## Lim xt 0 lim yt

and the predator population z evolving (Figure 9c) (0, 0, z^(t)) is locally asymptotically stable.

Figure 10, displays typical bifurcation diagrams for the prey populations x, y and the predator population z of the system [8] as p increases from 0 to 54 with initial value X(0) = (3, 4, 5). We can observe how the variable z(t) oscillates in a stable cycle. In contrast, the prey x(t) and y(t) rapidly decrease to zero. If the amount p of releasing species is smaller than pmax = 53.87 232 860, the prey-eradication solution becomes unstable and undergoes a transcritical bifurcation, then the prey and predator can coexist in a stable positive periodic solution when p <pmax = 53.87 232 860, and system [8] can be permanent. If the parameter increases further, the system will exhibit a wide variety of dynamic behavior. As the bifurcation parameter p increases, the bifurcation diagrams clearly show that system [8] has rich dynamics including period-doubling bifurcation, chaos bands, symmetry-breaking pitchfork bifurcations, period-halving bifurcations, crises (the phenomenon of 'crisis' in which chaotic attractors can suddenly appear or disappear, or change size discontinuously as a parameter smoothly varies), quasi-periodic oscillation, narrow periodic window, and wide periodic window, etc.

Because our focus is on parameter-dependent equations, it is natural to ask how dynamic behavior depends on bifurcation parameter p. In particular, we would like to know for which values of parameter one may expect chaotic behavior. So, we can compartmentalize the region p p [0, 54] into six parts considering the bifurcation.

### Equilibrium

As the bifurcation diagrams show, when 0< p <16.36, the system [8] has exactly a T period solution, that is, a stable equilibrium gives way to a simple limit cycle. That means the system is in 'the balance of nature' (Figure 10 and Figure 11a).

### Period-doubling bifurcation

For p « 16.36, there is an exchange of stability of period T to period 2 T, that is, bifurcation. This phenomenon is also called period doubling or flip bifurcation. When 16.36 <p < 19.55, there is a sequence of period-doubling bifurcations 2 T, 4 T, 8 T,..., finally into a region of apparent chaos (Figure 11 ).

Chaos

For 19.55 <p < 26.39, the system is in the chaotic region. In this case, the system has rich dynamical behavior. For p > 25.35, there is a cascade of period-halving bifurcations leading to a 4T period solution (Figure 12). When p slightly increases beyond ^25.5 935, there is a series of strong oscillations. In this case, the chaos suddenly disappears and suddenly appears again (Figure 13). When p« 26.2, the phenomenon of 'crisis' occurs, that is, the chaos suddenly disappears (Figure 14).

### Period-halving bifurcation

Similarly, from p « 26.35 to p ~ 36.0, there is a cascade of period-halving bifurcations leading to a quasi- T periodic solution till p« 30.90, and next there is a cascade of period-doubling bifurcations till p = 36.0 (Figure 15).

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Figure 11 Period-doubling bifurcation leads to chaos of system [8] with initial value X(0) = (3,4, 5). (a) Phase portrait of T period solution for p = 16.0; (b) phase portrait of 2T period solution for p = 16.5; (c) phase portrait of 4Tperiod solution for p = 19.0; (d) phase portrait of 8Tperiod solution forp = 19.35; (e) phase portrait of 16Tperiod solution forp = 19.5; and (f) phase portrait of the solution as it enters chaos region for p = 20.0.

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Figure 12 Period-halving bifurcation from chaos to cycles of system [2] with initial value X(0) = (3, 4, 5). (a) Chaos when p = 25.35; (b) phase portrait of 16T period solution for p = 25.42; (c) phase portrait of 8T period solution for p = 25.47; and (d) phase portrait of 4T period solution for p = 25.593.

Chaos

For 36.0 <p < 43.49, the system enters the chaotic region again. For p = 36, there is an exchange of stability of period to chaos, that is, the phenomenon of 'crisis' occurs when p > 36.0. It is easy to see that sufficiently small perturbations can suddenly vary the dynamic behavior of the system (Figure 16).

### Period-halving bifurcation

After these chaotic areas, that is, whenp >43.49, the solution to the system [8] undergoes a cascade of period-halving bifurcations from chaos to one cycle (Figure 17). In other words, the system [8] is permanent.

From Figures 10-17, we observe that a stable equilibrium gives way to a simple limit cycle. Further changes in the parameter p lead to a doubling of the period. Subsequent period-doubling occurs at an accelerating rate, until at the limit of 'infinite period' the system becomes chaotic. On the other hand, for certain ranges of the parameter values, the quasi-periodic attractor may be replaced by a limit cycle. The structure of the quasi-periodic attractor becomes steadily more complex, and

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