Abstract
In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.
Highlights
E dynamical behavior for the prey-predator system takes part in a significant role in theoretical ecology. ere are many ecologists feeling that the unique nonnegative steady state for a predator-prey model is globally asymptotically stable if it becomes stable asymptotically it is not always true
In order to explain the predation phenomenon, Holling [12] has proposed different types of functional responses such as type-I, type-II, and type-III. e response functions are the rate of prey consumption by the predator per unit time. e functional responses proposed by Holling [12] are the functions of prey density only. e response functions are independent of predator density
One of the widely studied functional responses is Holling type-II response function, which is characterized by decelerating intake rate. e rate of predation rises as prey density increases, but after a certain stage, the rate of predation remains constant prey density increases
Summary
Erefore, the positive interior steady state E∗(u∗, v∗) of (4) will be asymptotically stable if r0 − δ1 − 2cu∗ − (αv∗/(1 + βu∗)2) < 0 and otherwise unstable. We have made a relative investigation of the predator-prey model (2), by considering three different functional responses, which are type-I (model (19) with β 0, ξ 0), Holling type-II (already used in model (4) with ξ 0), and Beddington–DeAngelis. We investigate the sufficient condition for global asymptotic stability of the positive coexisting singular point E∗(u∗, v∗) by establishing appropriate Lyapunov function. When the prey birth rate r0 crosses a threshold value r0 r0∗ (δ1 + 2cu∗)(1 + (1/βu∗)) − (1/βu∗)(δ1 + cu∗), the predator-prey model (4) experiences Hopf bifurcation around the interior singular point E∗. The transversality condition holds, and the system (4) experiences Hopf bifurcation at r0 r0∗
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