Abstract
In this work, a prey-predator model with square root response function under a state-dependent impulse is proposed. Firstly, according to the differential equation geometry theory and the method of successor function, the existence, uniqueness and attractiveness of the order-1 periodic solution are analyzed. Then the stability of the order-1 periodic solution is discussed by the Poincaré criterion for impulsive differential equations. Finally, we show a specific example and carry out numerical simulations to verify the theoretical results.
Highlights
1 Introduction The herd behavior, such as of drifting herbivores observed in the savanna, is a scenario in which the predator can only interact with the prey along the outer corridor of the prey herd when the prey are attached by the predator
Unlike the discussion of the literature [1] as regards the existence of Hopf bifurcations, Bachchu et al investigated the nonexistence of periodic orbits, and the existence and uniqueness of limit cycles. They found the impact of herd behavior mechanism of prey population to the model system analytically. They analyzed how herd behavior of prey controls the dynamics of the model system near origin in an ecologically meaningful way
If yF1 < yP1+ < yF2, due to any two orbits are disjoint, the orbit passing through point P1+ tangents to the phase set N at point P1+, and does not intersect with the impulsive set M, the order-1 periodic solution is nonexistent (see Figure 4(c))
Summary
The herd behavior, such as of drifting herbivores observed in the savanna, is a scenario in which the predator can only interact with the prey along the outer corridor of the prey herd when the prey are attached by the predator. If yF1 < yP1+ < yF2 , due to any two orbits are disjoint, the orbit passing through point P1+ tangents to the phase set N at point P1+, and does not intersect with the impulsive set M, the order-1 periodic solution is nonexistent (see Figure 4(c)).
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