Abstract

A multitude of physical, chemical, or biological systems evolving in discrete time can be modelled and studied using difference equations (or iterative maps). Here we discuss local and global dynamics for a predator-prey two-dimensional map. The system displays an enormous richness of dynamics including extinctions, co-extinctions, and both ordered and chaotic coexistence. Interestingly, for some regions we have found the so-called hyperchaos, here given by two positive Lyapunov exponents. An important feature of biological dynamical systems, especially in discrete time, is to know where the dynamics lives and asymptotically remains within the phase space, that is, which is the invariant set and how it evolves under parameter changes. We found that the invariant set for the predator-prey map is very sensitive to parameters, involving the presence of escaping regions for which the orbits go out of the domain of the system (the species overcome the carrying capacity) and then go to extinction in a very fast manner. This theoretical finding suggests a potential dynamical fragility by which unexpected and sharp extinctions may take place.

Keywords: bifurcations, chaos, invariant sets, maps, nonlinearity, ecology

## 1. Introduction

Natural and artificial complex systems can evolve in discrete time, often resulting in extremely complex dynamics such as chaos. A well-known example of such a complexity is found in ecology, where discrete-time dynamics given by a yearly climatic forcing can make the population emerging a given year to be a discrete function of the population of the previous one [1]. Although early work already pointed towards complex population fluctuations as an expected outcome of the nonlinear nature of species interactions [2], the first evidence of chaos in species dynamics was not characterised until the late 1980s and 1990s [3, 4]. Since pioneering works on one-dimensional maps [5, 6], the field of dynamical complexity in ecology experienced a rapid development [5–7], with several key investigations offering a compelling evidence of chaotic dynamics in insect species in nature [1, 3, 4].

Discrete-time models have played a key role in the understanding of complex ecosystems, especially for univoltine species (i.e. species undergoing one generation per year) [5, 6]. Many insects inhabiting temperate and boreal climatic zones

Figure 1.

Two-species predator-prey dynamics can be studied with difference equations or maps when species generations are discrete (univoltine). (a) Here we display two insect species with univoltine generations at the North Hemisphere. The Heteroptera Picromerus bidens predates the butterfly Pararge aegeria by consuming the eggs (photos obtained from the Wikipedia). A simple model for this type of system is given by the map (1). (b) Some typical dynamics arising in discrete-time ecological systems for preys (green dots) and predators (blue dots): (upper panel) period-one fixed point and (lower panel) chaos.

behave as univoltine species, for example, Lepidoptera [8], Coleoptera [9], or Heteroptera [10] species, among others. For Lepidoptera, the populations of the butterfly Pararge aegeria are univoltine in its most northern range (e.g. northern Scandinavia). Adult butterflies emerge in late spring, mate, and die shortly after laying the eggs. Then, their offspring grow until pupation, entering diapause before winter. New adults emerge the following year, thus resulting in a single generation of butterflies per year [11].

Some predators feed on these univoltine insects. For example, Picromerus bidens (Heteroptera) predates on Pararge aegeria by consuming their eggs. Thus, both prey and predator display coupled yearly cycles (Figure 1(a)). This type of systems has been modelled using two-dimensional discrete-time models, such as the one we are introducing in this chapter, given by the map (1) (see Ref. [12] for more details on this model). As mentioned, the dynamical richness of discrete ecological models was early recognised [5, 6] and special attention has been paid to small food chains incorporating two species in discrete systems [12]. These systems, similarly to single-species maps, display static equilibria, periodic population oscillations, as well as chaotic dynamics (see, e.g. Figure 1(b)).

A crucial point that we want to address in this chapter is the proper characterisation of the invariant set in which the dynamics lives. This is of paramount importance for discrete-time systems since the iterates can undergo big jumps within the phase space and extinctions can occur in a very catastrophic manner if some iterate visits the so-called escaping regions. That is, catastrophic extinctions not caused by bifurcations but from topological features of the invariant sets may occur. Together with the characterisation of the invariant set, we provide a dynamical analysis of fixed points, local and global stability, as well as a numerical investigation of chaos.

## 2. Predator-prey map

We consider a food chain of two interacting species with predator-prey dynamics, each with nonoverlapping generations (see Figure 1(a)). The preys x grow logistically without the presence of predators population y, following the logistic map [6]. The proposed model to study such ecosystem can be described by the following system of nonlinear difference equations [12]:

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

$$
\begin{pmatrix} \mathbf{x}\_{n+1} \\ \mathbf{y}\_{n+1} \end{pmatrix} = T \begin{pmatrix} \mathbf{x}\_n \\ \mathbf{y}\_n \end{pmatrix} \qquad \text{where} \qquad T \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} = T\_{\mu,\beta} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} = \begin{pmatrix} \mu \mathbf{x} \begin{pmatrix} 1 - \mathbf{x} - \mathbf{y} \end{pmatrix} \\ \beta \mathbf{x} \mathbf{y} \end{pmatrix} \tag{1}
$$

is defined on the phase space given by the simplex:

$$\mathbf{S} = \{ (\mathbf{x}, \mathbf{y}) : \mathbf{x}, \mathbf{y} \ge \mathbf{0} \text{ and } \mathbf{x} + \mathbf{y} \le \mathbf{1} \}.$$

We will focus our analysis on the parameter regions, μ∈ð � 0, 4 and β ∈ð � 0, 5 , which contain relevant biological dynamics. State variables ð Þ <sup>x</sup>, <sup>y</sup> <sup>∈</sup> ½ � 0, 1 <sup>2</sup> denote population densities with respect to a normalised carrying capacity for preys (K ¼ 1). Observe that, in fact, if we do not normalise the carrying capacity, the term 1 � x � y in T<sup>μ</sup>, <sup>β</sup> should read 1 � x=K � y. As mentioned, preys grow logistically with an intrinsic reproduction rate μ> 0 without predators. Finally, preys' reproduction is decreased by the action of predators, which increase their population numbers at a rate β >0 due to consumption of preys.

## 3. Fixed points and local stability

The next lemma provides the three fixed points of the dynamical system defined by the map (1) for ð Þ μ, β ∈ð �� 0, 4 ð � 0, 5 and the parameter regions for which they belong to the simplex S.

Lemma 1.1. The dynamical system (1) on the simplex S has the following three fixed points (see Figure 2 (left)):


$$(\mu, \beta \,) \in \left[\frac{5}{4}, 4\right] \times \left[\frac{\mu}{\mu - 1}, 5\right].$$

The fixed point P<sup>∗</sup> <sup>1</sup> corresponds to co-extinctions, P<sup>∗</sup> <sup>2</sup> to predator extinction and prey survival, and P<sup>∗</sup> <sup>3</sup> to the coexistence of both populations.

Proof: It is a routine to check that P<sup>∗</sup> <sup>1</sup> , P<sup>∗</sup> <sup>2</sup> and P<sup>∗</sup> <sup>3</sup> are the unique possible fixed points of model (1). Thus, the first and the second statements of the lemma are evident.

We need to prove that P<sup>∗</sup> <sup>3</sup> belongs to the simplex S if and only if ð Þ μ, β ∈ 5 <sup>4</sup> , 4 � � � <sup>μ</sup> <sup>μ</sup>�<sup>1</sup> , 5 h i:

Observe that the inequalities μ>0 and β >0 directly give <sup>1</sup> <sup>β</sup> <sup>&</sup>gt;0, 1 � <sup>1</sup> <sup>μ</sup> � <sup>1</sup> <sup>β</sup> < 1, and 1 � <sup>1</sup> <sup>μ</sup> <sup>¼</sup> <sup>1</sup> <sup>β</sup> <sup>þ</sup> <sup>1</sup> � <sup>1</sup> <sup>μ</sup> � <sup>1</sup> β � �<1: So, the statement <sup>P</sup><sup>∗</sup> <sup>3</sup> ∈S is equivalent to <sup>1</sup> <sup>β</sup> < 1 and

$$0 \le 1 - \frac{1}{\mu} - \frac{1}{\beta} \Leftrightarrow \begin{cases} \mu > 1, \text{and} \\ \frac{1}{\beta} \le 1 - \frac{1}{\mu} = \frac{\mu - 1}{\mu} \Leftrightarrow \begin{cases} \mu > 1, \text{ and} \\ \beta \ge \frac{\mu}{\mu - 1}. \end{cases} \end{cases}$$

Figure 2.

The left picture shows the regions of existence of the fixed points P<sup>∗</sup> <sup>1</sup>,2,3. The centre (respectively right) picture specifies the regions of the parameter space (of course in its parametric domain of definition) where the fixed point P<sup>∗</sup> <sup>2</sup> (respectively P<sup>∗</sup> <sup>3</sup> ) has different local dynamics, together with the type of local dynamics displayed in each of the regions. The analogous picture for the point P<sup>∗</sup> <sup>1</sup> has been omitted for simplicity. The codification for the stability zones follows the next rules: Capital letters indicate stability—U indicates unstable, while AS indicates asymptotic stability. The subscripts show the type of stability: hyp = hyperbolic, node, attractingspiral, and repelling-spiral.

Clearly, the last two conditions give β ≥ <sup>μ</sup> <sup>μ</sup>�<sup>1</sup> <sup>&</sup>gt;1 which is equivalent to <sup>1</sup> <sup>β</sup> <1. On the other hand, <sup>μ</sup> <sup>μ</sup>�<sup>1</sup> <sup>≤</sup> <sup>β</sup> <sup>≤</sup>5 is equivalent to <sup>μ</sup><sup>≥</sup> <sup>5</sup> 4 :

In the next three lemmas, the different regions of local stability of these fixed points are studied. This study, standard in dynamical systems theory, is based on the computation of the eigenvalues of the Jacobian matrix at each fixed point and on the determination of the regions where their moduli are smaller or larger than 1. To ease the reading, the proofs have been deferred to the end of the section.

Lemma 1.2 (Stability of the point P<sup>∗</sup> <sup>1</sup> ) The fixed point P<sup>∗</sup> <sup>1</sup> is locally asymptotically stable (of attractor node type) if μ∈ ð Þ 0, 1 , with eigenvalues λ<sup>1</sup> ¼ μ< 1, λ<sup>2</sup> ¼ 0, and unstable (of hyperbolic type) if μ∈ð � 1, 4 . In that case its eigenvalues are λ<sup>1</sup> ¼ μ>1 and λ<sup>2</sup> ¼ 0.

Observe that in both cases, there is an eigendirection, corresponding to the yaxis, which is strongly attracting. As it often happens in many biological systems, its change of stability coincides with the "birth" of the fixed point P<sup>∗</sup> 2 .

Lemma 1.3 (Stability of the point P<sup>∗</sup> <sup>2</sup> ). Let us consider in the parameter region ð Þ <sup>μ</sup>, <sup>β</sup> <sup>∈</sup>½ �� 1, 4 ð � 0, 5 , the domain of existence of the fixed point <sup>P</sup><sup>∗</sup> <sup>2</sup> ∈ S, the curve

$$
\beta = \frac{\mu}{\mu - 1} \tag{2}
$$

(defined and contained in the domain for μ≥ <sup>5</sup> 4), and the vertical line μ ¼ 3. The curve and the line divide this domain into four regions (as shown in Figure 2 (centre)). Then, the local stability of system (1) in a neighbourhood of the fixed point P<sup>∗</sup> <sup>2</sup> is as follows: In the bottom-left region (brown), it is locally asymptotically stable (attractor of node type). In the top-right region (magenta), it is unstable (repelling of node type). In the bottom-right and top-left regions (in light blue colour), P<sup>∗</sup> <sup>2</sup> is also unstable, but of hyperbolic type. In the bottom part, the eigenvalues satisfy ∣λ1∣>1 and ∣λ2∣<1, while in the top part, these inequalities are reversed, ∣λ1∣< 1 and ∣λ2∣>1. As usual, the curves and lines defining the border

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

between these regions are characterised by a pass-through modulus 1 of some of the eigenvalues. Indeed, on the curve (2) (in blue colour, solid and dashed), one has λ<sup>2</sup> ¼ 1, and on the vertical line μ ¼ 3 (in red and green colours), one gets λ<sup>1</sup> ¼ �1. On the black point at the intersection of both curves, which has coordinates ð Þ¼ μ, β ð Þ 3, 1:5 , the eigenvalues are λ<sup>1</sup> ¼ �1 and λ<sup>2</sup> ¼ 1.

And last but not least, the following lemma establishes the different regions of stability for the point P<sup>∗</sup> <sup>3</sup> , the coexistence equilibrium.

Lemma 1.4 (Stability of the point P<sup>∗</sup> <sup>3</sup> ) Let us consider in the parameter region ð Þ <sup>μ</sup>, <sup>β</sup> <sup>∈</sup> <sup>5</sup> <sup>4</sup> , 4 � � � <sup>μ</sup> <sup>μ</sup>�<sup>1</sup> , 5 h i, the domain of existence of the fixed point <sup>P</sup><sup>∗</sup> <sup>3</sup> ∈ S, the above curve (2), and the following three curves:

$$
\beta = 2 \frac{\mu}{\mu - 1} \quad \text{(black)}, \tag{3}
$$

$$\beta = \frac{\mu}{2\left(\sqrt{\mu} - 1\right)} \quad (\text{red}), \tag{4}$$

$$
\beta = 3\frac{\mu}{\mu+3} \quad (\text{dashed magneta}).\tag{5}
$$

These curves divide this domain into four regions (see Figure 2 (right)):


We present now the proofs of Lemma 1.3 and Lemma 1.4. The one of Lemma 1.2 has been omitted since it consists on straightforward computations.

Proof of Lemma 1.3: The Jacobian matrix of T at the point P<sup>∗</sup> <sup>2</sup> is

$$DT\left(P\_2^\*\right) = \begin{pmatrix} 2-\mu & 1-\mu\\ 0 & \beta\left(1-\frac{1}{\mu}\right) \end{pmatrix},$$

being triangular, so its eigenvalues are <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup> � <sup>μ</sup> and <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>β</sup> <sup>1</sup> � <sup>1</sup> μ � �. They are both real and, since μ∈ð � 1, 4 , λ<sup>2</sup> is positive, and concerning λ1, one has j j λ<sup>1</sup> < 1 when μ∈ð Þ 1, 3 , j j λ<sup>1</sup> ¼ 1 when μ ¼ 3, and j j λ<sup>1</sup> > 1 when μ∈ð � 3, 4 : To determine more precisely the local stability of P<sup>∗</sup> <sup>2</sup> , we study the modulus of λ<sup>2</sup> on each of these intervals.

Case μ ∈ ð Þ 1, 3 . As we already said, in this case we have j j λ<sup>1</sup> < 1 and λ<sup>2</sup> >0. The curve λ<sup>2</sup> ¼ 1 is the curve (2) (in solid blue colour in Figure 2 (centre)). This curve intersects the line μ ¼ 3 at β ¼ 3=2 and the line β ¼ 5 at μ ¼ 5=4. On this curve the linearised system is stable but nothing can be said, a priori, about the nonlinear system. For the parameters β and μ for which β > <sup>μ</sup> <sup>μ</sup>�<sup>1</sup> , we have <sup>λ</sup><sup>2</sup> <sup>&</sup>gt;1 and, hence, P∗ <sup>2</sup> is unstable of hyperbolic type. In a similar way, for those parameters verifying β < <sup>μ</sup> <sup>μ</sup>�<sup>1</sup> , we get that both eigenvalues <sup>λ</sup>1,2 have modulus strictly smaller than 1. Hence, P<sup>∗</sup> <sup>2</sup> is asymptotically stable of node type.

Case <sup>μ</sup> <sup>¼</sup> <sup>3</sup>. Now the eigenvalues are <sup>λ</sup><sup>1</sup> ¼ �1 and <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup> <sup>β</sup> <sup>3</sup> . When <sup>β</sup> <sup>¼</sup> <sup>3</sup> 2 , λ<sup>1</sup> ¼ �1, and <sup>λ</sup><sup>2</sup> <sup>¼</sup> 1, so <sup>P</sup><sup>∗</sup> <sup>2</sup> is stable for the linearised system. Notice that ð Þ¼ μ, β ð Þ 3, 3=2 is exactly the intersection point of the curve (2) with the line <sup>μ</sup> <sup>¼</sup> 3. If <sup>β</sup> <sup>&</sup>gt; <sup>3</sup> <sup>2</sup>, then <sup>λ</sup><sup>1</sup> ¼ �1 and <sup>λ</sup><sup>2</sup> <sup>&</sup>gt;1, so <sup>P</sup><sup>∗</sup> <sup>2</sup> is unstable. Finally, if β < <sup>3</sup> <sup>2</sup>, then λ<sup>1</sup> ¼ �1 and λ<sup>2</sup> <1, and therefore P<sup>∗</sup> <sup>2</sup> is stable for the linearised system.

Case <sup>μ</sup> <sup>∈</sup> ð � <sup>3</sup>, <sup>4</sup> . Since j j <sup>λ</sup><sup>1</sup> <sup>&</sup>gt;1, the point <sup>P</sup><sup>∗</sup> <sup>2</sup> is always unstable. Moreover, as in the case μ∈ ð Þ 1, 3 , the modulus of λ<sup>1</sup> depends on the position of μ and β with respect to the curve (2) (λ<sup>2</sup> <sup>¼</sup> 1). Consequently, if <sup>β</sup> <sup>¼</sup> <sup>μ</sup> μ�1 , then <sup>λ</sup><sup>2</sup> <sup>¼</sup> 1 and <sup>P</sup><sup>∗</sup> <sup>2</sup> is unstable. If β > <sup>μ</sup> μ�1 , then λ<sup>2</sup> >1 and P<sup>∗</sup> <sup>2</sup> is unstable (of node type). Finally, if <sup>β</sup> <sup>&</sup>lt; <sup>μ</sup> μ�1 , then j j <sup>λ</sup><sup>2</sup> <sup>&</sup>lt;1 and <sup>P</sup><sup>∗</sup> <sup>2</sup> is an (unstable) hyperbolic point.

Proof of Lemma 1.4: The Jacobian matrix of T at the point P<sup>∗</sup> <sup>3</sup> is

$$DT\left(P\_3^\*\right) = \begin{pmatrix} 1 - \frac{\mu}{\beta} & -\frac{\mu}{\beta} \\ \beta \left(1 - \frac{1}{\mu} - \frac{1}{\beta}\right) & 1 \end{pmatrix}.$$

Then, the trace, the determinant of DT P<sup>∗</sup> 3 � � and the discriminant of the characteristic polynomial of this matrix are

$$\tau = \text{tr}DT(P\_3^\*) = 2 - \frac{\mu}{\beta}, \qquad D = \text{det } DT(P\_3^\*) = \mu \left(1 - \frac{2}{\beta}\right), \text{ and} \tag{6}$$

$$
\Delta = \pi^2 - 4D = \left(2 - \frac{\mu}{\beta}\right)^2 - 4\mu \left(1 - \frac{2}{\beta}\right) = \left(\frac{\mu}{\beta} + 2\right)^2 - 4\mu. \tag{7}
$$

The eigenvalues of DT P<sup>∗</sup> 3 � � are given by

$$
\lambda\_{1,2} = \frac{\pi \pm \sqrt{\Delta}}{2}. \tag{8}
$$

The curve determining whether the eigenvalues are real or complex is Δ ¼ 0, that is,

$$
\Delta = 0 \quad \Leftrightarrow \quad \left(\frac{\mu}{\beta} + 2\right)^2 = 4\mu \quad \Leftrightarrow \quad \frac{\mu}{\beta} = 2\left(\sqrt{\mu} - 1\right) \quad \Leftrightarrow \quad \beta = \frac{\mu}{2\left(\sqrt{\mu} - 1\right)},
$$

which corresponds to the red curve (4).

Observe that in the region above the red curve (4), Δ <0: So, the stability of P<sup>∗</sup> 3 in this region is determined by the modulus of

$$\lambda\_{1,2} = \frac{\pi \pm \mathrm{i}\sqrt{-\Delta}}{2} = \frac{\left(2 - \frac{\mu}{\beta}\right) \pm \mathrm{i}\sqrt{4\mu - \left(\frac{\mu}{\beta} + 2\right)^2}}{2}.$$

Precisely, we are interested on determining when j j λ1,2 ¼ 1 or, equivalently, when j j λ1,2 <sup>2</sup> <sup>¼</sup> <sup>1</sup>: We have

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

$$\left|\lambda\_{1,2}\right|^2 = \frac{\left(2 - \frac{\mu}{\beta}\right)^2 + 4\mu - \left(2 + \frac{\mu}{\beta}\right)^2}{4} = \frac{4\mu - 8\frac{\mu}{\beta}}{4} = \mu\left(1 - \frac{2}{\beta}\right).$$

Therefore, 1 ¼ j j λ1,2 <sup>2</sup> <sup>¼</sup> <sup>μ</sup> <sup>1</sup> � <sup>2</sup> β � � is equivalent to <sup>β</sup> <sup>¼</sup> <sup>2</sup><sup>μ</sup> <sup>μ</sup>�1, which is the black curve (3). This implies that, in the pink-coloured region above the black curve (3), displayed in Figure 2 (right), the point P<sup>∗</sup> <sup>3</sup> has complex eigenvalues with modulus greater than 1, and, consequently, it is unstable of repelling spiral type. Analogously, the green region corresponds to complex eigenvalues λ1,2, with (both) moduli smaller than 1. Here, P<sup>∗</sup> <sup>3</sup> is asymptotically stable of attracting spiral type.

In the region below the red curve (4), where Δ >0, both eigenvalues are real. They can be rewritten as

$$
\lambda\_{1,2} = \left(\mathbf{1} - \frac{\mu}{2\beta}\right) \pm \sqrt{\left(\frac{\mu}{2\beta} + \mathbf{1}\right)^2 - \mu},
$$

being λ<sup>1</sup> (respectively λ2) the eigenvalue corresponding to the + (respectively �) sign.

First we will show that j j λ1ð Þ μ, β <1 in the region delimited by the curves (4) and (2) (including the graph of the curve (4) and excluding the graph of the curve (2)). Observe that, since <sup>μ</sup> <sup>μ</sup>�<sup>1</sup> <sup>≤</sup> <sup>β</sup> and <sup>μ</sup>≤4, we have

$$\begin{aligned} \frac{\mu}{2\beta} - 2 &\leq \frac{\mu - 1}{2} - 2 < 0 \leq \sqrt{\left(\frac{\mu}{2\beta} + 1\right)^2 - \mu} \Leftrightarrow -1 < \left(1 - \frac{\mu}{2\beta}\right) + \sqrt{\left(\frac{\mu}{2\beta} + 1\right)^2 - \mu} \\ &= \lambda\_1. \end{aligned}$$

Furthermore,

$$\begin{aligned} \mathbf{1} &> \lambda\_1 = \left(\mathbf{1} - \frac{\mu}{2\beta}\right) + \sqrt{\left(\frac{\mu}{2\beta} + \mathbf{1}\right)^2 - \mu} \Leftrightarrow \sqrt{\left(\frac{\mu}{2\beta} + \mathbf{1}\right)^2 - \mu} < \frac{\mu}{2\beta} \Leftrightarrow \left(\frac{\mu}{2\beta} + \mathbf{1}\right)^2, \\\ & \quad -\mu < \left(\frac{\mu}{2\beta}\right)^2 \Leftrightarrow \mathbf{1} + \frac{\mu}{\beta} < \mu \Leftrightarrow \beta > \frac{\mu}{\mu - 1}, \end{aligned}$$

This proves that, indeed, j j λ1ð Þ μ, β < 1 in the region delimited by the curves (4) and (2), excluding the graph of the curve (2).

Now we study j j λ<sup>2</sup> : Observe that, clearly,

$$-\sqrt{\left(\frac{\mu}{2\beta} + 1\right)^2 - \mu} \le 0 < \frac{\mu}{2\beta} \quad \Leftrightarrow \quad \lambda\_2 = \left(1 - \frac{\mu}{2\beta}\right) - \sqrt{\left(\frac{\mu}{2\beta} + 1\right)^2 - \mu} < 1.1$$

Next, by using again that <sup>μ</sup> <sup>2</sup> <sup>β</sup> � 2< 0, we have

$$\begin{split}-\mathbf{1} &= \lambda\_2 = \left(1 - \frac{\mu}{2\beta}\right) - \sqrt{\left(\frac{\mu}{2\beta} + 1\right)^2 - \mu} \Leftrightarrow \frac{\mu}{2\beta} - 2 \\ &= -\sqrt{\left(\frac{\mu}{2\beta} + 1\right)^2 - \mu} \Leftrightarrow \sqrt{\left(\frac{\mu}{2\beta} + 1\right)^2 - \mu} = 2 - \frac{\mu}{2\beta} \Leftrightarrow \left(\frac{\mu}{2\beta} + 1\right)^2 - \mu \\ &= \left(2 - \frac{\mu}{2\beta}\right)^2 \Leftrightarrow 3\frac{\mu}{\beta} = \mu + 3 \Leftrightarrow \beta = 3\frac{\mu}{\mu + 3}.\end{split}$$

The last equality is curve (5) and, as shown in Figure 2 (right), it intersects the curve (2) at the point ð Þ¼ μ, β ð Þ 3, 3=2 , it is strictly increasing in the interval μ∈½ � 3, 4 , and intersects the line μ ¼ 4 at β ¼ 12=7 <2. By using the above chain of equivalent equalities, it is easy to check that <sup>λ</sup><sup>2</sup> <sup>&</sup>gt; � 1 if and only if <sup>β</sup> <sup>&</sup>gt;<sup>3</sup> <sup>μ</sup> <sup>μ</sup>þ<sup>3</sup> : Thus, the assertions (3) and (4) of the lemma follow straightforwardly.
