Conditioned chaos in seasonally perturbed predator-prey models

Ecological Modelling, 69 (1993) 79-97 Elsevier Science Publishers B.V., Amsterdam

79

Conditioned chaos in seasonally perturbed predator-prey models

Sergio Rinaldi a and Simona Muratori b

a CIRITA, Politecnico di Milano, 20133 Milano, Italy b Politecnico di Torino, 10129 Torino, Italy

(Received 24 February 1992; accepted 15 September 1992)

ABSTRACT

Rinaldi, S. and Muratori, S., 1993. Conditioned chaos in seasonally perturbed predator-prey models. Ecol. Modelling, 69: 79-97.

The problem of chaos in second-order predator-prey models with sinusoidally perturbed parameters is investigated in this paper. The analysis is carried out numerically, by means of a continuation method producing bifurcation curves in two-dimensional spaces. The results show that the phase separating two sinusoidal perturbations in time is a strategic variable for controlling the dynamics of the populations. In a large region of the parameter space, phase determines whether the behaviour of the system is chaotic or non-chaotic.

1. INTRODUCTION

The problem of chaos in population communities has received a great deal of at tention in the last 15 years. On one hand, we have many and detailed studies on the chaotic behaviour of mathematical models: populations with non-overlapping generations (May, 1974), host-parasi toid systems (Lauwerier and Metz, 1986; Bellows and Hassell, 1988; Hochberg et al., 1990), p reda to r -p rey communities (Inoue and Kamifukumoto, 1984; Schaffer, 1988; Toro and Aracil, 1988; Allen, 1989; Kuznetsov et al., 1992; Rinaldi et al., 1993), food chains (Hogeweg and Hesper, 1978; Hastings and Powell, 1991; Scheffer, 1991) and microbial systems (Kot et al., 1992). On

Correspondence to: S. Rinaldi, Dipartimento di Eiettronica e Informazione, Politecnico di Milano, Via Ponzio, 34/5, 20133 Milano, Italy.

0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

80 S. RINALDI AND S. MURATORI

the other hand, we have studies based on statistical analysis of recorded time series, showing some evidence of chaos (strange attractors) in nature or in laboratory experiments. Among these examples of strange attractors we can recall Nicholson's blowflies (Hassell et al., 1976; Brillinger et al., 1980), the Canadian lynx (Schaffer, 1984), some plant populations (Gutier- rez and Almiral, 1989), and measles epidemics (Schaffer and Kot, 1985; Kot et al., 1988; Olsen et al., 1988; Sugihara and May, 1990). But very few biological data sets are sufficiently long to allow the application of the numerical techniques used to estimate properties of the attractors underly- ing dynamic systems (Godfray and Blythe, 1990). For this reason, one could conjecture that during the next few years the research effort will be oriented toward the study of the chaotic properties of increasingly sophisti- cated models of population communities. In particular, we can expect that models will include increasingly detailed biological and physical aspects, that parameters will be varied within realistic ranges and that the analysis will be carried out by using the most powerful theoretical and computa- tional tools available for studying the dynamics of non-linear systems.

In this paper we study the problem of chaos in second-order continuous-time predator-prey models with periodically varying parameters. The first contribution on the subject is by Inoue and Kamifukumoto (1984) who varied the intrinsic growth rate of the prey and focused their discussion on the influence of the frequency of the forcing function. This is classical in mechanics and electronics, but has, unfortunately, very little relevance to ecology, where the period of the forcing function is fixed and known to be related to well-identified environmental cycles (e.g., daily light cycle, tide and moon cycles, seasons). Moreover, there is no interesting biological reason (see below) for varying only the intrinsic growth rate of the prey as done in Inoue and Kamifukumoto (1984) and later in Toro and Aracil (1988) and Allen (1989) for quite unusual predator-prey models. On the contrary, the discussion in Schaffer (1988) is much more interesting because the model is sound and the parameter which is periodically varied is the caloric content of the prey, which, indeed, can vary through seasons, for example, in plant-herbivore communities. Nevertheless, the analysis is carried out through simulation and this is possibly the reason why only one route to chaos, namely "torus destruction", has been detected by Schaffer. More detailed analyses (Kuznetsov et al., 1992; Rinaldi et al., 1993) have shown that the second route to chaos, namely "cascade of period dou- blings", is also present. In general, it occurs for relatively large amplitudes of the sinusoidal variations of the parameters, even if the predator-prey system does not autonomously cycle when the environment is constant. On the contrary, torus destruction is only possible if predator and prey populations autonomously oscillate when the system is unperturbed. This means

CHAOS IN PERTURBED PREDATOR-PREY MODELS 8 ]

that chaos can be present in a predator -prey community provided that the exogeneous and endogeneous sources of periodicities are, as a whole, sufficiently strong.

In all the above contributions on predator -prey models, only one parameter is periodically varied. This is rather naive since real populations are influenced by many and often independent environmental factors. In freshwater phytoplankton-zooplankton communities, for instance, light intensity, as well as temperature and nutrient concentration vary conspicu- ously during the year. For this reason, we analyse in this paper the behaviour of predator -prey models influenced by independent sources of periodicity. In particular, we try to understand how chaos can be reinforced or damped by suitably "controlling" the phase between different seasonalities. Information on this matter, although still quite abstract, is of great conceptual interest for interpreting the behaviour of ecosystems. It is also a good starting point for developing Decision Support Systems in the field of renewable resources management, where the time and intensity of stocking and harvesting must be well tuned with natural periodicity mechanisms in order to avoid undesirable modes of behaviour.

The paper is organized as follows. In the next section we describe the predator -prey model and we review the results on chaos induced by sinusoidal perturbations of a single parameter. In Section 3 we consider the case of two sinusoidally varying parameters Pi with average values Pi0 and amplitudes E~pi0, i = 1, 2, and we define regions of "conditioned chaos" in the two-dimensional parameter space (E 1, e2). By definition, for each point of this region the model can have either a regular (typically periodic) behaviour, or a chaotic behaviour, depending upon the value of the phase between the two sinusoids. In Section 4 we show how these regions can be computed by means of a standard continuation technique and in Section 5 we present the results of the analysis and we point out the main properties of the regions of conditioned chaos. In the last section we discuss the implications of our results and we suggest directions for further research.

2. THE MODEL AND ITS FIVE ELEMENTARY SEASONALITY MECHANISMS

The model we discuss in this paper is the classical Rosenzweig-MacAr- tur predator -prey model used in the last 20 years to interpret the behaviour of many predator -prey communities, namely

x) ay 1 A = x r 1 - - ~ b-+x ' (1)

= d , (2) Y e b + x

82 s. RINALDI AND S. MURATORI

where the six parameters r, K, a, b, d and e are positive and x and y are the numbers of individuals of prey and predator populations or suitable (but equivalent) measures of their density or biomass. The intrinsic growth rate r describes the exponential growth of the prey population at low densities, while the carrying capacity K is the prey biomass at equilibrium in the absence of predators. The function q ( x ) = a x / ( b + x ) appearing in Eqs. (1) and (2) is the type H functional response (Holling, 1965), where a is the maximum harvest rate of each predator, and b is the half-saturation constant, namely the density of prey at which the predation rate is half maximum. Finally, the parameter e in Eq. (2) is a simple conversion factor, called efficiency, that specifies the number of newly born predators for each captured prey, while d is the predator death rate per capita.

The analysis of the local stability of the equilibria of model (1),(2) (May, 1972) shows that there is a H o p f bifurcation at

ea + d K = b - - (3)

ea - d '

and a transcritical bifurcation at

d K = b - - (4)

ea - d "

The Hopf bifurcation is always supereritical (the computation of its Lia- punov number is relatively easy if one considers the orbitally equivalent system obtained by multiplying Eqs. (1) and (2) by (b +x)) and the asymptotic period of the appearing limit cycle is

T.=2 .

Moreover, the limit cycle does not bifurcate since it is unique (Cheng, 1981; Wrzosek, 1990). Thus, the parameter space is partitioned into three regions separated by the manifolds (3) and (4). For all combinations of the parameters there is a single attractor which is globally stable in the first quadrant. More precisely, for sufficiently high values of the carrying capacity K, the attractor is a stable limit cycle. For decreasing values of K this cycle shrinks and disappears through a Hopf bifurcation. Then the attractor is a stable equilibrium which is positive for intermediate values of K and trivial (absence of predator population) for low values of K.

Of course, if relevant environmental factors fluctuate seasonally the parameters of model (1),(2) must vary periodically. For simplicity we consider only sinusoidal perturbations and we normalize their period to 1 so that for any parameter Pi in Eqs. (1) and (2) we write

pi=Pio[1 + e i sin 2"rr(t - - , i ) ] ,

CHAOS IN P E R T U R B E D P R E D A T O R - P R E Y MODELS 83

where Pio is the average value of Pi, ei is the "degree of seasonality" (notice that e.iPio is the magnitude of the perturbation) and "/'i is the phase, namely the time at which the parameter Pi reaches its average value PiO from below. Obviously, 0 ~< e i ~< n l because Pi cannot be negative: ei = 0 corresponds to absence of seasonality, while •i = 1 means that the maximum value of the parameter is twice its average value.

Let us now consider the following five elementary seasonality mechanisms, each giving rise to periodic variations of a single parameter in Eqs. (1) and (2).

Prey intraspecific competition Surplus of prey mortality at high densities due to competition for special

niches or to epidemics can be enhanced in some seasons. If this is the case, the carrying capacity varies periodically, i.e., K = K0(1 + e/~ sin 2~-~-).

Caloric content o f the prey If the caloric content of the prey varies during the year, like in some

plant-herbivore communities, the energy available to the predator for reproduction varies consistently. Hence the efficiency varies periodically, i.e., e = e0(1 + E e sin 27r~').

Predator exploitation The periodic presence of a superpredator exploiting the predator com-

munity gives rise to periodic variations of the predator death rate, i.e., d = d0(1 + e d sin 27r~-). Phytoplankton-zooplankton communities with first-year class fish feeding on zooplankton during the summer, and t ree- insect pest systems controlled by migratory insectivores are examples of this class.

Predator and prey mimicry When the degree of mimicry of the prey (predator) is not constant

during the year or when variations of the habitat facilitate the escape or the capture of the prey in some specific season, the time spent by the predator to find one unit of prey varies periodically. This implies (Metz and Van Batenburg, 1985; Rinaldi et al., 1993) that the half-saturation constant varies in the same way, i.e., b = b0(1 + e b sin 27r~').

Predator resting time If the resting time of the predator fluctuates during the year, as in

populations characterized by some degree of diapause, the maximum harvest rate of the predator varies in the same manner (Metz and Van Batenburg, 1985; Kuznetsov et al., 1992), i.e., a --a0(1 + e a sin 27r~').

84 S. R1NALDI AND S. MURATORI

> ii: i;ii ?i!iiii!ii _ .. .:.-:-.::..I ::.:.. ?.:.-.i~!~!!!!,i ///~-:CNAOS.:.-iiI.:::I:::

/ H~.~-,::

~_~ H

f2 f, f .

I 0 0,5

DEGREE OF SEASONALITY £i

Fig. 1. The two regions of chaos of model (1),(2) when a single parameter is periodically varied: Pi = Pi0 (1 + ei sin 27rt). Curve h is a Hopf bifurcation curve. Curves f2 and f4 are period-doubling bifurcation curves. The other period-doubling bifurcation curves fs, f16 . . . . , are not shown but accumulate on f~.

For each one of the above mechanisms, the bifurcations of system (1),(2) have been computed (Rinaldi et al., 1993) by fixing the constant parameters at some reference value and by varying the mean value Pio and the degree of seasonality (?i of the periodically perturbed parameter. The five bifurcation diagrams turn out to be qualitatively similar in the parameter space (Ei, P;0). They point out the existence of strange attractors in two different subregions called CHAOS 1 and CHAOS 2 in Fig. 1. Point H on the vertical axis of Fig. 1 indicates the value of Pio for which model (1),(2) has a Hopf bifurcation in the constant parameter case (e i = 0). Below this point, the attractor of the unperturbed system is an equilibrium, while above it the attractor is a limit cycle. Curve h, rooted at point H, is a Hopf bifurcation curve separating periodic solutions of period 1 from quasi-periodic solutions (invariant tori). If Pio increases, the quasi-periodic solutions disappear on a bifurcation "set" through a homoclinic structure (torus destruction) and become genuine strange attractors in region CHAOS 1. Since the bifurcation set is not a regular curve, the boundary of region CHAOS 1 cannot be computed. Moreover, our numerical experiments have shown that the basins of attraction of the strange attractors obtained through torus destruction are often quite small, so that it is relatively unlikely that the system behaves chaotically even if the parameters (Ei, Pio) fall in region CHAOS 1. Finally, we like to stress that region CHAOS 1 is entirely above point H since curve h is always rising. This means that strange attractors of this kind can only be obtained in preda tor -prey

CHAOS IN PERTURBED PREDATOR-PREY MODELS 85

communities that behave on a limit cycle if the time-varying parameter Pi is frozen at its average value Pio"

On the contrary, region CHAOS 2 does not suffer of these limitations. Its boundary is a regular curve f~ which can be obtained numerically, since it is the accumulation curve of a sequence of period-doubling bifurcation curves f2, f4, f 8 , . . . , which can be computed by means of s tandard continuation procedures. In other words, approaching region CHAOS 2, the attractor which is initially a forced cycle of period 1 becomes of period 2, 4, 8 . . . . . and finally loses periodicity on curve f= and becomes a strange attractor afterwards. Curves f2 and f4 are very close to each other and curve f4 practically coincides with curve f®. Thus, in the following, the boundary fo~ of region CHAOS 2 will be approximated by curve f2. Moreover, all our numerical experiments indicate that the strange attractors of region CHAOS 2 are globally stable in the first quadrant. But even if this is not true in all cases, certainly the basins of attraction are very large and therefore the system can easily behave chaotically when the parameters (Ei, Pio) fall in region CHAOS 2. Finally, we would like to point out that the lowest part of region CHAOS 2 can be characterized (as in the case of Fig. 1) by values of Pi0 which are smaller than the value corresponding to the Hopf bifurcation in the constant parameter case. This means that even p reda to r -p rey communities which behave at equilibrium in a constant environment can be t ransformed into chaotic communities by sinusoidally varying a parameter , provided the amplitude of the perturbation is sufficiently high (examples of this kind are explicitly shown in Rinaldi et al. (1993)).

From now on we only consider chaos obtained through cascade of period-doubling bifurcations. Thus, if all the parameters of model (1),(2) are fixed at some nominal value and one of them is varied by adding to its nominal value Pio a sinusoidal perturbation (6.iPio sin 27rz), we can have three types of behaviour (see Fig. 1): Type O: for no value of E i the attractor is chaotic. Type 1: for 0 < e i < e i < 1 the attractor is chaotic. Type 2: for 0 < _e i < e i < gi <~ 1 the attractor is chaotic. Hence, depending upon the type of the system, we can have regular (i.e., non chaotic) behaviour for all degrees of seasonality, chaotic behaviour at high degrees of seasonality, or chaotic behaviour for intermediate degrees of seasonality.

3. MULTIPLE SEASONALITIES AND CONDITIONED CHAOS

We now assume that there are two independent sources of seasonality acting at the same time on the predator and prey populations. If we restrict

86 s, R I N A L D I A N D S. M U R A T O R I

our attention to the five elementary mechanisms described in the preceding section, we obtain 10 possible combinations (pairs). Each one of them is identified by two sinusoidally varying parameters

Pl =Pl0[ 1 + el sin 2 7 r ( t - ~1)],

PE =P20[ 1 + ez sin 2 7 r ( t - r 2 ) ].

Obviously, by properly selecting the origin of time, one of the two phase parameters can be annihilated. In other words, there is only one phase parameter of interest, namely

3" ~ ' / ' 2 - - T 1 ,

and this parameter varies between 0 and 1 as e I and e 2 do. Thus, if the mean values Pl0 and P20 are fixed, as well as all other parameters appearing in Eqs. (1) and (2), there are only three parameters qualifying the seasonalities, namely ca, e 2 and r.

Instead of computing and displaying the bifurcation surface delimiting the region of chaos in the three-dimensional space (el, e 2, "r), we still work with the two-dimensional space (el, e 2) and we use the phase r as a free (control) parameter. Of course, for any given value of ~', we could compute (and display) the bifurcation curves f~(~-) in the plane (el, e2). Obviously, such curves are rooted at \point e i or ~i on the ei-axis if the system is of type 1 or 2 with respect to Pi. But, more synthetically, we can partition the unit square 0 ~< e i ~< 1, i = 1, 2, into three subregions called "regular", "chaotic" and "conditionally chaotic". By definition, for no value of the phase ~-, give points in the first region rise to strange attractors obtained through cascade of period-doubling bifurcations, while points in the chaotic region correspond to strange attractors for all values of ~'. Consequently, in the third region, the asymptotic mode of behaviour of the system can be either chaotic or non-chaotic depending upon the phase ~'. Thus, for each point Q = (el, e 2) belonging to the region of conditioned chaos there is at least one value of r for which the corresponding bifurcation curve f~(~-) passes through Q. This means that the region of conditioned chaos is the union of all bifurcation curves fo~(z).

4. METHOD OF INVESTIGATION

We have seen in the preceding section that the region of conditioned chaos is the union of all bifurcation curves f~(~-) in the space (e 1, E2). As already said, these curves cannot be directly produced, but they can be fairly well approximated by the period-doubling bifurcation curves f2(~-). These bifurcation curves have been computed by means of a continuation method interactively supported by the program LOCBIF developed by A.

CHAOS1N P E R T U R B E D P R E D A T O R - P R E Y MODELS 87

Khibnik, Yu. Kuznetsov, V. Levitin and E. Nikolaev at the Research Computing Centre of the USSR Academy of Sciences at Pushchino. The method can be briefly described as follows (Khibnik, 1990a,b).

System (1),(2) with two sinusoidally varying parameters can be trans- formed into an autonomous three-dimensional system adding the equation i = 1 (t modl) . For this system Poincar6 section and Poincar6 (first return) map

(x(1), y(1))=P(1)(x(O), y(0))

can be defined (Arnold, 1982; Guckenheimer and Holmes, 1986). Obvi- ously, the map p(1) depends upon (e 1, e e, r). Fixed points of the second iterate of the map

(x(2), y(2))=P°)(P°)(x(O), y(O)))=P(2)(x(O), y(0))

correspond to periodic solutions (cycles) of Eqs. (1) and (2) with period 2. Moreover, a periodic solution of period 2 undergoes a period-doubling bifurcation if the Jacobian matrix of the map p(2) evaluated at the fixed point has a multiplier equal to - 1 (Arnold, 1982). Thus, the bifurcation curve f2(r) can be computed by projecting a one-dimensional manifold located in the four dimensional space (x, y, ca, e2) on the (el, e2)-plane. The manifold is defined by the two fixed point equations

(x, y)=p(2)(x, y) and by the bifurcation condition

det(A + I ) = 0

where A is the Jacobian matrix of p(2) at (x, y) and I is the 2 × 2 unit matrix. In the program LOCBIF, the bifurcation curves are computed by means of an adaptive predict ion-correction continuation procedure with tangent prediction and Newton correction. All relevant derivatives, as well as the Poincar6 map and its iterates, are evaluated numerically. Of course, bifurcation curves f2(al) in the space (r, E 2) and f2(~2) in the space (r, a 1) can also be produced with the same package.

In order to determine the set of conditioned chaos without computing too many bifurcation curves, we have followed the procedure illustrated in Fig. 2 for a hypothetical case in which the system is of type 0 with respect to Pl and of type 1 with respect to P2. The method starts by fixing the control parameter at will, say r = %, and by computing the bifurcation curve f2(%) with the continuation method starting from point 1 = (0, _%). Then, a point A = (¢IA, e2A) is selected on curve f2(r 1) and used to produce, by continuation from point (e2A, %), the bifurcation curve fZ(E1A) in the space (r, e2). Such a curve has a maximum and a minimum value of

88 S. RINALDI AND S. MURATORI

2 ,~,) 0.5 ~ - f z (,~s)

3 f2 (%)

I 0,5

DEGREE OF SEASOflALITY E 1

Fig. 2. The procedure used to genera te the boundary of the region of condi t ioned chaos in

the space (E 1, E2). Points 1, 2 . . . . , 5 belong to the boundary. The bifurcation curves f2("ri), i = 1 . . . . , 5 belong to the region of condi t ioned chaos.

e 2 for two distinct values of z, say z 2 and z 3. The maximum and minimum values of e z define points 2 and 3 in Fig. 2. By construction, these two points belong to the boundary of the region of conditioned chaos. Then, the two bifurcation curves f2(z2) and f2(z3) are computed by continuation, starting from points 2 and 3 (or from point 1). Notice that the curves f2(z2) and f2(z3) belong to the region of conditioned chaos: hence the boundary of that region is tangent to f2(z2) at point 2 and to f2G-3) at point 3. The procedure continues by selecting one point on each one of these two curves (see points B and C in Fig. 2) and these points are used to produce, by continuation, the bifurcation curves f2(e2B) and f2(E2c). Thus, two new points (4 and 5 in Fig. 2) of the boundary of the region of conditioned chaos are obtained, together with their corresponding values ~4 and z 5 of the control parameter. The bifurcation curves f2(z4) and f2(zs) passing through points 4 and 5 can therefore be computed by continuation.

The procedure can be iterated in an obvious way to obtain new points on the boundary of the region of conditioned chaos, and it is clear from Fig. 2 that the method converges very quickly. Indeed, the boundary of the region of conditioned chaos is already well identified in Fig. 2, since it must pass through five points and have a given tangent in four of these points.

5. ANALYSIS OF T H E RESULTS

Following the procedure outlined in the preceding section, we have computed the region of conditioned chaos for all pairs of elementary

CHAOS IN P E R T U R B E D P R E D A T O R - P R E Y MODELS

0 Ek E-b C Eb

0 0 ~e

1 E b C ~

0

Ee

1 ,E a C

C

Ee 1

E Eb

0 _6k Ek i 0 {k

/

1

89

Fig. 3. Conditioned chaos for different pairs of perturbed parameters. In region R (C), the system behaves regularly (chaotically) for all values of the phase. In the shaded region (conditioned chaos), the system is regular or chaotic depending upon the phase. The figures correspond to the following values of the parameters: e = 1, r -- d = 2~-, a = 47r, b = 0.3, K = 1 in ( i) . . . . . (iv), K = 1.2 in (v), K = 1.5 in (vi).

mechan isms (1), (2 ) , . . . ,(5) and for d i f fe ren t values of the parameters . Six examples are r epor t ed in Fig. 3, where R and C are the regular and chaot ic regions and the shaded region is the condi t ional ly chaotic one. In the first example (i) the system is of type 0 with respect to the first pa r ame te r ( K ) and of type 1 with respect to the second p a r a m e t e r (d). In examples (ii), (iii) and (iv) the system is of type I with respect to both parameters . Finally, in the last two examples the system is of type 2 with respect to the first p a r a m e t e r while with respect to the second pa ramete r , it is of type 1 in (v) and of type 2 in (vi).

90 s. RINALDI AND S, MURATORI

The first important fact pointed out by Fig. 3 is that the region of conditioned chaos is very large. This means that the two degrees of seasonality e 1 and is 2 are very often not sufficient to qualify the kind of behaviour (regular or chaotic) of the system. In other words, the phase z is a strategic control variable: for many degrees of seasonality (el, E 2) chaos can be present or absent depending upon the parameter r. This might be important information to be taken into account when analysing the dynamics of populations influenced by environmental factors with different phases. Typical examples are freshwater plankton communities where the phase T between temperature and light is increasing with the volume of the lake, and prey-predator systems located along the path of a migratory superpredator, where the death rate of the predator varies periodically with a phase dependent upon location.

As already said, the boundary of the region of conditioned chaos is rooted at the critical points e i and Ei o n the ei-axis. This implies that a system which behaves regularly with a degree of seasonality e 1 slightly smaller (bigger) than e 1 (gl), can be made chaotic by slightly perturbing a second parameter with a suitable phase. Similarly, a system which is chaotic with e 1 slightly greater (smaller) than e~ (gl) can be regularized by adding a second source of seasonality, provided the phase z is suitably tuned.

Another fact emerging from our analysis is that the points (el, e 2) are always interior points of the region of conditioned chaos. This means that when two sources of seasonality are active at the same time, chaos can be sustained by degrees of seasonality smaller than those strictly required by a single source, i.e.,

E1 ~__. 1, if2 ~__.2 •

But for the same reason, it is also true that the system can behave regularly even if

E1 ~ - f f l , if2 ~ - f f2 .

For example, in Fig. 3 chaos can be sustained by degrees of seasonality of the order of 0.2 but also fully avoided when the degrees of seasonality are equal to 1.

An unexpected result of the analysis is that the boundary of the region of conditioned chaos going from point (gl' O) to point (0, e 2) is very well approximated by the straight line

E1 E 2 - - + - - = 1 (5) -¢-1 -E2

in particular, for all combinations of parameters giving rise to relatively low values of e 1 and e 2. In other words, there is a kind of additivity property of the minimum degrees of seasonality needed to generate chaos.

C H A O S I N PERTURBED P R E D A T O R - P R E Y M O D E L S 91

Another property (not readable in Fig. 3) is that the points of the boundary (5) of the region of conditioned chaos are associated with almost the same value of the phase, say r = r*. This means that if the phase is equal to r* the chances to have chaotic dynamics at low degrees of seasonality are maximal. The same property does not hold, in general, for the rest of the boundary of the region of conditioned chaos. Thus, if one likes to transform a type 1 chaotic system with e I > gl into a regular system by using the lowest possible degree of seasonality e2, one must determine the phase r which, indeed, depends upon e r

>., g

g == c, . .

;>, g

E

O.q' O,q

0,2

G / - \ I , , :. ,,

,' ;"

• /):~ \

0,5 I PREY POPULATION x

0 o o i

,i\ ,: \

j J

i 0,5

PREY POPULATION x

©

O,U, 0,4

i "x \ \ \

®

0,2

i' /

o,', { o:, PREY POPULATION x PREY POPULATION x

®

0,2 °~ r -

=s ,<

0,2

== ml

==

N

==

= C~ -Io.6

i °gV U ,G/wy y VYY,-I 0 5 i0 15 20 25

TIME

Fig . 4. Poincar6 sections displaying strange attractors in (i), (ii) and (iii) and a periodic solution of period 4 in (iv). The t ime series in (v) corresponds to the strange attractor (i). Parameter values are as in Fig . 3. ( iv) and e 1 = e 2 = 0.5. The values of r corresponding to Figs . (i) . . . . . ( iv) a r e 0.60, 0.53, 0.52, 0.49.

92 s. R I N A L D I A N D S. M U R A T O R I

The fact that the lowest part of the boundary of the region of condi- t ioned chaos is almost a straight line (Eq. (5)) and that the phase r is constant on that line and not on the rest of the boundary can intuitively be explained by imagining that some sort of linearization with respect to the time varying parameters hold at low values of e i and by recalling that in linear systems phase and amplitude do not interfere.

Obviously, the Poincar6 section of a strange attractor corresponding to a point (el, e 2) of the region of condit ioned chaos varies with ~- until it degenerates into a finite number (2 k) of points representing a cycle of period 2 ~. One example is repor ted in Fig. 4 where the attractor is displayed for four different values of ~- on the Poincar6 section (where each point corresponds to a sample of the prey and predator populations at the beginning of each year for a period of 500 years). The figure shows how the attractor smoothly loses its fractal nature. Figure 4(v) shows the chaotic time series of prey and predator populations for a period of 26 years for the strange attractor of Fig. 4(i).

Of course, if the point (e 1, e 2) in the region of condit ioned chaos is just above the boundary (5), the strange attractor is only weakly fractal and appears as two separate lines on the Poincar6 section. An example is illustrated in Fig. 5, where the well-known "self-similarity" property of

5

W o-

0.3 0.045

0.2

0,1

0 0

Q

o15 PREY POPULATION x

@

. / .. /

. , / . J

J " ,,.,"

0,12 OJ13 PREY POPULATION x

0.040

0,055

0.030 O, 14

?=

,<

°"V Y Y,Yd v, L/ Y,Y Y lY YL/,q o o= 5 i0 15 20 25

TIME

Fig. 5. A Poincar6 section showing a strange attractor of system (1),(2) when the parameters a and e are periodically varying. The values of the parameters are like in Fig. 3(iv) with e I = e 2 = 0.2 and r = 0.81. The attractor in (ii) is a magnification of a small portion of the attractor in (i). Figure (iii) shows the corresponding prey (x) and predator (y) time series.


0,5

R

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Fig. 6. T h e r eg ion o f chaos (Fig. (i)) and a s t r ange a t t r ac to r (Fig. (ii)) o f sys tem (1),(2) w h e n K , a and b vary per iod ica l ly wi th t ime. T h e phase s o f a and b wi th r e s p e c t to K a re i nd i ca t ed by ~'a and zb. P a r a m e t e r va lues a re e = 1, r = d = 27r, a o = 4~r, b 0 = 0.3, K 0 = 1, e k = 0.4 and % = e b = 0.2. T h e s t r ange a t t r ac to r o f Fig. (ii) and the chao t i c t ime ser ies o f Fig. (iii) c o r r e s p o n d to ~'a = 0.5 and ~'b = 0.

fractal sets is also demonstrated: zooming on the attractor, each line splits into two lines and the process continues like this indefinitely. The time series of the strange attractors close to the boundary of the region of conditioned chaos are also quite regular. For example, the strange attractor of Fig. 5 is similar to a periodic solution of period 2.

Finally, we can say that the role played by the phase in controlling the behaviour of predator -prey communities is confirmed by the analysis of the bifurcations of model (1), (2) with three or more periodically varying parameters. In particular, if the seasonally varying parameters are three, there are two phase parameters, z 1 and ~'2, and the discussion can be very well organized in the two-dimensional space (z 1, z2). For example, Fig. 6(i) shows, in such a space, the regions C and R of chaotic and regular behaviour of a system with three periodically varying parameters, namely K, a and b. The boundary of the closed region C is the bifurcation curve f4 corresponding to the parameter values indicated in the caption. In order to show how the bifurcation diagram of Fig. 6(i) can be used, let us assume that z b = 0, i.e., that K and b vary in phase. This happens, for instance, if the prey is forced, when its carrying capacity is low, to search for food in

94 s. RINALDI AND S. MURATORI

areas where it can be more easily attacked by the predator. In fact, in such a case, b, which is proportional to the searching time of the predator, is minimal when K is minimal. Let us now assume that ~a = 1/2, i.e., that the variations of a and b are of opposite phase. Recalling that a is negatively correlated with the resting time, this is equivalent to say that the resting time of the predator is high when its searching time is high, an assumption that most biologists would agree upon. Since the point corresponding to ~-a = 1 /2 and % = 0 in Fig. 6 falls in region C, the conclusion is that, under the above assumptions, the behaviour of the two populations would be chaotic. On the contrary, the populations are expected to behave regularly if ~-a = "rb = 0, i.e., if the variations of the resting and searching times of the predator are negatively correlated.

6. CONCLUDING REMARKS

The problem of chaos in predator -prey communities with cyclic environment has been investigated in this paper. The analysis has been carried out by studying the bifurcations of a continuous-time second-order model with sinusoidally varying parameters. The studies performed up to now on this subject concern the ideal case in which only one source of seasonality affects the populations, so that only one parameter of the model is periodically varying in time. The main result of these studies (Rinaldi et al., 1993) is that a robust type of chaos due to cascade of period-doubling bifurcations is possible if the degree of seasonality E (i.e., the ratio between the amplitude of the sinusoidal perturbation and the mean value of the perturbed parameter) is sufficiently high. Typical values of E for the raising of chaos are 0.4-0.8.

Since real populations are affected by different sources of seasonality, we have analysed in this paper the case of two periodically varying parameters. The discussion is carried out with respect to two degrees of seasonality, E 1 and E 2, and with respect to the phase ~- separating the two sinusoidal perturbations in time. The main result of the analysis is quite appealing. By suitably selecting the phase ~-, chaos can be very easily produced at low degrees of seasonality (say, in the range 0.2-0.4), but also very easily avoided, even when the degrees of seasonality are very high. The question "are real populations chaotic?" is therefore far from being an- swered. Nevertheless, the discovery that the phase ~- is a variable that strongly influences the dynamic regime of the system is very interesting. In fact, knowing that even small variations of the phase can entail relevant changes in the dynamics (in the limit, from regular to chaotic) can be very useful for interpreting the behaviour of natural populations, or for tuning the time of harvesting or the time of stocking in renewable resources


management. Moreover, it might be that the phase ~- is important for understanding if natural populations (which are always influenced by periodically varying environmental factors) do or do not tend toward chaos on an evolutionary time scale (Ferri~re and Gatto, 1991; Gatto, 1992; Metz et al., 1992). Indeed, one could easily conceive that different alleles give rise to different phases and, therefore, different tendencies for the system to evolve toward chaotic or non-chaotic dynamics.

Although our analysis is quite detailed, still further research is needed on the subject. First of all, some properties concerning conditioned chaos should be better understood mathematically. In particular, it would be interesting to know if the additivity property of the minimum degrees of seasonality needed to generate chaos (see Eq. (5)) can be theoretically justified. Secondly, it would be useful to repeat the same kind of analysis for specific classes of predator-prey communities by using more detailed models and realistic values of the parameters. A first at tempt along this line is a recent study of chaos in plankton communities described by a general model composed by nutrient, phytoplankton, zooplankton and young fish (Doveri et al., 1993). Finally we believe that it would be interesting to generalise this study to more complex predator -prey models. Of course, three-stages food chains would be the obvious candidates for this. Nevertheless, food chains described by three differential equations can be chaotic even when parameters are constant (Hogeweg and Hesper, 1978; Hastings and Powell, 1991; Scheffer, 1991). Hence, it will probably be more reasonable to start with second-order food chains composed by variable prey and predator populations and by a constant number of superpredators. Such food chain models are characterized by homoclinic bifurcations (Kuznetsov et al., 1991; Scheffer, 1992) and should therefore have very rich and complex dynamics when they are periodically forced.

ACKNOWLEDGEMENTS

The authors would like to thank Yu.A. Kuznetsov and M. Scheffer for helpful discussions and constructive criticisms.

The work has been partially supported by Fondazione Eni Enrico Mattei, Milano, Italy.

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Conditioned chaos in seasonally perturbed predator-prey models

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