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

The Dynamics of a Predator-Prey Model Involving Disease Spread In Prey and Predator Cannibalism

In this article, dynamics of predator prey model with infection spread in prey and cannibalism in predator is analyzed. The model has three populations, namely susceptible prey, infected prey, and predator. It is assumed that there is no migration in both prey and predator populations. The dynamical analysis shows that the model has six equilibria, namely the trivial equilibrium point, the prey extinction point, the disease free and predator extinction equilibrium point, the disease-free equilibrium point, the predator extinction equilibrium point, and the coexistence equilibrium point. The first equilibrium is unstable, and the other equilibria conditionally local asymptotically stable. The positivity and boundedness of the solution are also shown. The analytical result is supported by numerical simulation. It is shown that in such a high cannibalization the coexistence equilibrium is locally asymptotically stable

The uneasy coexistence of predators and pathogens

Disease and predation are both highly important in ecology, and pathogens with multiple host species have turned out to be common. Nonetheless, the interplay between multi-host epidemics and predation has received relatively little attention. Here, we analyse a model of a predator-prey system with disease in both prey and predator populations and determine reasonable parameter values using allometric mass scaling relations. Our analysis focuses on the possibility of extinction events rather than the linear stability of the model equations, and we derive approximate relations for the parameter values at which we expect these events to occur. We find that if the predator is a specialist, epidemics frequently drive the predator species to extinction. If the predator has an additional, immune prey species, predators will usually survive. Coexistence of predator and disease is impossible in the single-prey model. We conclude that for the prey species, carrying a pathogen can be an effective weapon against predators, and that being a generalist is a major advantage for a predator in the event of an epidemic affecting the prey or both species

Intraguild Predation: Interactions between Predators, Pathogens, and Their Shared Resources in Crop Pest Communities

Crop pest management requires an understanding of the complex interactions among pest species that potentially damage crop yield and species that may be crucial for controlling pest species outbreaks. For example, predators, parasitoids, and pathogens are constantly interacting via their shared prey or hosts. Predators may prefer infected prey, which can be easier to catch; however, infected prey may be less nutritious or even lethal for predators. These interactions then dictate the short-term dynamics of host and pathogen as well as between prey and predator. How these dynamics change as the species in the system change either empirically or theoretically? is the underlying question in this dissertation. I conducted a meta-analysis to determine the effects of virus- and fungus-infected prey on predators. Examining experiments with one predator/parasitoid and one pathogen, I quantified life-history responses of predators consuming infected prey across published studies. Predators and parasitoids responded separately to infected prey. True predators had no preference, while parasitoids preferred healthy prey. Both predators and parasitoids had reduced fitness when reared in infected hosts. For example, if the host died from infection before the parasitoid completed development, the parasitoid also died. Predators also had a reduction in fitness when consuming infected prey (i.e., shorter lifespans, fewer offspring produced). I then used lab and field studies to expand on these results. In the lab, I reared a common agricultural predator, the spined-soldier bug (Podisus maculiventris) on a diet of either healthy or infected prey, which responded similarly to those in my meta-analysis. They suffered increased developmental time and decreased longevity. I also found that predators exhibited preference for infected prey while the prey were alive. When prey were frozen, the predators exhibited no preference for healthy or infected prey. This indicates that prey are likely easier to consume when they are infected. Field studies investigated how predators change disease transmission in their prey. I found that predators increased transmission by decreasing the prey\u27s heterogeneity in susceptibility to the disease. That is, the spread between the least susceptible and the most susceptible host decreased, which increase overall disease transmission. This research extended the results of the meta-analysis from the individual effects to population dynamics. Finally, I created two mathematical models to show how predators and pathogens interact across multiple generations as compared to the single generation in the field. These models compare and contrast the differences between predator response to prey and the effects the predator response has on disease dynamics. Predators that show a Holling type I response can lead to stable states, while predators exhibiting a type III response lead to cycles exemplified by boom and bust dynamics. Through a meta-analysis, field and lab studies, and a mathematical model, I explored the interactions between predators and pathogens when they attack the same prey or host species. By combining a variety of quantitative techniques to investigate a single question, my work adds important insight into how ecological interactions can help improve agricultural practices

Stability Analysis of a Stochastic Model for Prey-Predator System with Disease in the Prey

In this paper we consider a prey-predator system where the prey population is infected by a microparasite. Local as well as global stability properties of the interior equilibrium point are discussed. The stochastic stability properties of the model are investigated, suggesting that the deterministic model is robust with respect to stochastic perturbations

Modelling dispersal processes in impala-cheetah-lion ecosystems with infection in the lions

The study involved the predator-prey interaction of three species namely the predator (Cheetah Acinonyx jubatus), the super-predator (Lion Panthera leo), and their common prey (Impala Aepyceros melampus). The study area is the Kruger National Park. The predator being an endangered species, faces a survival problem. It is frequently killed by the super-predator to reduce competition for prey. The super-predator also scares away the predator o_ its kills. The prey forms the main diet of the predator. The plight of the predator motivated the author to formulate disease and reaction-diffusion models for the species interactions. The purpose of the models were to predict and explain the effect of large competition from the super-predator on the predator population. Important parameters related to additional predator mortality due to presence of super-predator, the disease incidence rate and induced death rate formed the focal points of the analysis. The dynamics of a predator-prey model with disease in super-predator were investigated. The super-predator species is infected with bovine Tuberculosis. In the study, the disease is considered as biological control to allow the predator population to regain from low numbers. The results highlight that in the absence of additional mortality on the predator by the super-predator, the predator population survives extinction. Furthermore, at current levels of disease incidence, the super-predator population is wiped out by the disease. However, the super-predator population survives extinction if the disease incidence rate is low. Persistence of all populations is possible in the case of low disease incidence rate and no additional mortality imparted on the predator. Furthermore, a two-species subsystem, prey and predator, is considered as a special case to determine the effect of super-predator removal from the system, on the survival of the predator. This is treated as a contrasting case from the smaller parks. The results show that the predator population thrives well in the total absence of its main competitor, with its population rising to at least twice the initial value. A reaction-diffusion three-species predator-prey model was formulated and analysed. Stability of the temporal and the spatio-temporal systems, existence and non-existence of stationary steady state solutions were studied. Conditions for the emergence of stationary patterns were deduced. The results show that by choosing the diffusion coeffcient d2 > _D 2 suffciently large, a non-constant positive solution is generated, that is, stationary patterns emerge, depicting dispersal of species. Predators were observed to occupy habitats surrounding prey. However, super-predators were observed to alternate their habitats, from staying away from prey to invading prey habitat. In the investigation, strategies to determine ways in which the predator species could be saved from extinction and its population improved were devised, and these included isolation of the predator from the super-predator

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

The Effect of Time Delay on Dynamical Behavior in an Ecoepidemiological Model

A delayed predator-prey model with disease in the prey is investigated. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. The effect of the two different time delays on the dynamical behavior has been given. Numerical simulations are performed to illustrate the theoretical analysis. Finally, the main conclusions are drawn

The Dynamics of Stage Structured Prey-Predator Model Involving Parasitic Infectious Disease

In this paper a prey-predator model involving parasitic infectious disease is proposed and analyzed. It is assumed that the life cycle of predator species is divided into two stages immature and mature. The analysis of local and global stability of all possible subsystems is carried out. The dynamical behaviors of the model system around biologically feasible equilibria are studied. The global dynamics of the model are investigated with the help of Suitable Lyapunov functions. Conditions for which the model persists are established. Finally, to nationalize our analytical results, numerical simulations are worked out for a hypothetical set of parameter values