Open access peer-reviewed chapter

On Dynamics and Invariant Sets in Predator-Prey Maps

By Blai Vidiella, J. Tomás Lázaro, Lluís Alsedà and Josep Sardanyés

Submitted: April 25th 2019Reviewed: September 6th 2019Published: November 25th 2019

DOI: 10.5772/intechopen.89572

Downloaded: 89

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, 6, 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 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) ).

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.

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]:

xn+1yn+1=TxnynwhereTxy=Tμ,βxy=μx1xyβxyE1

is defined on the phase space given by the simplex:

S=xy:xy0andx+y1.

We will focus our analysis on the parameter regions, μ04and β05, which contain relevant biological dynamics. State variables xy012denote population densities with respect to a normalised carrying capacity for preys (K=1). Observe that, in fact, if we do not normalise the carrying capacity, the term 1xyin Tμ,βshould read 1x/Ky. As mentioned, preys grow logistically with an intrinsic reproduction rate μ>0without predators. Finally, preys’ reproduction is decreased by the action of predators, which increase their population numbers at a rate β>0due 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 μβ04×05and the parameter regions for which they belong to the simplex S.

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

  • P1=00which belongs to the simplex Sfor every μβ.

  • P2=11μ0which belongs to the simplex Sfor every μβ14×05.

  • P3=1β11μ1βwhich belongs to the simplex Sfor every

Figure 2.

The left picture shows the regions of existence of the fixed points P 1,2,3 ∗ . 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 2 ∗ (respectively P 3 ∗ ) has different local dynamics, together with the type of local dynamics displayed in each of the regions. The analogous picture for the point P 1 ∗ 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, attracting-spiral, and repelling-spiral.

μβ544×μμ15.

The fixed point P1corresponds to co-extinctions, P2to predator extinction and prey survival, and P3to the coexistence of both populations.

Proof: It is a routine to check that P1,P2and P3are 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 P3belongs to the simplex Sif and only if μβ544×μμ15.

Observe that the inequalities μ>0and β>0directly give 1β>0,11μ1β<1,and 11μ=1β+11μ1β<1.So, the statement P3Sis equivalent to 1β<1and

011μ1βμ>1,and1β11μ=μ1μμ>1,andβμμ1.

Clearly, the last two conditions give βμμ1>1which is equivalent to 1β<1. On the other hand, μμ1β5is equivalent to μ54.

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 P1) The fixed point P1is locally asymptotically stable (of attractor node type) if μ01, with eigenvalues λ1=μ<1, λ2=0, and unstable (of hyperbolic type) if μ14. In that case its eigenvalues are λ1=μ>1and λ2=0.

Observe that in both cases, there is an eigendirection, corresponding to the y-axis, which is strongly attracting. As it often happens in many biological systems, its change of stability coincides with the “birth” of the fixed point P2.

Lemma 1.3 (Stability of the point P2). Let us consider in the parameter region μβ14×05,the domain of existence of the fixed point P2S, the curve

β=μμ1E2

(defined and contained in the domain for μ54), and the vertical line μ=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 P2is 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), P2is also unstable, but of hyperbolic type. In the bottom part, the eigenvalues satisfy λ1>1and λ2<1, while in the top part, these inequalities are reversed, λ1<1and λ2>1. As usual, the curves and lines defining the border 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 λ2=1, and on the vertical line μ=3(in red and green colours), one gets λ1=1. On the black point at the intersection of both curves, which has coordinates μβ=3,1.5,the eigenvalues are λ1=1and λ2=1.

And last but not least, the following lemma establishes the different regions of stability for the point P3, the coexistence equilibrium.

Lemma 1.4 (Stability of the point P3) Let us consider in the parameter region μβ544×μμ15,the domain of existence of the fixed point P3S, the above curve (2), and the following three curves:

β=2μμ1black,E3
β=μ2μ1red,E4
β=3μμ+3dashed magenta.E5

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

  1. The region at the top, coloured in pink and delimited by the curve (3), where the point P3is unstable of repeller spiral type (its Jacobian matrix has complex eigenvalues with λ1,2>1).

  2. The green-coloured zone, delimited by the curves (3) and (4), where P3is asymptotically stable of attracting spiral type with complex eigenvalues satisfying λ1,2<1.

  3. The region in brown colour, delimited by the curves (2), (4), and (5). Here the Jacobian matrix of P3has real eigenvalues with λ1,2<1,and P3is locally asymptotically stable of node type.

  4. The bottom region, in light blue, where λ1<1and λ2<1. Therefore, P3is unstable of hyperbolic type.

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 P2is

DTP2=2μ1μ0β11μ,

being triangular, so its eigenvalues are λ1=2μand λ2=β11μ. They are both real and, since μ14, λ2is positive, and concerning λ1,one has λ1<1when μ13, λ1=1when μ=3, and λ1>1when μ34.To determine more precisely the local stability of P2, we study the modulus of λ2on each of these intervals.

Case μ13. As we already said, in this case we have λ1<1and λ2>0. The curve λ2=1is the curve (2) (in solid blue colour in Figure 2 (centre)). This curve intersects the line μ=3at β=3/2and the line β=5at μ=5/4. On this curve the linearised system is stable but nothing can be said, a priori, about the nonlinear system. For the parameters βand μfor which β>μμ1,we have λ2>1and, hence, P2is unstable of hyperbolic type. In a similar way, for those parameters verifying β<μμ1,we get that both eigenvalues λ1,2have modulus strictly smaller than 1. Hence, P2is asymptotically stable of node type.

Case μ=3. Now the eigenvalues are λ1=1and λ2=2β3. When β=32, λ1=1, and λ2=1, so P2is stable for the linearised system. Notice that μβ=33/2is exactly the intersection point of the curve (2) with the line μ=3. If β>32, then λ1=1and λ2>1,so P2is unstable. Finally, if β<32, then λ1=1and λ2<1, and therefore P2is stable for the linearised system.

Case μ34. Since λ1>1, the point P2is always unstable. Moreover, as in the case μ13, the modulus of λ1depends on the position of μand βwith respect to the curve (2) (λ2=1). Consequently, if β=μμ1, then λ2=1and P2is unstable. If β>μμ1, then λ2>1and P2is unstable (of node type). Finally, if β<μμ1, then λ2<1and P2is an (unstable) hyperbolic point.

Proof of Lemma 1.4: The Jacobian matrix of T at the point P3is

DTP3=1μβμββ11μ1β1.

Then, the trace, the determinant of DTP3and the discriminant of the characteristic polynomial of this matrix are

τ=trDTP3=2μβ,D=detDTP3=μ12β,andE6
Δ=τ24D=2μβ24μ12β=μβ+224μ.E7

The eigenvalues of DTP3are given by

λ1,2=τ±Δ2.E8

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

Δ=0μβ+22=4μμβ=2μ1β=μ2μ1,

which corresponds to the red curve (4).

Observe that in the region above the red curve (4), Δ<0.So, the stability of P3in this region is determined by the modulus of

λ1,2=τ±iΔ2=2μβ±i4μμβ+222.

Precisely, we are interested on determining when λ1,2=1or, equivalently, when λ1,22=1.We have

λ1,22=2μβ2+4μ2+μβ24=4μ8μβ4=μ12β.

Therefore, 1=λ1,22=μ12βis equivalent to β=2μμ1, 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 P3has 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, P3is 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

λ1,2=1μ2β±μ2β+12μ,

being λ 1 (respectively λ 2) the eigenvalue corresponding to the + (respectively −) sign.

First we will show that λ1μβ<1in 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 μμ1βand μ4, we have

μ2β2μ122<0μ2β+12μ1<1μ2β+μ2β+12μ=λ1.

Furthermore,

1>λ1=(1μ2β)+(μ2β+1)2μ(μ2β+1)2μ<μ2β(μ2β+1)2μ<(μ2β)21+μβμβμμ1,

This proves that, indeed, λ1μβ<1in the region delimited by the curves (4) and (2), excluding the graph of the curve (2).

Now we study λ2.Observe that, clearly,

μ2β+12μ0<μ2βλ2=1μ2βμ2β+12μ<1.

Next, by using again that μ2β2<0,we have

1=λ2=(1μ2β)(μ2β+1)2μμ2β2=(μ2β+1)2μ(μ2β+1)2μ=2μ2β(μ2β+1)2μ=(2μ2β)23μβ=μ+3β=3μμ+3.

The last equality is curve (5) and, as shown in Figure 2 (right), it intersects the curve (2) at the point μβ=33/2, it is strictly increasing in the interval μ34,and intersects the line μ=4at β=12/7<2. By using the above chain of equivalent equalities, it is easy to check that λ2>1if and only if β>3μμ+3.Thus, the assertions (3) and (4) of the lemma follow straightforwardly.

4. Invariant set: where dynamics live and remain

A first natural question is whether and when Sis the domain of the dynamical system associated with model (1). This amounts asking whether and when Sis T-invariant (i.e. TSS). The complete answer to this question is given by the following proposition and corollary.

In this section, at some point we will consider μβas a function of β. So, for consistency, instead of using the simple notation T for the map from model (1), we will use the notation Tμ,βwhich emphasises the explicit dependence of T on the two parameters μand β.

Proposition 1.5 Tμ,βS=xyR+×R+:xμ+yβ14.

Remark 1.6 Indeed, we can say more: any point uvR+×R+such that

uμ+vβ<14

admits, exactly, two Tμ,βpreimages, and they belong to S. Moreover, if uvR+×R+is such that uμ+vβ=14,then 122vβSis the only Tμ,βpreimage of uv.

The line xμ+yβ=14joins the point μ40with 0β4.So, when β4,it is below the line x+y=1and when β>4it has points outside S. Consequently, from Proposition 1.5 we get

Corollary 1.7 The simplex Sis Tμ,β-invariant if and only if β4.

Remark 1.8 In fact, it can be easily shown that β4implies Tμ,βSSexcept when μ=β=4.

Proof of Proposition 1.5: We start by proving that

Tμ,βSxyR+×R+:xμ+yβ14.

Let xyS.We have Txy=μx1xyβxyand, μx1xy,βxy0because μ,β>0and, since xyS,x,y0and x+y1.So, we have proved that TxyR+×R+.To end the proof of the above inclusion, we have to show that μx1xyμ+βxyβ14.We have

μx1xyμ+βxyβ=x1xy+xy=x1x14.

Next we will show that for every uvR+×R+such that uμ+vβ14, there exists xySsuch that Txy=μx1xyβxy=uv(i.e. u=μx1xyand v=βxy).

If v=0, it is enough to take y=0and x such that μx1x=u.Observe that such point x exists because, in this case,

0u=μuμ+vβμ4.

Next we suppose that v>0.The fact that uR+together with uμ+vβ14implies that 0<vβ4.So, there exist two points 0<y12y+<1such that

βy1y=βy+1y+=v.

Since β>0and 0<yy+<1,the function zy=vβyfrom the interval yy+to 1y+1yis a decreasing homeomorphism (observe that we have zy; zy±=1y±). Moreover, since y12y+,we obtain 1y+121y(see plot above). Consequently,

βx1x:x1y+1y=vβ4.

Hence, there exists a point x=zy1y+1y(of course with yyy+) such that βx1x=βμu+vbecause vβμu+vβ4.Then, for these particular values of yand x=zy, we have βyx=vand

μx1yx=μx1xμyx=μββx1xβyx=μββμu+vv=u.

Next we consider the case β>4.We want to find an invariant subset of Sor, equivalently, the domain of definition of Tμ,βas a dynamical system.

We define the one-step escaping set ɛμ,βas the set of points zSsuch that Tμ,βzS(see Figure 7 for an example). Obviously, ɛμ,βSby definition.

The next proposition gives an estimate of the domain of definition of Tμ,βas a dynamical system (i.e. a Tμ,β-invariant subdomain of S) when β>4and μis small enough.

Proposition 1.9 For every β>4, there exists a unique value μ=μβ04for which the parabola y=1μx1xβμxand the line xμ+yβ=14intersect at a unique point (see Figure 3 ). Then, the set S\ɛμ,βis Tμ,β-invariant for every β>4and μμβ.

Figure 3.

Three examples of the domain S \ ɛ μ , β (in blue) with the one-step escaping set ɛ μ , β plotted in red for β = 5 and μ = 1.5 (left picture), μ = 2.340246528387 ⋯ (centre picture), and μ = 3.525 (right picture). The black region shows the set x y ∈ R + × R + : x μ + y β ≤ 1 4 } ⊃ T μ , β S ⊃ T μ , β S \ ɛ μ , β .

Proposition 1.9 together with Corollary 1.7 give the splitting of the parameter space according to the shape of the invariant set. Figure 4 and its caption give a graphical description of this splitting together with an account of some dynamical aspects in the different regions (see also Figures 5 and 6 ).

Figure 4.

The blue region is the one studied by Corollary 1.7: the set S is T μ , β -invariant. The blue point ( β = μ = 4 ) is, according to Remark 1.8, the unique point where T μ , β S = S . The red curve is μ ∗ β β for β ∈ 4 5 (see Remark 1.10). The green region union with the red curve corresponds to Proposition 1.9: S \ ɛ μ , β is T μ , β -invariant. The region at the left of the brown vertical line (i.e. μ < 1 ) corresponds to the parameters for which there exists global convergence to the fixed point P 1 ∗ (Theorem 1.13). The region between the line μ = 1 and the magenta curve φ β β with φ x ≔ 2 for x ∈ 0 2 , x x − 1 for x ∈ 2 5 , corresponds to the parameters for which there exists global convergence to P 2 ∗ (except for the escaping points and the preimages of P 1 ∗ —Theorem 1.14). The purple dots mark the values of the parameters of the dynamical pictures from Figure 5 , and the olive dots mark the values of the parameters of the dynamical pictures from Figure 6 .

Figure 5.

Plots of the set ∩ i = 0 ∞ T i S for β = μ = 3.412 (left picture), β = μ = 3.5485 (centre picture), and β = μ = 3.895 (right picture). In Figure 4 we can see the location in the parameter space that corresponds to these three dynamical pictures.

Figure 6.

(Top row) The invariant set S \ s ℛ μ , β for several values of β > 4 and μ > μ ∗ β . In Figure 4 we can see the location in the parameter space that corresponds to these three dynamical pictures. (Bottom row) Escaping regions with the number of iterates needed to go out of the domain represented in a gradient from 1 (black), 2 (dark violet), 3 (light violet) to 50 (yellow) iterates. Note the fractal nature of the invariant set and of the escaping regions (see movie1.mp4 for an animation of the invariant and escaping sets as a function of model parameters).

It is well known that the recurrent dynamics of a dynamical system STtakes place in the non-wandering set of T, ΩT,and ΩTi=0TiS(see, for instance, Lemma 4.1.7 from Ref. [13]). Moreover, both sets ΩTand i=0TiSare closed and invariant. Then, in the situation of the above proposition (especially in the light of the above remark), we have ΩTi=0TiSS.To understand the recurrent dynamics of ST, it is clearly interesting (and possible) to characterise the set i=0TiS(see Figure 9 for some examples for different parameter values).

Of course, as we have already implicitly said, in the region at the left and below the magenta curve (see Figure 4 ), one only can expect that i=0TiSwill be either P1or P1P2, and, hence, it does not draw much attention.

For β>4and μ>μβ, we also want to characterise the invariant set where the dynamics occur. To this end, we define the escaping set Rμ,βas the set of points zSsuch that Tμ,βnzSfor some n1.Clearly,

Rμ,β=n=0STμ,βnɛμ,β=Sn=0Tμ,βnɛμ,β.

As Figure 6 shows, the set S\Rμ,βis (not surprisingly) much more complicated than the sets Sand S\ɛμ,β.This prevents obtaining an analytic characterisation of it, as the one given in Proposition 1.11 for the set S\ɛμ,β.However, it is always possible (and easy) to obtain numerical approximations to this set for β>4and μ>μβto gain insight about its shape and topology. Observe (see Figure 6 ) that the invariant set S\Rμ,βcan be fractal.

Remark 1.10. From the proof of Proposition 1.9, it follows that μβis the unique root in the interval 04of the cubic equation:

μ3+α2bα3bμ2+α1bα3bμ+α0bα3b=0

with b=β4and

α3b=b2
α2b=2b3+8b2+16b+32
α1b=b4+16b3+96b2+320b+512,and
α0b=64b2+8b+16.

By means of the Tschirnhaus transformation

μ=zα2b3α3b=z+23b2b3+8b2+16b+32,

the above equation can be transformed into the following equivalent reduced form:

z3p3b4z+2q27b6=0E9

with

{p=3b4(α1(b)α3(b)α2(b)23α3(b)2)=b6+16b5+96b4+320b3+1536b2+4096b+4096,andq=27b62(α0(b)α3(b)α2(b)α1(b)3α3(b)2+2α2(b)327α3(b)3)=b9+24b8+240b7+512b63840b526112b488064b3245760b2393216b262144.

Since the linear coefficient of Eq. (9) is negative, it has three real roots, and, by using the trigonometric solution formula for three real root cases, we obtain

z=213p3b4cosarccos32q27b6123b4p33b4p343π=23b2pcosπarccosqpp343π=23b2pcosarccosqpp3,

and

μβ=zα2b3α3b=23b2pcosarccosqpp3+b3+8b2+16b+32.

To prove Proposition 1.9, we need a full characterisation of the one-step escaping set when β>4.This will be obtained in the next proposition.

Proposition 1.11. For every β>4,

ɛμ,β=xy:x12<141βand1μx1xβμx<y1x0

(see Figure 7 ).

Figure 7.

Two examples of the domain S in blue with the one-step escaping set ɛ μ , β plotted in red for β = 4.5 and μ = 2 (left picture) and β = 5 and μ = 3.75 (right picture). The one-step escaping set ɛ μ , β is vertically delimited by the curves 1 − μx 1 − x β − μ x < 1 − x on the interval with endpoints x ± ≔ 1 2 ± 1 4 − 1 β .

Remark 1.12. Observe that xyTμ,β1xyR+×R+:x+y=1if and only if μx1xy+βxy=1which, in turn, is equivalent to

y=1μx1xβμx.

Consequently,

xyR+×R+:y=1μx1xβμx=Tμ,β1xyR+×R+:x+y=1

and, hence, ɛμ,βis the set of points xywith x12<141βwhich are between the line u+v=1and its Tμ,βpreimage (in particular they belong to S).

Proof of Proposition 1.11: By assumption we have β>4μ.So, additionally, we have βμ>0.We denote

x12141βandx+12+141β

so that x12<141βis equivalent to xxx+.Thus, since

0<5125x<12<x+5+125<1,

x12<141βimplies x01.Hence, μx1x1and 1μx1xβμxare well defined and non-negative.

To simplify the notation and arguments in the proof, we denote

Eμ,β:={(x,y):|x12|<141βand1μx(1x)(βμ)x<y1x}={(x,y):x(x,x+)and1μx(1x)(βμ)x<y1x}.

Then, the proposition states that Eμ,βØand ɛμ,β=Eμ,β.

We start by proving that

1μx1xβμx<1xif and only ifxxx+,E10

which implies that the set Eμ,βis a non-empty subset of S,because xx+01and 01μx1xβμx. To prove (10) observe that

1μx1xβμx=1x1βx1xβμx=0βx1x=1.

On the other hand, xand x+are the two solutions of the equation βx1x=1.Hence, 1μx1xβμx=1xif and only if xxx+.Moreover,

1μx1xβμxx=12=4μ2βμ<12=112

because β>4.So, (10) holds because 12xx+.

Next we will show that Eμ,βɛμ,β.For every xyEμ,βS, we have Tμ,βxy=μx1yxβxywith μx1yx,βxy0.So,

μx1yx+βxy=μx1x+βμxy>μx1x+βμx1μx1xβμx=1.

Consequently, Tμ,βxyS,and hence xyɛμ,β.

To end the proof of the lemma, we show the other inclusion: ɛμ,βEμ,β,which is equivalent to S\Eμ,βS\ɛμ,β.From above (see again Figure 7 ) and the fact that for every point xySwe have μx1yx,βxy0,the inclusion S\Eμ,βS\ɛμ,βcan be written as

{(x,y):x[0,1]\(x,x+)and0y1x}{(x,y):x(x,x+)and0y1μx(1x)(βμ)x}=S\Eμ,βS\εμ,β={zS:Tμ,β(z)S}={(x,y)S:μx(1yx)+βxy1}.

Let us first consider a point xysuch that x01\xx+and y01x.Since xand x+are the two solutions of the equation βx1x=1,it follows that x01\xx+is equivalent to βx1x1.Thus, β>μgives

μx1yx+βxyβx1yx+βxy=βx1x1.

Now we consider a point xysuch that xxx+and 0y1μx1xβμx.In this case, in a similar way as before, we have

μx1yx+βxy=μx1x+βμxyμx1x+βμx1μx1xβμx=1.

Proof of Proposition 1.9: We will use the characterisation of the set ɛμ,βgiven by Proposition 1.11. We start by showing the existence of μ=μβ.

Fix β>4.Clearly, the parabola y=1μx1xβμxand the line xμ+yβ=14intersect if and only if

1μx1xβμxβ4xβμ=0

for some xR+.This equation is equivalent to

4μ2+4ββμx24μ2+βμβμx+4μ4μβμx=0

which, in turn, is equivalent to

4μ2+4ββμx24μ2+βμβμx+4μ=0.

Thus, the parabola y=1μx1xβμxand the line xμ+yβ=14intersect at a unique point if and only if the discriminant of the above quadratic equation is zero:

0=4μ2+βμβμ216μ4μ2+4ββμ=μββ8+16μ32β2β4+32μ2+ββ3+64μ64β2.

We need to study the polynomial

P˜0μββ8+16μ32β2β4+32μ2+ββ3+64μ64β2=α3bμ3+α2bμ2+α1bμ+α0b,

where the coefficients αib,with the change of variables β=4+bwith b01,are

α3bββ8+16=b2>0
α2b2β2β4+32=2b3+8b2+16b+32<0
α1bββ3+64=b4+16b3+96b2+320b+512>0
α0b64β2=64b2+8b+16<0.

To do it we consider the following sequence of polynomials:

P0μP˜0μα3b=μ3+α2bα3bμ2+α1bα3bμ+α0bα3b,P1μ13P0μμ=μ2+2α2b3α3bμ+α1b3α3b,P2μ9α3bemP0μP1μ=6α1bα3b2α2b2μ9α0bα3b+α1bα2b,andP3μP3emP1μP2μ=81α0b2α3b2+12α1b354α0bα1bα2bα3b+12α0bα2b33α1b2α2b236α1b2α3b224α1bα2b2α3b+4α2b4=192b10+6912b9+113664b8+1069056b7+6438912b6+b27131904b5+86507520b4+214695936b3+383778816b2+415236096b+201326592b12+32b11+448b10+3712b9+22528b8+118784b7+>0536576b6+1900544b5+5767168b4+15204352b3+29360128b2+33554432b+16777216

where emPQdenotes the remainder of the division of Pby Q(i.e. Pmodulo Q). Since P3μ0for every b, it follows that gcdP0μP1μ=1,and hence P0μand P1μdo not have common roots. In other words, all roots of P0μare simple. Consequently, since α3b>0for every b,the equation P˜0μ=0is equivalent to P0μ=0,and the above sequence is a Sturm sequence for the polynomial P0μ.The following formulae show this Sturm sequence evaluated at μ=0and μ=4,and the signs of these values:

P00=α0bα3b<0,P10=α1b3α3b>0,P20=9α0bα3b+α1bα2b=2b7+24b6+240b5+1088b4+2816b3+7680b2+18432b+16384<0,P30=P3>0,P04=64+16α2bα3b+4α1bα3b+α0bα3b=64b2+16α2b+4α1b+α0bb2=4b3+32b2+128b+256b>0,P14=16+42α2b3α3b+α1b3α3b=48b2+8α2b+α1b3b2=b3+16b+643b>0,P24=46α1bα3b2α2b29α0bα3b+α1bα2b=b224α1b+9α0b+α2bα1b+8α2b=2bb6+20b5+176b4+704b3+1536bb+1+2048<0,andP34=P3>0.

So, the sign sequences of the Sturm sequence evaluated at μ=0and μ=4are P00P10P20P30=++which has three changes of sign and P04P14P24P34=+++which has two changes of sign. Consequently, the polynomial P0μ(and hence the polynomial P˜0μand in turn the above discriminant) has a unique root μ=μβ04.Moreover, since P00<0and P04>0,the discriminant is negative for every μ0μand positive for every μμ4.This implies that the parabola y=1μx1xβμxand the line xμ+yβ=14do not intersect whenever μ<μand intersect at a unique point when μ=μ(see Figure 3 ). Moreover, for μsmall enough and an arbitrary x01, we have

1μx1xβμxβ4xβμ>0.

Consequently, since the parabola y=1μx1xβμxand the line xμ+yβ=14do not intersect for μ<μ,it follows that

1μx1xβμx>β4xβμand1μx1xβμxβ4xβμ

for every β>4,μ<μ, and x01.On the other hand, by Proposition 1.11, the one-step escaping set ɛμ,βis above the parabola y=1μx1xβμxand, by definition, it is contained in S.Consequently, for every β>4and μμ,

xyR+×R+:xμ+yβ14ɛμ,β=Ø,

which is equivalent to

SxyR+×R+:xμ+yβ14S\ɛμ,β.

On the other hand, for every β>4μ,Tμ,βS\ɛμ,βTμ,βSand, by definition, Tμ,βS\ɛμ,βS.Then, by Proposition 1.5, for every β>4and μμ,

Tμ,βS\ɛμ,βSTμ,βS=SxyR+×R+:xμ+yβ14S\ɛμ,β.

This proves that the set S\ɛμ,βis Tμ,β-invariant for every β>4and μμβ.

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 P1is 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

limnTnxy=00=P1

for every xyS\Rμ,β(the non-escaping set of T) and μ01.

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, P1=00is the unique fixed point of T in Swhen μ01.

We define

φx2forx02,xx1forx25,

a continuous non-increasing map from 05to 542.

Theorem1.14 (Global asymptotic stability for 1<μ<φβFor every parameter point βμ05×1φβand xyS\Rμ,β, we have either

Tnxy=00=P1for somen0,orlimnTnxy=1μ10=P2.

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, P1and P2are the unique fixed points of Tin Sfor every βμ05×1φβ. Moreover, P2is 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, βwas greater than or equal to 2.5. To recycle the proof of Theorem 19 from Ref. [14] for Theorem 1.14 in the case β2,the conditions 1<μ<ββ12and αμμ1μ<1β12,used in that proof, must be replaced, respectively, by 1<μ<φβ=2and αμ<121β,which play the same role.

6. Chaos

Discrete-time systems can display chaotic behaviour at low dimensions. One example is the well-known logistic model which describes the dynamics of a single species with nonoverlapping generations and intraspecific competition [6]. This system is known to undergo the so-called Feigenbaum (period-doubling) route to chaos [15]. In order to identify the chaotic regions in map (1), we compute the full spectrum of Lyapunov exponents using the algorithm described in Ref. [16], pp. 74–80. Figure 8(a) displays a bifurcation diagram obtained by iteration for increasing values of β. Notice that the fixed point P3becomes unstable and a Neimark-Sacker bifurcation takes place. This bifurcation has been detected with the Lyapunov exponents shown in Figure 8(b) , with Λ1,2=0at the bifurcation value. After this bifurcation the first Lyapunov exponent is 0 and the second one is negative. Then the dynamics are governed by attracting invariant curves; further increase of βinvolves the entry into the chaotic regime, where the first Lyapunov exponent, Λ1(in black), becomes positive. Notice the presence of hyperchaotic attractors, with Λ1,2>0.

Figure 8.

Bifurcation diagram for Eq. (1) using β as control parameter and μ = 2.1 . (a) Dynamics on the attractor for predators at increasing β . The violet and orange lines show the values for fixed points P 2 ∗ and P 3 ∗ , respectively. The vertical blue lines display bifurcations. (b) Spectrum of Lyapunov exponents Λ 1 , 2 , computed for the same range of parameter β . Notice that a Neimark-Sacker bifurcation takes place and the fixed point P 3 ∗ 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, Λ 1 . 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.

Enlarged views of the Lyapunov exponents in the parameter space μβare represented in Figure 9 , as well as four examples of the sets i=0TiSfound in the regions labelled with letters in Figure 9(A) .

Figure 9.

(A) Enlarged view of the framed region in grey colour in Figure 8(c) displaying the first Lyapunov exponent, Λ 1 , in the parameter region ( μ , β ∈ 2.5,4 ). In the orange-red regions, the dynamics are chaotic with Λ 1 > 0 . (B) Second Lyapunov exponent, Λ 2 , within the range μ ∈ 2.8,4 and β ∈ 3 4 . The orange-red regions correspond to the hyperchaotic regimes since Λ 1 , 2 > 0 . Lower row of pictures: four plots of the set ∩ i = 0 ∞ T i S found in the regions labelled with the white numbers in panel (A), period-6 fixed point (a), using μ = 3.25 , β = 3.25 , and three examples of strange chaotic attractors, (b) μ = 3.7 , β = 3.2 , (c) μ = 3.8 , β = 3.5 , and (d) μ = 3.7 , β = 3.95 . In all of the phase portraits, we plot the fixed points P 1 ∗ (red), P 2 ∗ (blue), and P 3 ∗ (orange). See movie3.mp4 to visualise the dynamics of Eq. (1) for increasing parameter μ and setting β = 3.9 .

7. Conclusions

In this chapter we have analysed the dynamics of a predator-prey dynamical system in discrete time (see also [12]). We have provided conditions for the global stability of the fixed points corresponding the co-extinctions of the predator-prey as well as for the extinction of predators and survival of preys. For some parameter regions, we have identified hyperchaos (i.e. more than one positive Lyapunov exponent; see [17]). A deep analysis of the existence and properties of the invariant set has been provided for a wide region of the parameter space containing the most biologically relevant dynamics. We have identified the presence of escaping zones in the phase space at which species populations go out of the domain (e.g. they overcome the carrying capacity) and then the iterates become negative, meaning that populations go to extinction. By means of iteration, we have characterised a very complicated shape of the escaping regions, presumably with a highly entangled, fractal topology. These escaping regions could be responsible for species extinctions evolving in discrete time. Although early experimental research allowed to identify deterministic chaos in insect populations [3], as far as we know, no empirical proofs about this phenomenon have been described.

Acknowledgments

We want to thank Ricard Solé, Sergi Valverde, and Tomás Alarcón for useful comments. LlA has been supported by the by Spain’s “Agencia Estatal de Investigación” (AEI) grant MTM2017-86795-C3-1-P. JTL has been partially supported by the Catalan grant 2017SGR1049 and the MINECO grant MTM2015-65715-P and PGC2018-098676-B-100 (AEI/FEDER/UE). BV was funded by the PR01018-EC-H2020-FET-Open MADONNA project. This work has been also partially funded by the CERCA Program of the Generalitat de Catalunya and by the MINECO grant MDM-2014-0445 within the “María de Maeztu” Program. JS has been funded by a “Ramón y Cajal” contract RYC-2017-22243 and by the MINECO grant MTM2015-71509-C2-1-R and AEI grant RTI-2018-098322-B100.

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Blai Vidiella, J. Tomás Lázaro, Lluís Alsedà and Josep Sardanyés (November 25th 2019). On Dynamics and Invariant Sets in Predator-Prey Maps, Dynamical Systems Theory, Jan Awrejcewicz and Dariusz Grzelczyk, IntechOpen, DOI: 10.5772/intechopen.89572. Available from:

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