Abstract
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.
Highlights
Persistence, structure, and dynamics of biological species have received great attention from ecologists as well as mathematical biologists
In order to study the dynamic behavior of model system (1), we consider a set of parameter values as r1 = 4 8, r2 = 4, K1 = 95, K2 = 99, b13 = 0 27, b23 = 0 35, a1 = 1 4, a2 = 0 25, α12 = 0 02, α21 = 0 01, β31 = 1 45, β32 = 1 25, δ = 5 3, γ = 0 005
We found that the carrying capacity K1 of the first prey N1 does not affect the dynamics of the system (see Figure 3(a))
Summary
Persistence, structure, and dynamics of biological species have received great attention from ecologists as well as mathematical biologists. The significance of top predator interference and gestation delay is studied by Jana et al [16] on a three-species food chain model involving intermediate and top predator populations, and they observed the subcritical Hopf bifurcation phenomena. Wang and his group explored the dynamics of different stochastic epidemic models like SIRS and FIV with different infectious forces under intervention strategies and environmental variability [17,18,19,20].
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