α The mutual interference of predator and 0<α<1.
In recent years, mutual interference of predator has been studied extensively by some researchers [11 13].
In 1971, Hassell [11] proposed the concept of mutual interference of predator species and introduced the mutual interference constant m ((0 < m le 1)).
Figure 1(c)–(d) imply that different values of mutual interference of predator m and the prey refuge θ bring different dynamic properties for system (1.2). Figure 2 and Fig. 3 show that impulsive period T heavily affects the dynamic behavior of system (1.2).
If we change the mutual interference of the predator m and the prey refuge constant θ, the other parameters are the same as in Fig. 1, then we get another dynamic portrait that is very different from Fig. 1.
(c) dynamic behavior of system (1.2) with the mutual interference of the predator (m=0.8) and the prey refuge (theta=0.7), (d) phase portrait of (x t)), (y(t)), (z(t)), other parameters are the same as in Fig. 1(a).
Theorem 2.1, Theorem 3.1, Theorem 4.1, and simulations show that dynamic properties of (1.2) depend on the mutual interference of the predator m, the prey refuge θ, delays (tau_{i}) ((i = 1,2,3)), and the parameter (d_{i}) ((i = 1,2,3)) of pests or predator which represent the regular harvest or deaths from spraying pesticide.
Figure 1(c) and Fig. 1 (d) show that the mutual interference of the predator m and the prey refuge constant θ affects the dynamic behaviors of system (1.2). Figure 1 The dynamic behavior of system (1.2) with initial values of (x 0)=1), (y(0)=1), (z(0)=1), (a) time series of (x t)), (y(t)), (z(t)), which are simulated 100 cycles in the interval ([0,200]).
Ω is a bounded domain in (R^{n}) with smooth boundary ∂Ω, (f(S,u) = frac{S}{1 + m_{1}S + k_{1}u}), (g(S,upsilon ) = frac{S}{1 + m_{2}S + k_{2}upsilon}) are Beddington-DeAngelis functions; (m_{i}), (i = 1,2) are the Michaelis-Menton constants; (k_{i}) ((i = 1,2)) model mutual interference of between predators.
It is assumed that predator feeding rate decreases by higher predator density even when prey density is high, and effects of predator interference in feeding rate remain important all the time whenever an individual predator is handling or searching for a prey at a given instant of time.
(I3): It is assumed that ω is the effects of the capture rate, (k_{1}) is the handing time and (k_{2}) is the magnitude of interference among predator (zooplankton) [18].
