The objective of this paper is to study the dynamical behaviour systematically of an ecological system with Beddington-DeAngelis functional response which avoids the criticism occurred in the case of ratio-dependent functional response at the low population density of both the species. The essential mathematical features of the present model have been analyzed thoroughly in terms of the local and the global stability and the bifurcations arising in some selected situations as well. The threshold values for some parameters indicating the feasibility and the stability conditions of some equilibria are also determined. We show that the dynamics outcome of the interaction among the species are much sensitive to the system parameters and initial population volume. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions are also derived with the use of both the normal form and the central manifold theory (cf. Carr [1]). Numerical illustrations are performed finally in order to validate the applicability of the model under consideration.Mathematics Subject Classification: 92D25, 92D30, 92D40.
In this study, a predator–prey population model has been described with disease in the predator species. This is a three-dimensional study comprising of a prey and a predator taken in two different groups such as susceptible and infected predator species. Dynamical behavior of the spread of the disease having the potential to become epidemic has been discussed. Parametric conditions are determined for the control of disease outbreak. Some basic properties like boundedness, persistence of the system have been ensured. Minimal conditions are framed, in such way that the disease can be naturally controlled. Different qualitative behavior like stability, bifurcation and numerical simulations has been performed. Substantial numerical simulations have been carried out in order to validate the obtained theoretical results.
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