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arXiv:0811.3657v1 [q-bio.PE] 22 Nov 2008

General Mechanisms for Inverted Biomass

Pyramids in Ecosystems

Hao Wang a,∗, Wendy Morrison b, Abhinav Singh c,

Howard (Howie) Weiss a

aSchool of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

bSchool of Biology, Georgia Institute of Technology, Atlanta, GA 30332, USA

cSchool of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Abstract

Although the existence of robust inverted biomass pyramids seem paradoxical, they

have been observed in planktonic communities, and more recently, in pristine coral

reefs. Understanding the underlying mechanisms which produce inverted biomass

pyramids provides new ecological insights, and for coral reefs, may help mitigate

or restore damaged reefs. We present three classes of predator-prey models which

elucidate mechanisms that generate robust inverted biomass pyramids. The ﬁrst

class of models exploits well-mixing of predators and prey, the second class has a

refuge (with explicit size) for the prey to hide, and the third class incorporates the

immigration of prey. Our models indicate that inverted biomass pyramids occur

when the prey growth rate, prey carrying capacity, biomass conversion eﬃciency,

the predator life span, or the immigration rate of prey ﬁsh is suﬃciently large. In the

second class, we discuss three hypotheses on how refuge size can impact the amount

of prey available to predators. By explicitly incorporating a refuge size, these can

more realistically model predator-prey interactions than refuge models with implicit

Preprint submitted to Elsevier 22 November 2008

refuge size.

Key words: inverted biomass pyramids, coral reef, predator-prey model, refuge,

immigration.

1 Introduction

The biomass structure is a fundamental characteristic of ecosystems (Odum,

1971). The shape of biomass pyramids encodes not only the structure of com-

munities, but also integrates functional characteristics of communities, such

as patterns of energy ﬂow, transfer eﬃciency, and turnover of diﬀerent compo-

nents of the food web (Odum, 1971; Reichle, 1981; Del Giorgia et al., 1999).

A trophic pyramid is a graphical representation showing the energy or biomass

at each trophic level in a closed ecosystem. Energy pyramids illustrate the

production or turnover of biomass and the energy ﬂow through the food chain,

while biomass pyramids illustrate the biomass or abundance of organisms at

each trophic level. When energy is transferred to the next higher trophic level,

typically only 10% is used to build new biomass (Pauly and Christensen, 1995)

and the remainder is consumed by metabolic processes. Hence, in a closed

ecosystem, each trophic level of the energy pyramid is roughly 10% smaller

than the level below it, and thus inverted energy pyramids cannot exist.

A standard biomass pyramid is found in terrestrial ecosystems such as grass-

land ecosystems or forest ecosystems, where a larger biomass of producers sup-

∗Corresponding author: Telephone: 1-404-894-3949; Fax: 1-404-894-4409;

Email address: wanghao@math.gatech.edu.

2

port a smaller biomass of consumers (Dash, 2001). Although they appear to be

rare, inverted biomass pyramids exist in nature. They have been observed in

planktonic ecosystems (Del Giorgia et al., 1999), where phytoplankton main-

tain a high production rate and are consumed by longer lived zooplankton

and ﬁsh. Recently, inverted biomass pyramids have also been observed in pris-

tine coral reefs in the Southern Line Islands and Northwest Hawaiian Islands

(Friedlander and Martini, 2002; Sandin et al., 2008), where the benthic coral

cover provides refuge for prey ﬁsh (Figure 1). At least one prominent researcher

suspects that an apparent inverted biomass pyramid exists on a reef oﬀ the

North Carolina coast, and he speculates this is due to signiﬁcant immigration

of prey ﬁsh into the reef (M. Hay, pers. comm., 2008). In this manuscript, we

introduce three classes of predator-prey models to study how inverted biomass

pyramids can arise via these three distinct mechanisms.

2 Well-Mixed Mechanism

Most predator-prey models implicitly assume that predators and prey are

well mixed, and many incorporate a Holling-type predation response (Holling,

1959a,b). Although the “well mixed” assumption is usually far from being

satisﬁed when prey are animals, it appears to be a reasonable assumption

for phytoplankton-herbivore interactions in aquatic ecosystems, and we ﬁrst

discuss the existence of inverted biomass pyramids in this setting.

We begin by considering the standard Lotka-Volterra predator-prey model

with mass-action predation response (Lotka, 1925; Volterra, 1926), described

by the system

3

dx

dt =ax −bxy, (1)

dy

dt =cbxy −dy, (2)

where

x: prey biomass density, y : predator biomass density,

a: prey growth rate, b : per capita predation rate,

c: biomass conversion eﬃciency, d : predator death rate.

The interior equilibrium point (x∗, y∗) = d

cb,a

b!is neutrally stable (a center),

at which the predator:prey biomass ratio is

y∗

x∗=ac

d.(3)

The ratio is greater than 1 if and only if ac > d. We obtain our ﬁrst result in

biomass pyramid theory:

Result 1 For the model (1)-(2), if ac > d (the prey growth rate multiplied by

the conversion eﬃciency is greater than the predator death rate), the biomass

pyramid is inverted; otherwise, the biomass pyramid is standard.

Result 1 provides a rigorous foundation for the belief expressed by some biolo-

gists that inverted biomass pyramids result from the high growth rate of prey

and low death rate of predators (Del Giorgia et al., 1999). Result 1 further

suggests that the biomass conversion eﬃciency can signiﬁcantly inﬂuence the

shape of the biomass pyramid.

We now incorporate a general well-mixed predation response into the predator-

prey model, which is described by the system

4

dx

dt =ax −f(x)y, (4)

dy

dt =cf(x)y−dy, (5)

where

f(x) : predation response function.

At the interior equilibrium point (ˆx, ˆy), the ratio ˆy/ ˆx=a/f (ˆx), where f(ˆx) =

d/c. Thus the predator:prey biomass ratio is

ˆy

ˆx=ac

d.(6)

This interior equilibrium point is attracting when the system (4)-(5) is eventu-

ally bounded and has no stable limit cycles. The system is eventually bounded

if there is a bounded region where all solutions eventually enter into and stay

in. Result 1 remains valid for this extended model whenever the interior equi-

librium point is stable. Actually, whenever the prey grow exponentially, Result

1 is robust to variations in refuge-dependent predation patterns.

We now incorporate logistic prey growth into the preceding predator-prey

model, which is described by the system

dx

dt =ax 1−x

K−f(x)y, (7)

dy

dt =cf(x)y−dy, (8)

where

K: prey carrying capacity,

and the predation functional response f(x) is a strictly increasing function.

Any reasonable predation function must satisfy this monotone condition, which

5

all three Holling-type functions do. The monotonicity implies that the inverse

function f−1exists (Inverse function in Wikipedia, internet), and thus the x-

component of the interior equilibrium point can be solved from cf (x) = das

˜x=f−1(d/c). The biomass ratio at the interior equilibrium point (˜x, ˜y) can

be written as

˜y

˜x=ac

d"1−f−1(d/c)

K#.(9)

This formula modiﬁes the biomass ratio in model (1)-(2) and model (4)-(5) by

the factor 1 −f−1(d/c)

K. This interior equilibrium point is attracting when the

predator-extinction equilibrium (K, 0) is unstable and there exist no stable

limit cycles. For instance, if f(x) = bx

η+xis a Holling type II functional

response, then the interior equilibrium point is globally attracting whenever

ηd

b−d< K < η(b+d)

b−d. Under the stability condition, we obtain the new

result:

Result 2 For the model (7)-(8), if ac

d"1−f−1(d/c)

K#>1, the biomass pyra-

mid is inverted; otherwise, the biomass pyramid is standard.

The new condition for the inverted biomass pyramid depends additionally on

the prey carrying capacity K. We see that the predator:prey biomass ratio is

an increasing function of the prey growth rate (a), the conversion eﬃciency

(c), and the prey carrying capacity (K), while the biomass ratio is a decreasing

function of the predator death rate (d). As a conclusion, we have the following

result:

Result 3 The increase of prey growth rate, the conversion eﬃciency, the prey

carrying capacity, or the predator life span facilitates the occurrence of inverted

biomass pyramids.

6

Result 3 is robust whenever the predation function is an increasing function

of prey density. We will see in the next section that the same relations hold

for refuge-dependent predation functions.

3 Refuge Mechanism

Seeking refuge from predators is a general behavior of most animals in nat-

ural ecosystems (Cowlishaw, 1997; Sih, 1997) where the refuge habitats can

include burrows (Clarke et al., 1993), trees (Dill and Houtman, 1989), cliﬀ

faces (Berger, 1991), thick vegetation (Cassini, 1991), or rock talus (Holmes,

1991). Some ecologists even believe that refuges provide a general mecha-

nism for interpreting ecological patterns (Hawkins et al., 1993), speciﬁcally the

extent of predator-prey interactions (Huﬀaker, 1958; Legrand and Barbosa,

2003; Rossi et al., 2006). Aquatic ecologists have recently observed inverted

biomass pyramids in pristine coral reefs, where the benthic coral cover pro-

vides the refuge for prey ﬁsh (Friedlander and Martini, 2002; Sandin et al.,

2008).

In Singh et al. (2008), we needed to introduce a refuge with explicit size.

Although the Holling type III functional response oﬀers the prey a refuge at

low population density (Murdoch and Oaten, 1975), the refuge is only implicit,

and one can not specify the size of the refuge. Some authors include an explicit

refuge size into their models by multiplying the prey density by 1−r, where 0 ≤

r < 1 is a proxy of the refuge size (McNair, 1986; Sih, 1987; Hausrath, 1994;

Kar, 2005; Huang et al., 2006; Kar, 2006; Ko and Ryu, 2006). This procedure

has two fundamental drawbacks. The ﬁrst is that for these modiﬁed predation

response functions, the switch point, where the predation rate starts to quickly

7

increase, critically depends on both the proxy refuge size and the proxy half-

saturation constant (independent of the refuge size). The latter dependence is

undesirable. For our model, it is important that the switch point be a function

of only the refuge size. The second drawback is that, unlike the Holling-type

functional responses which are mechanistically derived from basic biological

principles, we have seen no derivation in the literature and we are unable to

mechanistically derive these functional forms from basic biological principles

to incorporate a refuge.

We now introduce a family of predator-prey models with explicit refuge size,

which we call refuge-modulated predator-prey (RPP) models. An im-

portant feature of this family is that the switch points for the functional

responses depend solely on the size of the refuge. We group these models into

three classes, RPP Types I, II, and III, depending on the mechanistic depen-

dence of prey availability for predators on the refuge size. All previous refuge

models assume the mechanism behind our Type I class.

3.1 Refuge-Modulated Predator-Prey Models

In our recent work (Singh et al., 2008), we modeled the biomass of ﬁsh in coral

reefs. Small ﬁsh ﬁnd refuge in coral reefs by hiding in holes where large preda-

tors cannot enter (Hixon and Beets, 1993). This ﬁeld observation motivated us

to incorporate a refuge into the standard predator-prey model, where the coral

reef refuge size inﬂuences the pattern of predation response. We introduced

the following family of models:

8

dx

dt =ax 1−x

K−f(x, r)y, (10)

dy

dt =cf(x, r)y−dy, (11)

where

r: refuge size,

f(x, r) : refuge-dependent predation response,

and f(x, r) is a strictly increasing function of prey biomass density x. For each

ﬁxed r, the function fr(x) = f(r, x) is strictly increasing in x, and thus its

inverse f−1

rexists. We solve for the x-component of the interior equilibrium

point from cf (x, r)−d= 0 as ¯x=f−1

r(d/c). The biomass ratio at the interior

equilibrium point (¯x, ¯y):

¯y

¯x=ac

d"1−f−1

r(d/c)

K#.(12)

For each ﬁxed refuge size r, the relationships between the biomass ratio and

other parameters are the same as in the well-mixed predator-prey models.

This provides the robustness of Result 3. Additional hypotheses are needed

to determine the relationship between the biomass ratio and the refuge size.

Although the ﬁeld observation in Hixon and Beets (1993) suggests that prey

ﬁsh hide in refuge places from predators, it is still unclear how the refuge

regulates the prey availability for predators. In the next subsection, we propose

three hypotheses all motivated from biological considerations.

9

3.2 Hypotheses on Refuge Eﬀects

We model three biological hypotheses on how prey availability for predators

depends on the refuge size (Figure 2). We call these models RPP (Refuge-

modulated Predator-Prey) Type I, Type II, and RPP Type III. All predation

functions depend on the maximum predation rate b, the refuge size r, the

minimum predation rate regulator β, and the slope regulator ξ.

RPP Type I: This model assumes that the prey availability for predators

decreases as the refgue size increases. Prey hide in the refuge, but trade-

oﬀ protection (i.e. increased survival) for a decrease in growth or reproduc-

tion due to lower quality resources within the refuge (Persson and Eklov,

1995; Gonzalez-Olivares and Ramos-Jiliberto, 2003; Reaney, 2007). Thus an

increase in the size of the refuge protects more of the prey and results in less

prey available to the predator. Thus, the prey availability for predators is the

prey density outside the refuge and f(x, r) is a decreasing function of refuge

size r. We choose the following representive function that can be ﬁtted to

empirical data:

f(x, r) = b

1 + βe−ξ(x−r).(13)

The variable xis the total prey density (per unit area) and thus the prey

availability for predators is x−r.

The parameter βcaptures the minimum predation rate as follows: when no

prey and no refuge are available, the predation rate is b

1 + β, which is the

minimum predation rate. We should choose βsuﬃciently large such that

1

1 + β<< 1, since it is reasonable to have a small predation rate when no

prey are available. The parameter ξdetermines the slope of the predation

10

curve when xis close to r. The prey density at the interior equilibrium point

is ¯x=f−1

r(d/c) = r−1

ξln "1

β bc

d−1!#. The interior equilibrium point only

exists when bc > d. To see this, if bc ≤dand β > 0, then dy

dt <(bc −d)y≤0,

and thus predators go extinct since all solutions tend to the boundary equi-

librium point (K, 0). Biologically, when the maximum predation rate multi-

plied by the conversion eﬃciency is less than the predator death rate, one

would expect that predators cannot persist. Under the conditions that βis

suﬃciently large and bc > d, the term 1

ξln "1

β bc

d−1!# is negative. Hence,

¯x=f−1

r(d/c) = r−1

ξln "1

β bc

d−1!#=r+1

ξln βd

bc −d>0for suﬃciently

large β.

The evidence for the presence of trade-oﬀs (survival vs. growth or reproduc-

tion) with the use of refuges is plentiful (Lima and Dill, 1990; Persson and Eklov,

1995; Reaney, 2007). However, we are aware of only one experimental example

that shows a decrease in growth rate of the predator in response to the use of

a refuge by the prey (Persson and Eklov, 1995).

RPP Type II: This model assumes that the prey availability for predators

is independent of the refuge size (in the sense of density, per unit area), i.e.

f(x, r) is a constant function of r. Prey biomass within the refuge may increase,

but the amount available to the predators remains the same. We choose our

representative RPP Type II predation function:

f(x, r) = b

1 + βe−ξ x .(14)

The variable xis the prey density, and the parameters b,β, and ξhave the

same meanings as in RPP Type I. The x-component of the interior equilibrium

11

point is ¯x=f−1

r(d/c) = 1

ξln βd

bc −d>0for bc > d and suﬃciently large β.

We know of no biological examples of RPP Type II. However, we believe

that RPP Types I and III are the extremes of a continuum, suggesting that

condition can exist where the prey available to predators is not aﬀected by

the area within the refuge.

RPP Type III: This model assumes that the prey availability for predators

increases as the refuge size increases. This may occur when resources such as

food and mating sites are available within the refuge, allowing the prey to

increase in numbers until some limiting resource forces a number of the prey

to emigrate from the refuge in search for new habitat. The number of immi-

grants should be positively related to refuge size. Thus, f(x, r) is an increasing

function of r. Our representative RPP Type III predation function looks quite

similar to our Type I predation function, but the parameters require diﬀerent

interpretations:

f(x, r) = b

1 + βe−ξ(x+r).(15)

The variable xis the exterior (out of refuge) prey density (per unit area), and

x+ris the total prey density (per unit area). This model assumes that the

refuge stores a substantial amount of prey and constantly provides food to

predators, and thus the prey availability is the total prey density (per unit

area), i.e. x+r.

For bc > d and βsuﬃciently large such that 1

ξln βd

bc −d>0, the x-component

of the interior equilibrium point is ¯x=f−1

r(d/c) = −r+1

ξln βd

bc −d>0for

r < ¯r. The threshold refuge size ¯r=1

ξln βd

bc −d>0 for bc > d and suﬃciently

12

large β. Because the refuge size in the model is measured by density (per unit

area), it is biologically reasonable to assume a threshold maximum value for

the refuge size.

The Elk Refuge in Yellowstone National Park is one example of a RPP Type

III. The Elk Refuge provides protection (and food) to the elk during winter in-

creasing survival to 97% (Lubow and Smith, 2004). The surviving elk migrate

out of the refuge and provide a source of food for predators in Yellowstone

National Park and surrounding areas (Smith, pers. comm., 2008). Our RPP

Type III is also analogous to spillover and larval export hypotheses in marine

protected areas (MPA) (Ward et al., 2001). MPAs are areas of the ocean that

are protected from ﬁshing (i.e. man is the predator). The ﬁsh within these

MPAs are hypothesized to increase the number of ﬁsh (prey) available out-

side the protected area through two mechanisms. The ﬁrst, spillover, occurs

when adult ﬁsh become crowded within the MPA and immigrate into the sur-

rounding area. The second occurs when the ﬁsh within the MPA increase their

reproductive output, increasing the number of recruits available to surround-

ing areas (larval export). While support for the spillover hypothesis is present

(though limited spatially), it is much harder to prove the beneﬁts of larval

export (Ward et al., 2001).

We now make a couple of general remarks about the RPP-type functional

responses. We always assume that f(0, r)>0 and small, i.e. for each ﬁxed r,

the predation rate at zero prey density is positive, but minimal. For Holling-

type responses, f(0, r) = 0. We believe our choice is reasonable, since when the

main prey species are no longer available, predators may temporarily switch

to alternative lower quality food sources (Warburton et al., 1998). Thus, one

13

must choose βsuﬃciently large such that 1

1 + β<< 1. If we ﬁt this predation

function to empirical data, ξmay need to be chosen large, depending on the

size of β. When the prey availability is high, f(x, r) is close to the maximum

predation rate b. Mathematically, the refuge size rsolely determines the shift

of the predation curve.

3.3 Dependence of Biomass Ratio on the Refuge Size

In this subsection, we use (12) to analyze the eﬀects of the refuge size on the

predater:prey biomass ratio. It is evident that the biomass ratio in (12) is a

decreasing function of f−1

r(d/c).

For RPP Type I, the term f−1

r(d/c) = r+1

ξln βd

bc −dis an increasing function

of the refuge size r. Thus, the predator:prey biomass ratio at the interior

equilibrium point is a decreasing function of the refuge size r.

For RPP Type II, the predator:prey biomass ratio is independent of the refuge

size.

For RPP Type III, the term f−1

r(d/c) = −r+1

ξln βd

bc −d, is decreasing as the

refuge size rincreases. Thus, the predator:prey biomass ratio at the interior

equilibrium point is an increasing function of the refuge size r.

The following results immediately follow from these observations:

Result 4 For the model (10)-(11), if ac

d"1−f−1

r(d/c)

K#>1, the biomass

pyramid is inverted; otherwise, the biomass pyramid is standard.

14

Result 5 For RPP Type I, the decrease of the refuge size facilitates the oc-

currence of inverted biomass pyramids. For RPP Type II, the refuge size has

no eﬀects on biomass pyramids. For RPP Type III, the increase of the refuge

size facilitates the occurrence of inverted biomass pyramids.

As an illustrative example, data from Kingman and Palmyra (Sandin et al.,

2008) suggests that the predator-prey biomass ratio is an increasing function

of the refuge size (equivalent to the benthic coral cover), and thus the appro-

priate predation response function is RPP Type III. RPP Type III may be

biologically appropriate if increases in refuge size either increase recruitment

or increase the survival of recruits (Shulman, 1984; Doherty and Sale, 1985).

After the surviving recruits grow into juveniles or adults, they leave the refuge

and provide an increase in the food available to the predators.

4 Immigration Mechanism

Reef ecologists observed signiﬁcant immigration of prey ﬁsh in a North Car-

olina reef (M. Hay, pers. comm., 2008). We consider two types of immigration:

(i) immigrating prey ﬁsh stay in the coral reef and adapt to survive in the new

habitat; (ii) immigrating prey ﬁsh leave the coral reef if they are not eaten by

hungry predators, i.e. they provide additional food to predators but do not

add to the local prey population. In this section, we incorporate both types

of immigration into the Lotka-Volterra predator-prey model:

(i)

15

dx

dt =ax −bxy +ι, (16)

dy

dt =cbxy −dy; (17)

(ii)

dx

dt =ax −bxy, (18)

dy

dt =cb(x+ι)y−dy; (19)

where ιis the constant immigration rate. For case (i), the predator:prey

biomass ratio at the interior equilibrium point (˜x, ˜y) is

˜y

˜x=ac

d+ιc2b

d2.(20)

For case (ii), the predator:prey biomass ratio at the interior equilibrium point

(ˆx, ˆy) is

ˆy

ˆx=ac

d−ιcb.(21)

In both immigration cases, the biomass ratios are increasing functions of the

immigration rate ι. This remains true when we incorporate these two immi-

gration eﬀects into Holling type or RPP type models. As a conclusion, we

obtain the following robust result:

Result 6 The immigration of prey facilitates the occurrence of inverted biomass

pyramids.

5 Discussion

We develop a theory of biomass pyramids. Our major contributions can be

summarized as follows. First, when prey grow exponentially, the biomass pyra-

mid is inverted if and only if the prey growth rate multiplied by the conversion

16

eﬃciency is greater than the predator death rate. Second, the increase of prey

growth rate, the conversion eﬃciency, the prey carrying capacity, or the preda-

tor life span robustly facilitates the development of inverted biomass pyramids.

Third, based on plausible biological hypotheses, we introduce a new series of

predator-prey models (called RPP type models) which explicitly and naturally

incorporates a prey refuge. Fourth, depending on the nature of an ecosystem,

the occurrence of inverted biomass pyramids can be positively or negatively

related to, or independent of, the refuge size. Fifth, the immigration of prey

facilitates the occurrence of inverted biomass pyramids.

We propose three new refuge-dependent predation functions with explicit

refuge size, which capture the three essential biological hypotheses on the

refuge (Figure 2). The three can be combined into one function

f(x, r) = b

1 + βe−ξ[x−(2−i)r],(22)

where iis the index of RPP type, that is, i= 1 for RPP Type I, i= 2 for

RPP Type II, and i= 3 for RPP Type III.

Some, but not all, of the prey that hide in the refuge are available to predators.

Thus, there should be a discount rate for the refuge size in the predation

function of either RPP Type I (assume no prey in the refuge are available) or

RPP Type III (assume all prey in the refuge are available). We incorporate

this discount rate into the general refuge-dependent predation function:

f(x, r) = b

1 + βe−ξ(x+ηr),(23)

where −1≤η≤1. This model is close to RPP Type I if −1≤η < 0, close to

RPP Type II if η= 0, and close to RPP Type III if 0 < η ≤1. We call ηas

the refuge-eﬀect parameter.

17

What characteristics of the prey might lead to RPP Type I versus RPP Type

III? As stated above, RPP Type I will occur when the use of the refuge

results in strong trade-oﬀs between survival and reproduction or growth. Most

previous theoretical models assume that the hypothesis for RPP Type I is

the case; however, we hypothesize that RPP Type III will occur when the

prey have the ability to reproduce within the refuge and/or when the refuge

increases prey survival through a population bottleneck.

Prey animals seek refuges to hide from predators and thus it is sometimes

necessary to explicitly incorporate the refuge mechanism into the predation

function of predator-prey models. The family of RPP-type models explicitly

incorporating the refuge size can more accurately describe realistic predator-

prey interactions in ecosystems. We believe that RPP-type models provide

the next generation of models for predator-prey interactions. In the coming

Winter, we plan to test these models via microcosm experiments.

Acknowledgement

We would like to thank Mark Hay for insightful comments and helpful dis-

cussions, Alan Friedlander and Bruce Smith for their useful feedback and

references to our questions. We also would like to thank Lin Jiang for his sug-

gestions and allowing us to perform refuge experiments in his lab in the near

future.

18

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Figure 1. This ﬁgure is reproduced from Sandin et al. (2008). At Kingman

coral reef, it was recently discovered that apex predators constitute 85% of

the total ﬁsh biomass. The biomass pyramid is clearly inverted in this pristine

coral reef. This is in sharp contrast to most reefs where the prey biomass

substantially dominates the total ﬁsh biomass.

Figure 2. Three biological hypotheses for the eﬀects of the refuge size on the

prey availability for predators. Type I: the prey availability for predators is

a decreasing function of the refuge size, because the refuge provides places

for prey to hide from predators. Type II: the prey availability for predators is

independent of the refuge size in the sense of density (per unit area), because

in a number of cases prey biomass is proportional to the refuge size. Type III:

the prey availability for predators is an increasing function of the refuge size,

because the refuge both provides prey to predators and stores prey for latter

consumption by predators.

24

Fig. 1. The pristine coral reef at Kingman.

Fig. 2. Three possible biological hypotheses for the eﬀects of the refuge size on the

prey availability for predators.

25