A “strategic” model of material cycling in a closed ecosystem

A “strategic” model of material cycling in a closed ecosystem

A “Strategic” Model of Material Cycling in a Closed Ecosystem R. M. NISBET, J. McKINSTRY, AND W. S. C. GURNEY Department of Applied Physics, Univer...

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A “Strategic” Model of Material Cycling in a Closed Ecosystem R. M. NISBET, J. McKINSTRY,

AND

W. S. C. GURNEY

Department of Applied Physics, University of Strathclyde, Glasgow, G4 ONG, Scotland Received 26 August 1982; revised 16 November 1982

ABSTRACT The aim of this paper is to reconcile the observed vulnerability of self-sustaining (materially closed) experimental ecosystems with demonstrations of virtually unconditional stability in mathematical models incorporating material recycling. We prove deterministic local stability in a generalized version of a model previously investigated by two of us (Nisbet and Gurney), but show that, except with rather narrowly specified parameter values, the system is likely to be extremely sensitive to external perturbations.

INTRODUCTION Folklore has it that energy flow and material cycling play a central role in determining the structure of communities and ecosystems, though there is no consensus on the dynamic significance of these fundamental processes. Thus, for example, the high level of nutrient recycling in tropical rain forest is highlighted by many authors, but its pertinence to questions of stability, resilience, or succession is largely a matter of speculation (see e.g. May [25] and references therein). There are few experimental pointers to the effects of material recycling on ecosystem stability. Potentially, the most instructive experiments are those involving closed systems, but the majority of materially closed microcosms [5, 14, 181 or mesocosms [ 1 l] are run for too short a time for material cycling to significantly affect stability; in such experiments interest is normally focused on the transient dynamics (e.g. a “spring” bloom of phytoplankton and its aftermath). However a few brave individuals have battled with long-running systems-Hansen [12] cites Folsome’s 15-year-old marine microbiological systems [ 191; Maguire’s freshwater microcosms, which have survived for four years [20], and a synthetic, brackish-water system of his own, aged two. The high degree of experimental skill required to sustain such systems is a clear indicator of their vulnerability, and it seems natural to ask whether dynamic instability is in any sense generic to closed systems. MATHEMATICAL

BIOSCIENCES

64:99-l

OElsevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017

13 (1983)

99 0025-5564/83/03099 + 15$03.00

100

R. M. NISBET ET AL.

The standard repertoire of “strategic” ecological models (logistic, LotkaVolterra, etc.) has little to offer in answer to this rather basic question. Ulanowicz [33], May [22], and Hirata and Fukao [15] have contributed to the development of an ambitious model describing the simultaneous processes of energy flow and material cycling in a closed system; their analysis suggests that stability occurs only in the special circumstances of specific energy (i.e. stored potential energy per unit biomass) increasing up a trophic chain. By contrast, two of us (Nisbet and Gurney [28]) published a study of a simpler model describing solely the cycling of elemental matter in a closed system involving n trophic levels. We demonstrated unconditional local stability of any steady state in which all the standing crops are positive, a result which was subsequently significantly strengthened by the work of Hirata [16] who proved global stability for the case n = 2, and local stability for a variant of the model with an improved representation of the recycling process. In short, the literature on closed ecosystem models is thin, and there are no clear indications whether closed systems ought “normally” to be robust or fragile. In this paper we try to sharpen our understanding of the role played by material cycling in stabilizing a closed system. We start by proving unconditional local stability in a generalized version of the original Nisbet-Gurney model (incorporating a more general representation of primary production than was used in the original work), thereby establishing that the original result was not a mathematical fluke. We then argue that a deterministically stable system may still be dynamically fragile in that small external perturbations may produce a response large enough to destroy the system. We quantify this effect by defining a “susceptibility index” involving the intensity of fluctuations of standing mass in a trophic level, and study the behavior of this index in a special two-level model. We find that the system is only robust in the face of external disturbances if rather special conditions are met, a result which goes some small way towards reconciling the known vulnerability of real closed systems with the deterministic stability of many simple closed system models.

LOCAL STABILITY Nisbet and Gurney [28] proved local stability in a model involving a linear chain of n trophic levels (labeled 1,2,. . . , n) as shown in Figure 1, with X, denoting the number of moles of a particular element in the i th trophic level. Consumption of members of level i by members of level i + 1 was represented by a Lotka-Volterra interaction, implying that the flux of material from level i to level i + 1 is of the form Ai,i+ ,xixi+ r, where Aj I+, is a positive constant. They argued that in the context of “strategic” investigations, the details of the dynamics of death and decomposition could be ignored, and they assumed that elemental material bound in the i th trophic level was converted

101

STRATEGIC MODEL OF MATERIAL CYCLING x,u(x,,x,) .

k’x 1 Axx

121

v

_

X n-1

.

+

2

k n- ,x,_,

I

1

1

I

A

FIG. 1.

1

n-l,n

x n-1 x n

Standing crops and material fluxes in the proposed model of an n-level trophic

chain.

to inorganic elemental material at a rate k,xi. The principal weakness of their model was that the subsequent uptake of material from the inorganic “pool” was also modeled by a Lotka-Volterra term-a totally implausible assumption if either the inorganic pool (x0) or the standing mass of producers (x,) is very large. We now revisit that model, but tackle its critical weakness by modeling the rate of uptake of inorganic material as a general term, which we write in the form x,U(x,, x,) and which is subject only to the obvious constraints

102

R. M. NISBET ET AL.

that for all positive x0, x, , g&O,

The dynamics tial equations,

(1)

g,0.

0

1

of the system is then described by the set of n + 1 differen-

jc,= -x,U(x,,x,)+

n c kiXi,

(2)

i=l

(3)

~,=~,~(~0,~,)-~,*~,~2-~,~,,

2
i, = A,_, , iXi-~Xi-Ai,i+,x*xi+~-kiX1, k, = A,_ i,nx,-IX,

(4) (5)

- knxn,

which clearly represent a closed system in which the total amount critical element in the system,

of the

is conserved. Thus only n of the n + 1 equations are necessary for a full description, and we shall ignore the equation for jc,, replacing x0 wherever necessary by T -Cr= ,xi. We now further restrict our investigation to situations where the form of the specific uptake function U(x,, x,) is such that the system described by Equations (3)-(5) has a single, feasible steady state in which all the standing crops are positive. We investigate the fate of small perturbations from the steady states XT and find, after some routine manipulation, that the stability matrix (May [24], Nisbet and Gurney [29], and many references therein) is - r, - r, B,, =

-p,-r.

-r.

-r.

q2

0

-p2

0

0

q3

0

-P3

... a** ...

7

(7)

where

P~=x;A~,~+,,

qi=A;-,,ix,+,

(8)

the * on a partial derivative indicating that it is to be evaluated at the steady state. The matrix B, differs from the stability matrix for the original model

STRATEGIC MODEL OF MATERIAL CYCLING

103

(Nisbet and Gurney [28, Equation (2)]) only in the (1,1) element, from which a positive quantity (r,) has been subtracted. This suggests that this model should be “at least as stable” as its predecessor, and in fact it is not difficult to prove that the eigenvalues of B, all have negative real parts and hence that the steady state is locally stable. Details of the proof are given in the Appendix. SUSCEPTIBILITY

TO EXTERNAL

DISTURBANCES

Deterministically stable systems may still be highly vulnerable to external perturbations-a feature responsible for much confusion and controversy in other areas of theoretical ecology (e.g. models of niche overlap [23] and of the length of food chains [31])-so we now attempt to assess vulnerability in the model proposed in the preceding section. In population dynamics there is a simple criterion for the vulnerability of a species, namely its likelihood of extinction, but in community and ecosystem studies there is no analogous, unambiguous indicator of robustness. Two measures of susceptibility to external disturbances have been used in both theoretical and experimental work-the characteristic return time (CRT) to equilibrium [4, 13, 17, 21, 23, 31, 341, and the intensity of fluctuations in a variable environment [26, 29, 30, 321. Unfortunately the CRT has serious defects as an indicator of stability: there are situations where a small CRT enables a population to change its density rapidly in response to enviromnental variation, thereby producing large fluctuations [27], and situations where a large CRT indicates the approach of dynamic instability and correspondingly large fluctuations [21, 23, 311. Furthermore, Vincent and Anderson [34] have shown that the CRT is not a proper measure of the vulnerability (in the sense of Goh [9, lo]) of linear food chains. Of the two measures, the fluctuation intensity is thus the more generally useful, and we shall use it in the present work; however, we should remember that we are modeling fluctuations in material content of a particular trophic level and that the relationship between fluctuation intensity and extinction time of a single population (Bartlett [2], Nisbet and Gurney [29, p. 2481) is inapplicable. Nevertheless, it remains true, at least for small experimental systems, that large fluctuations are a necessary precursor of the destruction of the system, and we therefore choose to define an index of susceptibility in terms of fluctuation intensities, confident that we are providing at least a crude characterisation of the vulnerability or resilience of the system. We now follow a well-worn (but still suspect) path and assume that the uptake function U(x,, x,) in our model depends on some external parameter 9 as well as on x0 and x,. We denote by +* the long-time average value of + and define f(t)=+(t)-+*.

(9)

R. M. NISBET ET AL.

104

We similarly denote by t,(t) the fluctuation of the material content of level i from its steady-state value. Provided the fluctuations are small, the Fourier transforms i,(w), f(w) of ti(t) and f(t) respectively are then related by the equation

where the transfer function

T,(w) =

T,(w) is given by (cf. Reference [29], p. 99)

{(iol-Bn)p’}i,x?( f$)*.

The intensity of fluctuations (i.e. the mean squared fluctuation) by (cf. Reference [29], pp. 246-248)

(11) is then given

(12) where S+(o) is the spectral density of the driving fluctuations. Having claimed that fluctuation intensity is a defensible measure of system vulnerability, we now qualify this assertion in two ways. We note firstly that fluctuations are regarded as “large” or “small” by comparing them with the corresponding steady-state standing masses (XT), and secondly that the size of fluctuations must be related to the size of the disturbance producing them. In a strategic investigation, we do not want to cover the full range of possible forms for the spectrum of the driving fluctuations; instead, we argue that considerable insight is obtainable from the response of the system to white noise, and thus we indulge in the popular fiction that we can regard S+(w) as a constant, which we denote simply by S. With this simplification, our two qualifications lead us to define an index ofsusceptibility as (13) which is of course easily calculated once the transfer function is known. High values of p, for all i indicate a stable, nonvulnerable system, while low values of any of the indices indicate the likelihood of large fluctuations and extreme susceptibility to external disturbances.

A MODEL WITH TWO TROPHIC

LEVELS

The preceding section yielded a recipe for a susceptibility index, but no information on how it might behave in our closed system model. In the

STRATEGIC MODEL OF MATERIAL CYCLING

105

general case of n trophic levels, it is unlikely that we can obtain much useful insight, as even the analysis of simple n-level linear-chain models is difficult, and the interpretation of the results controversial [31, 341. We therefore restrict our investigation to a model of material cycling in a closed system consisting of two trophic levels (producers and consumers) and a reservoir of inorganic nutrient. The dynamics are described by the three differential equations k, = k,x, + k,x,

- x,U(x,,x,),

(14)

k,=x,U(x,,x,)-A,,x,x,-k,x,, k, = A,,x,x,

(15)

- k2x2.

(16)

We choose as our specific uptake function a variant on the form proposed by de Angelis et al. to model feeding in a trophic model [l], namely

AOIXO

wxo~xl)=

x

A

01

I+-

There are three readily interpreted tional form:

Aolxoxl

0

gnm +

~

(17)

cn,

regimes covered by this choice of func-

(a) When both x0,x, are small, U + A,,x, and uptake of inorganic material is described by a Lotka-Volterra term as in Nisbet and Gurney [28]. (b) When x0 is large and x, is small (plentiful nutrient, few producers), is the maximum specific uptake rate peru+ gIlI,, illustrating that g,, mitted by the producer physiology. (c) When both x0 and x, are large, the total flux of the critical element from the reservoir to the producers (x,U) approaches G,,, and the rate of primary production is set by factors (such as availability of energy) external to the model. We can scale Equations (14)-(16) by choosing T (the total amount of material in the system) and k; ’ as our base units of material and time respectively. With this choice of units, the equations take the form k. = F, + LF, -

k, =

MFo F, I+ G,F, + G,F,F,

M&F,

l+ G,Fo + G,F,F,

i;2 = WF,F, - LF,,



(18)

- WF,F, - F,, (20)

where F, (i = 0, 1,2) is the fraction of the total amount of material (T) in the

R. M. NISBET ET AL.

106 system bound in level i, and the other dimensionless are M=k,

Ao,T

Wzk,

I G,=-,

controlling

parameters

A,*T

AO,T

I G2=G

gmax

(21)

AJ2 max

M (proporThe parameters M, G1, and G, describe primary production; tional to A,,) represents an upper bound on the primary productivity of the system, while G, and G, describe respectively the degrees of self-limitation and external control. The parameter L is simply the ratio of loss rates from the two levels, while W is a measure of the effectiveness of consumer predation (directly comparable with M). There is only one feasible steady state, at which F;=L/W

(22)

and G,L2 + G2L2

$+L+G,L_F; =

G,L + G2L2

G2L3

-_-_w

w

-

w,

w

W2

1

2(G,L

+ G,L’/W) G,L2

W+L+G,L-W+W----

G2L2

ML ML2 w-_-_--.W2 + __G,L2 w

G2L3

G,L+

w2

W

L

G,L

w

w

G2L2 + -G2 L3 W2 W3

G2L2

-

)I’

2

w2

“2

(23)

the solution analogous to (23) but with the positive square root being excluded, as it produces a negative value of F,* ( = 1 - Ff’ - Fj+). As our choice of uptake function U has

au

>O

ax,

-1

and

$$O 1

(24)

STRATEGIC

MODEL

OF MATERIAL

107

CYCLING

for all x,,, x,, local stability of the steady state is guaranteed. Examination of the behavior of F, and & as F,, F, --* 0 indicates that there is no domain of attraction of a steady state involving zero values for either F, or F2 or both. We can thus safely conclude that provided the parameters yield positive values for FF and F;, the steady state is globally stable. We have studied the effects of external “driving” of the uptake function by allowing the parameter G,, (which, as we have seen, sets the external limitations on the uptake of material by the producers) to vary with time. As explained in the previous section, we chose the simplest form of “random” fluctuation and set G,,(r) where f(t)

represents

=G,‘L(l+f(t))

Gaussian

1000

(25)

white noise with (constant)

spectral density

/

/ lo3

I

W

/

lo*/

/

100

/ /

/ /

--

/-

/

/

---

‘O/ 5/

-

-

10

\_-=-

--

1

00 FIG. 2. Contours of constant susceptibility index in the M - W parameter closed system with L = 1, G, = 0.2, G2 = 0.2. (a) Contours of constant p,.

space for a

108

R. M. NISBET

ET AL.

S. The susceptibility indices p, and p2 were calculated analytically, but the resulting expressions are messy and singularly uninstructive. However, we can readily demonstrate graphically the dependence of these indices on the parameters M (representing “efficiency” of primry production) and W (representing “efficiency” of consumer predation), a typical example being reproduced in Figures 2(a) and (b), where we have plotted contours of constant p, and p2 for a particular choice of values for the other parameters. To assist interpretation we illustrate in Figure 3(a) the corresponding dependence of the steady-state masses, and in Figure 3(b) the dependence of the “coherence numbers” of the fluctuations (a measure of the extent to which the fluctuations appear quasicyclic [29, p. 2541) on M and W. From these graphs it is clear that there are only two situations which produce high susceptibility indices (implying nonvulnerability):

(a) High M, low W (efficient primary production, weak predation). The standing mass in level 1 is higher than that in level 2, and the “driven”

1000

W

100

50/-/ --

10

1

FIG. 2. (b).

Contours

of constant

p2

STRATEGIC MODEL OF MATERIAL CYCLING

109

1000

100

W

10

1

1

10

M

100

FIG. 3. (a). Scaled standing masses and (b) contours of constant terms of M and W for the case L = 1, G, = 0.2, G, = 0.2.

1000

coherence

number

in

fluctuations are noncyclic. Essentially, the recycling of material from level 1 stabilizes the system and the flow through level 2 is a dynamically unimportant perturbation. (b) Low M, high W (slow primary production, intense predation). The flow of material into level 1 regulates growth at this level, and the dynamical behavior of the system resembles that of the well-known damped LotkaVolterra model [8, 291 in that driven fluctuations may well be quasicychc. The broad picture outlined above is not sensitive to the choice of parameter values, nor even to which parameter “wobbles.” One of us (McKinstry [26]) has performed a wide variety of computations with different parameter selections, all consistent with the general theme that only in rather special regions of parameter space are fluctuation intensities small, and is system resilience to be expected.

R. M. NISBET ET AL.

110 1ooc

1oc

W i \ 1c

\’ \’ \ \‘-

eb 8 n = 1.0 g______*_______.

-__-_-

overdamped 1

100

10

1000

M FIG. 3. (b).

DISCUSSION The present investigation had the aim of reconciling the obvious fragility of experimental self-sustaining ecosystems with the stability of the NisbetGurney model and its many cousins. We cannot claim to have unambiguously resolved this problem, but the computations on the two-level model suggest that, notwithstanding the occurrence of deterministic stability under very general conditions, a high susceptibility to external disturbances is likely to be the rule rather than the exception. Closed-system research, however, has only esoteric application (e.g. to extended life-support systems for use on Mars), and we can reasonably ask if our models or their many relatives can provide insight with wider potential application. The only materially closed natural system is the entire biosphere, and the present models are clearly irrelevant to discussions of global biogeochemical cycles, where recycling involves time scales many orders of magni-

STRATEGIC MODEL OF MATERIAL CYCLING

111

tude longer than those involved in ecological interactions [3]. However there are many ecologists who will argue that the extent of material recycling within an open system, quantifiable via a cycling index [6, 71, significantly affects stability, and De Angelis [35] has performed some model calculation which support this contention. May [25] has speculated that cycling indices should increase as succession proceeds, while pollution and other external disturbances may destabilize by reducing the amount of recycling in a mature ecosystem. The robustness of such speculations could be assessed by “opening up” the present models “conservatively” (i.e. adding balanced input and output terms) and studying variations in dynamic behavior from the pattern presented in this paper. One of us (McKinstry [26]) has made a preliminary study of such models which highlighted the importance of the detailed form of the output (harvesting or washout) terms, but which pointed to the likelihood of multiple steady states (and hence considerable mathematical complexity) in many ecologically plausible situations. The careful investigation of such “open ecosystem” models is an obvious challenge well worth further effort.

APPENDIX.

PROOF MODEL

OF LOCAL

STABILITY

FOR

THE

n-LEVEL

We require to show that all the eigenvalues of the matrix B, [defined in the main text, Equation (7)] have negative real parts. We define C, to be the matrix obtained by setting r, equal to zero in the definition of B,. We also define a matrix An _ , of dimension n - 1 by 0

A,_, =

-pz

0

q3

0

-P3

0

q‘j

0

*** .**

(Al)

-.-

It is then a routine algebraic exercise to show that det(B, - AI) = det(C, - XI)-

r,det(A,_,

- XI).

(A2)

We now prove that the roots of det(B, - XI) cannot be zero or pure imaginary. The proof is facilitated by defining two polynomials !,,(A)=(-l)“det(C,-XI)

(A3)

and g,_,(X)=(-l)“-‘det(A,_,-XI),

(A4)

112

R. M. NISBET

and proceeds by reductio ad absurdum. We assume that + conjugate roots. Then

io

are

ET AL.

a pair of

(A5)

f,(io)=-r,g,_,(iw)=-r,g,~,(-iw)=f,(-iw),

the middle equality being a consequence of the fact that the roots of A,_, are pure imaginary. But we know from the analysis of the original NisbetGurney model [28] that the roots of f,(X) have negative real parts, implying that f,( io) and f,( - iw) cannot be equal. This contradicts the original hypothesis, and hence the roots of det(B, - AZ) cannot be zero or pure imaginary. However, the roots of a polynomial are continuous functions of the polynomial coefficients, and hence B, must always be stable or always unstable, irrespective of the value of r, (except if we pass through r, = 0, where the argument in the preceding paragraph is invalid). But for small positive r,, the matrix B, is stable, a result easily proved by first-order perturbation theory on C,. Thus B,, is stable for all positive values of r,. We thank Peter Maas for introducing us to the mysterious world of experimental closed-system ecology. REFERENCES 1

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