Untangling Complex Systems, page 27
(continuous black trace) and − v
*
(
)
C ln ( C Css ) (dashed grey trace) in (b), and trend of P
RV within a cycle in (c).
The Emergence of Temporal Order in the Ecosystems
123
production is produced, and it is counterbalanced by the entropy production that is consumed in the
second half.
This behavior reminds us that of coupled chemical reactions. Having negative values of entropy
production is possible. However, if we consider the sum of all the contributions, the net result is
either a positive or a total null entropy production.
5.4 MORE ABOUT PREDATOR-PREY RELATIONSHIPS
The Lotka-Volterra model studied so far accounts for the existence of cyclic variations in popula-
tions of predators and prey. However, it has some unsatisfactory features. For example, according
to the steps in mechanism [5.1], in the absence of C, the population of H should grow exponentially,
and this is not realistic. A model closer to reality describes the growth of H, in the absence of C, like that of a population in the presence of limited food resources.4
TRY EXERCISES 5.1 AND 5.2
The Lotka-Volterra model has other weak points. For example, it assumes that the rate of prey con-
sumption is a linear function of the prey density; every predator assault has a positive outcome; the prey
assimilated by the predators are all turned into new predators, and all the species are uniformly distrib-
uted in the environment. These are rough assumptions of what we can encounter in real ecosystems.
A more general system of differential equations (Harrison 1979) for the predator-prey interaction
has the following shape:
dH = h( H)− f ( H ) a( C)
dt
. [5.16]
dC = g( H) b( C)− c( C)
dt
In the first differential equation of [5.16], the term h( H ) represents the intrinsic growth rate of the prey including all factors except predation; the term −
f ( H ) a( C ) is the rate of prey consump-
tion per predator, and f ( H ) is called the functional response of the predator. In the second equa-
tion appearing in equations [5.16], the term − c ( C ) is the intrinsic rate of decrease of the predator, whereas the term g ( H ) b( C ) represents the predator rate of increase in the presence of H and g ( H ) is called the numerical response of the predator. Holling (1959), 5 studying the predation of small mammals on pine sawflies, identified three types of functional responses. Type I is linear with H,
as in Lotka-Volterra model and the example of exercise 5.1 (see Figure 5.5). It is verified with passive predators like spiders. The number of flies captured in spider net is proportional to fly density.
Type II describes a situation in which the number of prey consumed per predator initially rises
almost linearly as the density of prey increases, but then it bends and finally levels off (see Figure 5.5).
Type II is a valid model when a predator has to accomplish two distinct actions to reach its final goal:
• Capturing a prey, which means wandering, detecting, chasing, catching and subduing it;
• Handling the prey, which means killing, eating, and digesting it.6
4 For more information, see also the Logistic Map in Chapter 10. For the moment, try exercises 5.1 and 5.2.
5 Crawford Stanley Holling (born in 1930 in Theresa, New York) is one of the conceptual founders of ecological economics, which is referred to as both an interdisciplinary field of academic research that aims to address the interdependence and coevolution of human economies and natural ecosystems over time and space.
6 A further refinement of Type II functional response is presented by Jeschke et al. (2002). In their paper, the authors introduce digestion of the prey by a predator as a process influencing the hunger level of predators and hence the probability of searching for new prey. It emerges that the asymptotic maximum predation rate (i.e., asymptotic maximum number of prey eaten per unit time, for prey density approaching infinity) is determined by either the handling time or the digestion time.
124
Untangling Complex Systems
2
Type I
Type II
)
f( H
Type III
0
H
0
FIGURE 5.5 The three types of functional responses f( H) as functions of prey density.
The mechanistic model for Type II functional response is
kc
→
C H
CH
kh
+ ←
(
) →
C
2 ,
[5.17]
k− c
where:
C captures H with a rate constant k , c
( CH) denotes the prey captured by the predator.
H can escape before being killed and eaten by C, with a rate constant k– . The rate constant of hanc
dling is k . The expression of the functional response can be achieved by writing how H and ( CH) h
change over time according to the elementary steps of [5.17]:
dH = − k
− (
)
c HC + k c CH
dt
[5.18]
d ( CH ) = k
− (
)
( )
c HC − k c CH − kh CH .
dt
If we apply the steady-state approximation for ( CH) and we exploit the mass balance equation
C 0 = C + C
( H ), where C 0 is the total density of predators, we obtain:
k
[ 0 −
]
c HC
kcH C
( CH )
( CH ) =
=
. [5.19]
k− c + kh
k− c + kh
The definition of ( CH) becomes
kcHC
( CH ) =
0
. [5.20]
k− c + kh + kcH
The Emergence of Temporal Order in the Ecosystems
125
By inserting equation [5.20] and the mass balance for a predator into the first differential equation
of system [5.18], we achieve:
dH
khHC
= − k
0
0
)
c H C − ( CH
k− c CH
. [5.21]
dt
+
( ) = − k− c + kh + H
kc
The Type II functional response will be
khH
khH
f ( )
II H
=
=
. [5.22]
k− c + kh + H K + H
kc
It is a rectangular hyperbolic function having a concave down graph (see Figure 5.5). When H is low and much smaller than K, the functional response increases linearly with H, that is f ( )
II H
≈ khH/ K.
When the prey density is high and much larger than K, f ( )
II H
≈ kh. In other words, at high densities,
prey is easy to capture, and the rate of their consumption is almost exclusively affected by the time
required for pursuing the “handling stage.” When a Type II functional response is combined with
a numerical response proportional to the rate of prey consumption, the system of two differential
equations describing the evolution of the number of prey and predator is:
dH
H
khHC
= k
1 F 1 −
H
= H
−
dt
K 1 F
K + H
. [5.23]
dC
k′ hHC
=
− k C
3
= C
dt
K + H
In equation [5.23], the predators convert the predation into offspring with an efficiency k′ h ≠ kh and experience density-independent mortality at a rate k . K
/ − )
1 = ( k 1 k
.
3
1
The evolution of the prey and predator system can be depicted in a phase space defined by C and
H (see Figure 5.6). The equations representing the null-clines
H = 0 and
C = 0 are a concave-down
parabola and a vertical line, respectively:
k 1 FK
k 1 F
K
k
C =
+
−
H
1 H 2
1
[5.24]
k
− −
h
kh
K 1 F
kh
k 3 K
H =
. [5.25]
k′ h − k 3
The parabola intercepts the C axis at C = k 1 FK kh, and the H axis at H = K 1 F; its vertex7 is at H = ( K
′
1 F − K ) 2
/ (see Figure 5.6 and equation [5.24]). If ( k 3 K ( kh − k 3)) ≤ K 1 F, there is a stationary state represented by the intersection between the parabola and the vertical line. When the vertical
isocline is to the right of the vertex of the parabola, that is when
k
(
− )
3 K
K 1 F K
>
, [5.26]
k′ h − k 3
2
7 The vertex of the parabola can be found by calculating the derivative of C with respect to H and determining when it is null: dC dH = k
1
2
0.
1 F
k ( −
1
) −
−1
=
h
K K F
Hk
kh
126
Untangling Complex Systems
C
k 1 FK
kh
H
k 3 K
K 1 F − K
k 3 K
K
k′
1 F
h − k 3
2
k′
A
h − k 3 B
FIGURE 5.6 Phase-plane for a Type II functional response and a numerical response proportional to the
rate of prey consumption. The spiraling arrows represent dynamics in the neighborhood of stationary states
indicated by the intersections of straight lines (A and B) with the parabola. The vertical dashed line indicates
the vertex of the parabola. All the intersection points on the left of the vertex represent unstable solutions.
The corresponding stationary states are stable. Any oscillations in predator and prey densities will
eventually be damped out, and both species will coexist at the levels H and C (defined by the
ss
ss
intersection points).
When the vertical isocline is to the left of the vertex or intersects the vertex, the stationary states
are unstable. A prey-predator system that starts close to a stationary state exhibits diverging oscilla-
tions that spiral out toward a limit cycle. The existence of the limit cycle can be shown by using the
Poincaré-Bendixson theorem8 (Strogatz 1994). This theorem says that if a trajectory is confined to a closed, bounded region that contains no fixed points, then the trajectory must eventually approach
a closed orbit. This type of closed curve corresponds to sustained oscillations in time, with unique
amplitude and frequency, irrespective of initial conditions. It should be distinguished from oscilla-
tions of the Lotka-Volterra type, which present an infinity of amplitudes and frequencies depending
on the initial conditions. Numerical solutions of [5.23] show that the further the stationary state is to
the left of the vertex of the prey isocline, the faster the oscillations diverge and the larger the ampli-
tude of the limit cycle. Once the trajectory is close to the limit cycle, the system exhibits continued
oscillations. Whether the predator and prey can coexist oscillating in their number, depends on how
close these oscillations come to the boundaries C = 0 and H = 0 (Harrison 1995).
TRY EXERCISES 5.3, 5.4 AND 5.5
Finally, there is the Type III functional response, which has a sigmoid shape curve (see Figure 5.5). When the prey density is pretty low, the rate of predation is really slow. At intermediate densities, it grows
sharply, and at very high densities, saturation occurs. A functional response having a Type III shape
emerges when (1) the ecosystem is heterogeneous, and the prey is not uniformly distributed across the
space, or (2) the predator has to do practice to become competent in catching and killing prey.
8 The Poincaré-Bendixson theorem applies only in two-dimensional systems. In higher dimensional systems ( n ≥ 3) trajectories may wander around forever in a bounded region without settling down to a fixed point or a closed orbit. In some cases, the trajectories are attracted to a complex geometric object called a strange attractor. A strange attractor is a fractal object (see Chapter 10) on which the motion is chaotic because it is aperiodic and sensitive to tiny changes in initial conditions. The Poincaré-Bendixson theorem excludes chaos in two-dimensional phase space.
The Emergence of Temporal Order in the Ecosystems
127
1. When a kind of prey is distributed unevenly in patches, the predator may encounter one
of such wealthy areas and have many options ( n > 1) to chase: all of them in proximity.
For example, a kind of prey is browsing only in a few leaves of a tree. When its preda-
tor lands in one of those leaves, which are wealthy in prey, it has many chances to catch
and eat its favorite food, up to complete satiation. If the predator fails to find his pre-
ferred kind of prey after searching in many leaves, it may switch to retrieve another type
of prey, just to avoid dying of starvation. Guppy is an example of switching predator.
Usually, it hunts fruit flies on the water’s surface. When fruit flies decrease in number,
guppies change from feeding on the fruit flies to feeding on tubificids living in deep
water (Murdoch 1977).
2. Type III functional response is the only type of functional response for which prey mortal-
ity can increase with prey density raising from very low values. This property accounts
for a natural improvement of a predator’s hunting efficiency as prey density increases. For
example, many predators respond to chemicals emitted by prey (the so-called kairomones)
and increase their activity. Polyphagous vertebrate predators (e.g., birds) can switch to the
most abundant prey species by learning to recognize it visually. When a predator finds
a type of prey infrequently, it has no experience to develop the best ways to capture and
kill that species of prey. For instance, many felines kill prey larger than themselves by
inserting canines between vertebrae and disarticulating the spine. They must learn exactly
where to bite and how to kill big prey before it injures them.
A mechanism modeling the Type III functional response is
kc
→
C nH
CH
kh
+
←
(
) n →
C
2 ,
[5.27]
k− c
where:
n is a number larger than 1 and the kinetic constants k , k , k have the same meaning encountered c
–c
h
in [5.17]
The differential equations describing how H and ( CH ) change over time are
n
dH = − nk CHn
− (
)
c
+ nk c CH
dt
n
[5.28]
d ( CH )
n
= k C n ( −
)(
)
c H − k c + kh
CH
dt
n
If we apply the steady-state approximation for ( CH ) and the mass balance for the predator
n
( C
)), we obtain
0 = C + ( CHn
k C
n
c 0 H
( CH ) n =
. [5.29]
k
n
− c + kh + kcH
Introducing this definition of ( CH ) in the first differential equation of the system [5.28], we obtain
n
that
1 dH
k
n
n
n
ck C
h
H
k C
h
H
k C
h
H
0
0
0
= −
= −
= −
. [5.30]
n dt
k
n
n
− +
− c + kh + kcH
k c kh + Hn
K + H
kc
128
Untangling Complex Systems
Type III functional response will be
k
n
hH
f ( )
III H
=
. [5.31]
K + H n
Equation [5.31] confirms that Type III functional response is a sigmoid function. When the prey
density is very low, f ( H) is a function of H to the power of n. On the other hand, when H is III
high enough to be much larger than K, f ( H) becomes constant and roughly equal to the rate of III
