5. Global dynamics for low values of μ

In this section we investigate global dynamics of the fixed points for low prey's growth rates. This will be done in the next two theorems. In the first one, we show that the fixed point P<sup>∗</sup> <sup>1</sup> is globally asymptotically stable when the intrinsic growth rate of the preys is smaller than 1. See Figure 4 for a view of these results in the parameter space.

Theorem 1.13 (Global asymptotic stability for μ<1) We have

$$\lim\_{n \to \infty} T^n(\mathbf{x}, \boldsymbol{\chi}) = (\mathbf{0}, \mathbf{0}) = P\_1^\*$$

for every ð Þ x, y ∈SnR<sup>μ</sup>, <sup>β</sup> (the non-escaping set of T) and μ∈ ð Þ 0, 1 .

The proof of this theorem goes "mutatis mutandis" along the same lines as the proof of Theorem 15 from Ref. [14] by using that, by Lemmas 1.2, 1.3, and 1.4, P∗ <sup>1</sup> ¼ ð Þ 0, 0 is the unique fixed point of T in S when μ∈ ð Þ 0, 1 :

We define

$$\rho(\mathbf{x}) \coloneqq \begin{cases} 2 & \text{for } \mathbf{x} \in [0, 2], \\ \frac{\mathbf{x}}{\mathbf{x} - \mathbf{1}} & \text{for } \mathbf{x} \in [2, 5], \end{cases}$$

a continuous non-increasing map from 0, 5 ½ � to <sup>5</sup> <sup>4</sup> , 2 � �:

Theorem1.14 (Global asymptotic stability for 1< μ< φ βð Þ For every parameter point ð Þ β , μ ∈ ½ �� 0, 5 ð Þ 1, φ βð Þ and ð Þ x, y ∈SnR<sup>μ</sup>, <sup>β</sup> , we have either

$$\begin{aligned} T^\bullet(\varkappa, \boldsymbol{\jmath}) &= (\mathbf{0}, \mathbf{0}) = P\_1^\ast \quad \text{for some } n \ge 0, \text{or} \\ \lim\_{n \to \infty} T^\bullet(\varkappa, \boldsymbol{\jmath}) &= \left(1 - \boldsymbol{\mu}^{-1}, \mathbf{0}\right) = P\_2^\ast. \end{aligned}$$

#### Figure 8.

Bifurcation diagram for Eq. (1) using β as control parameter and μ ¼ 2:1. (a) Dynamics on the attractor for predators at increasing β . The violet and orange lines show the values for fixed points P<sup>∗</sup> <sup>2</sup> and P<sup>∗</sup> <sup>3</sup> , respectively. The vertical blue lines display bifurcations. (b) Spectrum of Lyapunov exponents Λ<sup>1</sup>,2, computed for the same range of parameter β . Notice that a Neimark-Sacker bifurcation takes place and the fixed point P<sup>∗</sup> <sup>3</sup> becomes unstable, and after an ordered dynamics with invariant curves and periodic fixed points, the dynamics enters into chaos. The chaotic region displays a wide range of hyperchaos, with two positive Lyapunov exponents. (c) Two-parameter phase diagram displaying the ordered and chaotic dynamics by plotting the first Lyapunov exponent, Λ1. Note that chaos is found for large values of μ and β (shown in dark yellow-orange-red colours). See movie2.mp4 for an animation of the dynamics of Eq. (1) at increasing both parameters μ and β . The video displays the bifurcation diagram for β and the corresponding attractors obtained numerically.

On Dynamics and Invariant Sets in Predator-Prey Maps DOI: http://dx.doi.org/10.5772/intechopen.89572

As before, the proof of this theorem goes "mutatis mutandis" as the proof of Theorem 19 from Ref. [14]) taking into account that, by Lemmas 1.2, 1.3, and 1.4, P∗ <sup>1</sup> and P<sup>∗</sup> <sup>2</sup> are the unique fixed points of T in S for every ð Þ β , μ ∈½ �� 0, 5 ð Þ 1, φ βð Þ . Moreover, P<sup>∗</sup> <sup>2</sup> is the unique locally asymptotically stable fixed point of T in this parameter region. The difference between this theorem and Theorem 19 from Ref. [14] is that, in that paper, β was greater than or equal to 2.5. To recycle the proof of Theorem 19 from Ref. [14] for Theorem 1.14 in the case β ≤2, the conditions 1<μ< <sup>β</sup> <sup>β</sup> �<sup>1</sup> <sup>≤</sup> 2 and αμ<sup>≔</sup> <sup>μ</sup>�<sup>1</sup> <sup>μ</sup> < <sup>1</sup> <sup>β</sup> ≤ <sup>1</sup> <sup>2</sup> , used in that proof, must be replaced, respectively, by 1<μ<sup>&</sup>lt; φ βð Þ¼ 2 and αμ <sup>&</sup>lt; <sup>1</sup> <sup>2</sup> ≤ <sup>1</sup> <sup>β</sup> , which play the same role.
