub-populations such as () susceptible prey and () infected prey population. It is also assumed that a predator may consume both susceptible prey as well as infected prey. Also here, the Crowley-Martin type functional form has been taken to consume the prey (both susceptible and infected) population by the predator. It also considers two types of time delays in this model, () one is disease transmission delay when susceptible prey moves to infected prey stage and () other is predator maturation delay. As the predator consumes both susceptible and infected prey, so we have incorporated the fact in this model that disease in the prey species also effect on predator population growth. Here, positivity and boundedness of solutions of our proposed model have been discussed. Then calculating different equilibrium points, the stability of the system have been discussed around those. In this paper, the Hopf bifurcation analysis has been done for non-delayed system with respect to the crowding coefficient. Finally, some numerical simulations have been presented to validate our theoretical findings.

Nonlinear Analysis: Modelling and Control

The present investigation deals with the disease in the prey population having significant role in curbing the dynamical behaviour of the system of prey-predator interactions from both ecological and mathematical point of view. The predator-prey model introduced by Cosner et al. has been wisely modified in the present work based on the biological point of considerations. Here one introduces the disease which may spread among the prey species only. Following the formulation of the model, all the equilibria are systematically analyzed and the existence of a Hopf bifurcation at the interior equilibrium has been duly carried out through their graphical representations with appropriate discussion in order to validate the applicability of the system under consideration.

International Journal of Computing Science and Mathematics, 2016

This paper presents a nonlinear mathematical model of prey-predator interaction in which the prey is infected by an infectious disease while assuming that the disease is not transmitted to predator though the rates of predation can be different for the susceptible and infected preys. We also assume that only susceptible prey population contributes in the reproduction. The infective population competes with susceptible population to population growth towards the carrying capacity and here the disease transmission follows the standard incidence. The basic reproduction numbers both in absence and presence of the predator are computed and the equilibria of the mathematical model are obtained. Our results show that there is a possibility of two coexistence equilibria for some set of parameters but only one of them can be locally asymptotically stable. We also observed that the system undergoes 'Hopf-bifurcation' when the maximum predation rate β crosses a threshold value. Finally, the numerical simulation is performed and that supports the analytical findings.

IOP Conference Series: Materials Science and Engineering, 2019

A predator-prey model with disease in both populations is proposed to illustrate the possibility of disease transmission between prey and predator through contact and predation. We used saturated incidence rate which takes behavioural changes of healthy population into consideration when disease spreads around them. The existence of eight non-negative equilibrium points is analysed and their local stability has been investigated. Numerical simulations are given to illustrate analytic results.

Journal of Applied Mathematics and Computing, 2019

The objective of the present paper is to investigate the dynamics of an ecoepidemiological system with predator's hyperbolic mortality and Holling type II functional response. The local stability, global stability of the ecosystem near biologically feasible equilibria have been thoroughly investigated. The boundedness and positivity of solutions for the model are also derived. Threshold values for a few parameters, which determine the feasibility and stability of some equilibria are calculated and a threshold is identified for the disease to die out. The existence of Hopf bifurcation around the coexistence equilibrium is shown. Finally, numerical illustrations are performed in order to validate some of the important analytical findings.

Adv. Syst. Sci. Appl., 2021

An eco-epidemiological model representing the interactions between prey and predator populations affected by a disease in an ecosystem is presented. The model is governed by a five-dimensional nonlinear system of ordinary differential equations coupling both ecological and epidemiological features of interacting populations. The well-posedness of the model is established with respect to positivity and boundedness of solutions. Conditions for asymptotic stability of different equilibrium points are extensively investigated to determine the existence and coexistence of prey and predator species using local linearization and Lyapunov functions techniques. Additionally, the analysis of the model is extended to assess the effects of three timedependent control functions, such as disease prevention, treatment and alternative resource for predator, on the population dynamics of the prey-predator coexistence in the system.

2020

A predator-prey system with nonlinear incidence rate and refuging in prey is proposed to describe behavior change of certain infected diseases on healthy prey when the number of infected prey is getting large, while predator can predate prey by accessing refuging in prey. Therefore, this paper discusses the dynamics behavior predator-prey model with the spread of infected disease that is denoted by nonlinear incidence rate and adding prey refuge. We find the existence of eight non-negative equilibrium in the model, which their local stability has been determined. Furthermore, we also observe the prey refuge properties in the model. We find that prey refuge can prevent extinction in prey populations. In the end, some numerical solutions are carried out to illustrate our analytic results. For future work, we can investigate the harvesting effect in both populations, which is disease control in the predator-prey model with the spread of infected disease.

World Academy of Science, Engineering and Technology, International Journal of Mathematical and Computational Sciences, 2017

Nonlinear transmission of disease between infected and uninfected prey was studied using a prey-predator eco-epidemiological model. The interaction of predators with infected and uninfected prey species depends on their numerical superiority. Harvesting of both uninfected and infected prey was carried out, and stability analysis was carried out for equilibrium values. Using the parameter μ, the death rate of infected prey as a bifurcation parameter, it is shown that Hopf bifurcation could occur. The theoretical results are compared with numerical results for different sets of parameters.

2015

In this research a predator-prey model involving disease in a prey and prey refuge has been proposed and analyzed. It is assumed that only the prey species is divided into two classes infected and susceptible and disease transmitted by contact between a prey species. The existence, boundedness , permanence of the model has been investigated. The local and global stability conditions of all possible equilibrium points are established. Finally, numerical simulation is carried out to study the global dynamics of the model.

2003

We propose a model to describe the interaction between a diseased sh population and their predators. Analysis of the system is performed to determine the stability of equilibrium points for a large range of parameter values. The existence and uniqueness of solutions is established and solutions are shown to be uniformly bounded for all nonnegative initial conditions. The model predicts

Mathematical Methods in the Applied Sciences, 2003

The present paper deals with the problem of a classical predator-prey system with infection of prey population. A classical predator-prey system is split into three groups, namely susceptible prey, infected prey and predator. The relative removal rate of the susceptible prey due to infection is worked out. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability. It is shown that local asymptotic stability of the system around the positive interior equilibrium ensures its global asymptotic stability. We prove that there is always a Hopf bifurcation for increasing transmission rate. To substantiate the analytical ÿndings, numerical experiments have been carried out for hypothetical set of parameter values. Our analysis shows that there is a threshold level of infection below which all the three species will persist and above which the disease will be epidemic.

Differential Equations and Dynamical Systems, 2020

In the proposed model, a prey-predator system with migration in both species along with the disease in prey population is considered. To study the dynamics among healthy predator, susceptible prey and infected prey, Holling type II functional response is incorporated. The model has been examined using local stability analysis around the existing equilibrium points and by considering migration in susceptible prey as the control parameter, bifurcation conditions are discussed in brief. Basic reproduction number is determined which affirms the disease to be endemic or not and global stability analysis has been implemented using geometric approach. At the end, numerical simulation to support theoretical results is exhibited.

In this paper, we present and analyze a spatio-temporal eco-epidemiological model of a prey predator system where prey population is infected with a disease. The prey population is divided into two categories, susceptible and infected. The susceptible prey is assumed to grow logistically in the absence of disease and predation. The predator population follows the modified Leslie-Gower dynamics and predates both the susceptible and infected prey population with Beddington-DeAngelis and Holling type II functional responses, respectively. The boundedness of solutions, existence and stability conditions of the biologically feasible equilibrium points of the system both in the absence and presence of diffusion are discussed. It is found that the disease can be eradicated if the rate of transmission of the disease is less than the death rate of the infected prey. The system undergoes a transcritical and pitchfork bifurcation at the Disease Free Equilibrium Point when the prey infection rate crosses a certain threshold value. Hopf bifurcation analysis is also carried out in the absence of diffusion, which shows the existence of periodic solution of the system around the Disease Free Equilibrium Point and the Endemic Equilibrium Point when the ratio of the rate of intrinsic growth rate of predator to prey crosses a certain threshold value. The system remains locally asymptotically stable in the presence of diffusion around the disease free equilibrium point once it is locally asymptotically stable in the absence of diffusion. The Analytical results show that the effect of diffusion can be managed by appropriately choosing conditions on the parameters of the local interaction of the system. Numerical simulations are carried out to validate our analytical findings. (D. Melese).

In this article, the dynamical behavior of a three dimensional continuous time eco-epidemiological model is studied. A prey-predator model involving infectious disease in predator population is proposed and analyzed. This model deals with SI infectious disease that transmitted horizontally in predator population. It is assumed that the disease transmitted to susceptible population in two different ways: contact with infected individuals and an external sources. The existence, uniqueness and bounded-ness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The local bifurcation analysis and a Hopf bifurcation around the positive equilibrium point are obtained. Finally, numerical simulations are given to illustrate our obtained analytical results.