Bifurcation Analysis of an Impulsive Three Species Predator Prey Model

Recently, the field of research on chaotic impulsive differential equations about biological control seems to be a new increasingly interesting area, to which some scholars have paid attention. Here, we illustrate our new work about impulsive predator-prey system with Watt-type functional response. The ecosystem is of the form described in [8]:

Ay = -S2y }t =(n + l- 1)T Az = -&3z x (nT+) = x(nT) y(nT+) = y(nT) \t = nT z(nT+) = z(nT) + p with ax = x(t+)—x(t), where, x(t+) = lim x(t). ay =

y(t+)—y(t), az = z(t+)—z(t), x(t), and y(t) are functions of time representing population density of prey species, and z(t) is population size of predator species. All parameters are positive constants, hi (i = 1,2, 3) are intrinsic rates of increase or decrease, a, ft are parameters representing competitive effects between two prey, ai (i = 1,2) are coefficients of decrease of prey species due to predation, d (i = 1,2) are equal to the transformation rate of predator, 0< l <1, 0 < 6i < 1 (i = 1,2, 3), T is the period of the impulsive effects, n p N, N is the set of non-negative integers. p >0 is the bifurcation parameter.

Here, we consider the following set of parameters for our analysis: T = 6, a1 = 1, a2 = 1.1, h1 = 4.1, b2 = 4.5, b3 = 0.6, d1 = 9, d2 = 10, c = 0.5, m = 0.2, a = 0.1, ft = 0.15, S1 = 0.2, 62 = 0.15, 63 = 0.0001, l = 0.5 and the initial value (x(0), y(0), z(0)) = (3, 4, 5). Figure 9 displays the dynamic behavior of system [8] with the time series. In Figure 9, we can see, when p>pmax = 53.87 232 860, the two prey populations

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