In this paper, we investigate the effect of mutual interference on the dynamics of a predator-prey system with gestation delay. It has been observed that there is stability switches and system becomes unstable due to the combine effect of mutual interference and time delay. We determine the conditions under which the model system becomes globally asymptotically stable around the non-zero equilibria. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. Computer simulations have been carried out to illustrate different analytical findings.
In this paper, we consider a Gause-type model system consisting of two prey and one predator. Gestation period is considered as the time delay for the conversion of both the prey and predator. Bobcats and their primary prey rabbits and squirrels, found in North America and southern Canada, are taken as an example of an ecological system. It has been observed that there are stability switches and the system becomes unstable due to the effect of time delay. Positive invariance, boundedness, and local stability analysis are studied for the model system. Conditions under which both delayed and nondelayed model systems remain globally stable are found. Criteria which guarantee the persistence of the delayed model system are derived. Conditions for the existence of Hopf bifurcation at the nonzero equilibrium point of the delayed model system are also obtained. Formulae for the direction, stability, and period of the bifurcating solution are conducted using the normal form theory and center manifold theorem. Numerical simulations have been shown to analyze the effect of each of the parameters considered in the formation of the model system on the dynamic behavior of the system. The findings are interesting from the application point of view.
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