Ecological Modelling 135 (2000) 127 – 134 www.elsevier.com/locate/ecolmodel
Non-unique population dynamics: basic patterns Veijo Kaitala a,b,c,*, Janica Ylikarjula a,b, Mikko Heino c a
Department of Biological and En6ironmental Science, Uni6ersity of Jy6a¨skyla¨, Box 35, FIN-40351 Jy6a¨skyla¨, Finland b Systems Analysis Laboratory, Helsinki Uni6ersity, of Technology, Box 1100, FIN-02015 HUT, Helsinki, Finland c Department of Ecology and Systematics, Di6ision of Population Biology, Uni6ersity of Helsinki, Box 17, FIN-00014 Helsinki, Finland Received 25 March 1999; received in revised form 2 May 2000; accepted 6 June 2000
Abstract We review the basic patterns of complex non-uniqueness in simple discrete-time population dynamics models. We begin by studying a population dynamics model of a single species with a two-stage, two-habitat life cycle. We then explore in greater detail two ecological models describing host – macroparasite and host – parasitoid interspecific interactions. In general, several types of attractors, e.g. point equilibria vs. chaotic, periodic vs. quasiperiodic and quasiperiodic vs. chaotic attractors, may coexist in the same mapping. This non-uniqueness also indicates that the bifurcation diagrams, or the routes to chaos, depend on initial conditions and are therefore non-unique. The basins of attraction, defining the initial conditions leading to a certain attractor, may be fractal sets. The fractal structure may be revealed by fractal basin boundaries or by the patterns of self-similarity. The fractal basin boundaries make it more difficult to predict the final state of the system, because the initial values can be known only up to some precision. We conclude that non-unique dynamics, associated with extremely complex structures of the basin boundaries, can have a profound effect on our understanding of the dynamical processes of nature. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Bifurcation diagram; Chaos; Fractals; Parasitism; Population dynamics
1. Introduction Non-uniqueness poses a challenge for predicting and controlling the dynamics in various areas of engineering and environmental sciences. Nonunique, coexisting non-trivial attractors have been observed in a differential equation model of a Van * Corresponding author. Tel.: +358-14-2602296; fax: + 358-14-2602321. E-mail address:
[email protected] (V. Kaitala).
der Pol-Duffing electronic oscillator (Gomes and King, 1992), in a one-dimensional cubic iterative mapping (Testa and Held, 1983) and in the twodimensional mapping (Peitgen et al., 1992; He´non, 1976). In ecological models, comparable patterns have been observed in an age-structured single-species model (Wilbur, 1996) and in twospecies interactions such as host–parasite (Kaitala and Heino, 1996), host–parasitoid models (Kaitala et al., 1999) and predator–prey models (Ives et al., in press). With non-unique attractors
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we can examine the initial points leading to certain attractor as the number of iterations approach infinity. These points are called the basin of attraction for this particular attractor. Sometimes the basin boundaries are fractal sets, which can make the identification of the final point extremely difficult. Fractal basin boundaries have been described for example for the dynamics of pendulum (Peitgen et al., 1992), in spatial logistic models (Hastings, 1993) and in competition models (De Feo and Ferrie´re, 2000). We review in this paper the basic patterns of complex non-uniqueness in relatively simple discrete-time models. We first study the dynamic properties of an age-structured model for a single species. We then analyse ecological models describing host–macroparasite and host – parasitoid interspecific interactions.
2. Two-stage life cycle model The well-known exponential form for describing density dependence in single-species population dynamics can be formulated as a one-dimensional mapping known as the Ricker (1954) model as follows Nt + 1 = RNt e − aNt,
(1)
where Nt is the population size at time t, t = 0, 1, 2, 3, … and R and a are constant parameters. The initial value N0 is fixed and given. This model describes the simplest one-stage life cycle as exemplified, e.g. by annual species. The dynamic properties of the Ricker model have been thoroughly analysed. May (1976) has shown that its properties can be summarised as a bifurcation diagram. As population growth ratio R increases, the dynamics changes from stable point equilibrium to chaos through a period-doubling Feigenbaum cascade. Wilbur (1996) extended the Ricker model to account for two-stage amphibian life cycle. In particular, the model has two state variables, the densities of eggs and metamorphs (adults). During one time unit, eggs develop into new metamorphs, and metamorphs give birth to new eggs. Here it is assumed that competition in reproduction occurs
among the adults and that survival to metamorphosis is density-dependent because of competition among eggs and larvae. The model can be presented as follows Mt + 1 = pEt e − aEt,
(2)
Et + 1 = RMt e − bMt,
(3)
where M and E are densities at larval and adult stages, respectively, and a, b, p and R are constant parameters. Eqs. (2) and (3) can be rescaled (t% = t/2 and E%= aE) and reduced to a one-dimensional map E%t% + 1 = RpE%t%e − E%t %(1 + p(b/a)e
− E%t %)
.
(4)
The introduction of sequential density dependence can give this map a bimodal shape. Thus, the possibility arises for the existence of alternative attractors for same parameter values but different initial states. These alternatives may be stable points, limit cycles, and chaotic attractors (Wilbur, 1996). Parameters a, b, and R affect crucially the dynamics of the two-stage life cycle model. We illustrate the complexity of the dynamics arising from the non-uniqueness of the attractors by studying the basins of attraction of the initial condition E %0 as a function of population growth ratio, R (Fig. 1). For example, for p= 1, b/a=14 and R in the range 146–525, (Eq. (4)) has three non-trivial equilibrium points. Thus, there is a possibility of coexisting attractors. The basins of attraction, indicated by different colours in Fig. 1, are the sets of initial points leading to a certain type of an attractor. Different initial values for the population dynamics may lead to completely different types of attractors. For R= 146, a stable point attractor coexists with a period-4 attractor. The former attractor is the largest fixed point of the map, the latter is located around the smallest fixed point. When the initial condition E %0 is increased, the resulting dynamics alternates between these two attractors. The pattern is maintained for increasing values of R, although the alternative attractors change: the smaller attractor undergoes a period-doubling cascade to eight-cycle followed by a period-doubling reversal to two-cycle,
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whereas the larger attractor experiences a perioddoubling cascade to chaos. However, at about R = 240 the non-uniqueness of the attractors disappears when the chaotic attractor loses its stability and two-cycle becomes the global attractor. When the value of R is increased further, an area of complex switches between two attractors is again observed. An interesting illustration of the fractal properties of the basins of attraction is seen for R= 420. In addition to clear black and white ranges we can observe areas with very sensitive dependence of the attractor to the value of E %0. Consecutive enlargements of the basins of attraction reveal a complex Cantor set-like structure. Here the basins of attraction are fractal sets, expressed as the possible lack of strict boundaries between two basins of attraction, and the possible patterns of self-similarity. The fractal character has the effect that the uncertainty in the initial points leads to greater uncertainty in the final state. Therefore, fractal basin boundaries make it impossible to specify the asymptotic behaviour of the system if the initial values are not known precisely.
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3. Ecological interspecific interactions
3.1. Host–parasite interaction We next study the discrete-time dynamics of a parasitised host population (Kaitala and Heino, 1996; Kaitala et al., 1997). This model is motivated by the attempts to understand the complicated dynamics of small rodents living in seasonal environments. The parasites are known to harm their hosts, for example, by decreasing the expected reproductive rate of the parasitised hosts. Some host individuals develop an immune system for protecting them against the parasites by eliminating initial infections and resisting reinfections. Other individuals, lacking this immune response, are susceptible to parasite infections. The costly immune systems may decrease the host reproductive rate or survival. In particular, let P and H denote the population sizes of the parasite and the host, respectively, and let o denote the frequency of immunised individuals in the host population. Parasites are assumed to attack all host individuals at the same intensity. However, we assume a perfect immunity, such that parasites are able to reproduce only in non-immunised host individuals. The frequency of non-immunised individuals in the population is 1− o. The host dynamics are given as Ht + 1 =
[r St (1−o)+ rIo]Ht 1+ (Ht /bH)gH
(5)
where r St and rI are lifetime fecundities of the susceptible and immune host individuals in absence of intra-specific competition, and bH \ 0 and gH ] 1 are constant parameters. The fecundity of susceptible hosts decreases exponentially with increasing average load of parasitism and parasite reproductive rate (rP), yielding Fig. 1. The basins of attraction in the two-stage life cycle model. In the grey region there is only one population dynamics attractor. However, for many values of population growth ratio R two alternative attractors coexist. Black and white indicate the initial states attracted to the attractor with higher and lower mean density, respectively. The parameter values — b/a=14 and c = 1.
r St = r0 exp − crP
Pt Ht
(6)
where r0 is the reproductive rate in the absence of parasites and c is a scaling parameter. Further, the fecundity of the immunised individuals is given as
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Fig. 2. Bifurcation diagrams for parasite (a and b) and host (c and d) population dynamics. For o[0.094, 0.47] the population dynamical attractor is non-unique: When the simulations are initialised with the values H= P= 10 (a and c) we observe a period-doubling cascade from 3-cycle to chaos, followed by a period-doubling reversal to 3-cycle again. For the initial values H =P = 1 (b and d) a fixed point attractor and a quasiperiodic attractor are seen to be alternative attractors for most of the periodic and chaotic attractors in a and c. For o\ 0.47 the population dynamical attractor is unique. The parameter values — gH = gP = 2.5, r0 = 12.5, bH = 100, c = 0.1, d= 0.25 rP =10 and bP = 2.
rI = r0(1− d)
(7)
where d, 05 d51, is constant. This describes the fact that the immune system is costly decreasing the reproductive potential of the immune individuals by a constant fraction d. The parasite dynamics are given as Pt + 1 =
rP(1−o)Pt 1+ (Pt /bPHt )gP
(8)
where rP, bP \ 0 and gP ]1 are constant parameters. Fig. 2 illustrates the bifurcation diagrams of the host and parasite population sizes in the host – parasite interaction Eqs. (5) – (8), with respect to the proportion of immune hosts o. The simulations are initialised from the values H =P = 1 and H = P = 10. For each of 640 values of o, the population dynamics were first iterated for 1000 generations to let the population dynamics reach an attractor and the next 200 values were plotted. Coexistence of
hosts and parasites is observed for o [0.094, 0.90]. The bifurcation diagram is non-unique, different initial states may produce a qualitatively different long-term behaviour of the population dynamics. In Fig. 2a and c we observe a period-3 cycle for o close to 0.1. When the proportion of immune hosts increases, we observe a period-doubling to 6, 12, 24, …. For o between 0.13 and 0.38, the dynamics are chaotic with periodic windows, followed by a period-doubling reversal to a 3-cycle. At o : 0.47 the period-3 attractor suddenly changes to period-5 attractor. Fig. 2b and d show that when simulations are initialised from a different initial condition, the dynamical system may have a different attractor. In fact, for o[0.094, 0.32] a point equilibrium is always an alternative attractor for the system and for o[0.32, 0.47] the alternative attractor is a quasiperiodic one with frequent frequencylockings. The two bifurcation diagrams show that the attractor of the dynamics for a certain parameter combination may not be unique. Alternative attractors to each other are at least the following: locally stable point versus 3-, 6- and 12-cycles, locally stable point vs. chaotic attractor, quasiperiodic cycle versus 3-, 6- and 12-cycles, quasiperiodic cycle versus chaotic attractor and various cycles with different periods. The attractor depends both on the bifurcation parameter o and the initial values. Unique attractors can be observed on both sides of the potentially chaotic region, for oB 0.094 (trivial attractor H= P= 0) and for o\ 0.47 (quasiperiodic, fixed point and period-2 attractors). For o=0.15 the two attractors are a fixed point attractor and a three-piece chaotic attractor, illustrated in Fig. 3a. For o= 0.42 the dynamics follow either a quasiperiodic attractor or a period-6 attractor. In both cases the basins of attraction show a complex pattern, making predictions of the nature of the dynamics on basis of initial conditions alone difficult. The basins of attraction have the fractal property of self-similarity: magnification of lowerleft corner yields a series of similar patterns (Fig. 4). However, the basin boundaries are clearly defined, but for some parameter values fractal basin boundaries can also be observed. The results show strikingly that, depending on the initial value, the population dynamics may
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Fig. 3. The basins of attraction and the corresponding attractors for the host – parasite dynamics in Fig. 2. (a) For o= 0.15 the system has two attractors: chaotic attractor consisting of three separate pieces (C) and a fixed-point attractor (F). Points on the shaded region of H, P-space are attracted to the chaotic attractor, other points are attracted to the fixed point. (b) For o =0.42 the alternative attractors are period-6 attractor (P) and quasiperiodic attractor (Q). Parameter values as in Fig. 2.
either be stable or highly complex and that strange attractors may coexist with more simple but nontrivial attractors in the same dynamical system.
intrinsic growth rate of the host population, a the instantaneous search rate, T the total time initially
3.2. Host– parasitoid interaction Host–parasitoid dynamics provide another example of two-species interspecific interactions. We analysed a simple discrete-time model and observed that the complexities that we have found in the host –parasite model can also be observed in this model. Indeed, the complex dynamics patterns, arising in this model, also include non-unique dynamics with multiple attractors and basins of attraction with fractal properties (patterns of selfsimilarity and fractal basin boundaries). In addition, we observed rare features such as supertransients and chaotic transients. Consider the discrete-time dynamics of host – parasitoid interaction, where the dynamics are given as (Holling, 1959; Royama, 1971; Rogers, 1972) aTPt Nt + 1 = Nt exp r(1−Nt ) − , (9) 1 +aThNt
Pt + 1 = Nt 1−exp
−aTPt 1 +aThNt
n
,
n
(10)
where Nt and Pt denote the host and parasitoid population sizes in generation t, respectively, r the
Fig. 4. Series of magnifications from the Fig. 3a (o=0.15). The basins of attraction show self-similarity; the magnifications from the lower-left corner show an emergence of series of similar figures. Parameter values as in Fig. 2.
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Fig. 5. Bifurcation diagram of the host population in the host – parasitoid interaction for r= 2.82, 0.035 a 50.055, T = 100, Th =1 and initial values N0 =0.5, P0 = 0.5. For each a the dynamics were first iterated 2500 times to reach the attractor and then the next 300 values were plotted.
available for searching and Th the handling time. This model is especially suitable for describing insect arthropod populations. The host–parasitoid model may produce stable, periodic, quasiperiodic or chaotic dynamics. Here we are interested in sustained coexistence of both species. We consider the dynamics of the host population in terms of constant parameter values of r, T and Th and use a as a bifurcation parameter. We assume that r =2.82, T = 100 and Th =1. For small and large values of the instantaneous search rate a, i.e. a B 0.016 or a \0.130 parasitoids go extinct and the host population is described by the Moran – Ricker model (Moran, 1950; Ricker, 1954). Stable coexistence becomes possible at a=0.016 (Fig. 5) and after these equilibrium attractors a Hopf bifurcation occurs at a : 0.033 followed by a quasiperiodic region. This range includes a major period-4 frequency-locking, after which another Hopf bifurcation and quasiperiodic range occurs in each of these four periodic components. At a : 0.0474 a uniform chaotic region begins. Also in this model we can observe non-uniqueness in the bifurcation diagram. This interesting feature is revealed by clear discontinuities at 0.0431BaB0.0438. For example, for a =0.0433
the alternative attractors are two periodic attractors — a 4-cycle and a 16-cycle. Fig. 6 illustrates the basins of attraction for these two alternative attractors. The self-similarity of the patterns and the fractal feature of some basin boundaries are apparent.
Fig. 6. The basins of attraction for the two alternative attractors — the black and white areas are the basins of attraction for the period-4 and period-16, respectively. The patterns of self-similarity and fractal basin boundaries are readily visible. The parameter values — r =2.82, a = 0.0433, T= 100 and Th =1.
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4. Concluding remarks We have reviewed the basic patterns of nonuniqueness in two-dimensional maps using a single-species age-structured model and two-species host –parasite and host – parasitoid models as examples. We observed that various types of attractors may coexist, giving rise to non-unique bifurcation diagrams. The non-uniqueness is further complicated by the fact that the basins of attraction may be fractals, which is revealed by fractal structures of the basin boundaries; e.g. Grebogi et al. (1987) or by the patterns of selfsimilarity; e.g. Kaitala and Heino (1996). Ecologists have studied the qualitative properties of population dynamics ever since Robert May published the seminal results of the perioddoubling route to chaos in single-species population models (May, 1974, 1976; May and Oster, 1976). In particular, they have debated on the existence of complicated dynamics in natural populations (Berryman and Millstein, 1989; Stone, 1993; Rohani et al., 1994) and the practical and theoretical possibilities to detect such dynamics in population data (Sugihara and May, 1990; Cazelles and Ferrie´re, 1992; Hastings et al., 1993; Stone, 1993). A crucial feature characterising the chaotic population dynamics is the sensitivity of the dynamics with respect to the initial population level, also referred to as the ‘butterfly effect’. Nevertheless, the practical and theoretical options for detecting complicated dynamics in population data are still a major challenge in ecological research (Sugihara and May, 1990; Cazelles and Ferrie´re, 1992; Stone, 1993; Hastings et al., 1993; Cohen, 1995; Kaitala and Ranta, 1996). However, we may witness a major breakthrough soon (Costantino et al., 1997; Cushing et al., 1999). Nevertheless, this work may be jeopardised by the fact that dynamics may become non-unique even in two-dimensional mappings. The sensitivity of the trajectories to small disturbances in the initial conditions has usually been used to detect chaotic dynamics. However, with multiple attractors if the initial point is not known precisely, it may be difficult to predict in which of the attractors the trajectory will finally
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settle down. This task becomes even more difficult, if the basin boundaries are fractals. Nonunique dynamics, associated with extremely complex structures of the basin boundaries, can give us a hint how complex the processes in nature can possibly be.
Acknowledgements This work was supported by the Academy of Finland and NorFA. We thank M. Doebeli, W.M. Getz, P. Lundberg, E. Ranta and J. Ripa for numerous discussions on different aspects of interspecific interactions.
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