Untangling complex syste.., p.28
Untangling Complex Systems, page 28
handling, k .
h
When a Type III functional response with n = 2 is combined with a numerical response pro-
portional 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
k
2
hH C
= k
1 F 1 −
H
= H
dt
−
K
2
1 F
K + H
[5.32]
dC
k′ 2
hH C
=
− k C = C
dt
K + H 2
3
In equation [5.32], k′ h represents the predator’s efficiency at turning prey into predator offspring, whereas k is its mortality rate. The phase space for such system is shown in Figure 5.7. The grey 3
curve represents the nullcline
H = 0, whereas the continuous black vertical lines are three possible
values of the nullcline
C = 0. The intersection points between the two nullclines,
H = 0 and
C = 0,
represent stationary states. Their stability changes depending on the values of the rate constants. We
distinguish three regions in the phase space. Stationary states in the region labeled as (a) are stable
like those in the region (c). On the other hand, the stationary states of the intermediate region (b) are
unstable, and when the system is pushed away from them, it evolves to a limit cycle.
TRY EXERCISE 5.6
C• = 0
C• = 0
C
H• = 0
(a)
(b)
(c)
H
FIGURE 5.7 Phase-plane for a Type III functional response and a numerical response proportional to the
rate of prey consumption. The intersection points between the two nullclines in regions (a) and (c) are stable,
whereas those in (b) are unstable and the system evolves up to a limit cycle. The spiraling arrows represent
the dynamics.
The Emergence of Temporal Order in the Ecosystems
129
5.5 OTHER RELATIONSHIPS WITHIN AN ECOSYSTEM
Within an ecosystem, we not only have predator-prey interplays, but also many more symbiotic9
relationships. A list is in Table 5.1. The type of relationship depends on the mutual effects exerting on both protagonists, reminding that an effect can be either positive (+), or negative (−), or null (0).
The antagonism relationship may occur among species and also among members of the same
species. Antagonists may compete for the same type of food, living space, and mate. Antagonism
may cause the death of some competitors in fighting or when deprived of food for a long time. The
severity of antagonism depends on the extent of the similarity of resource requirements of different
organisms and the shortage of supply in the habitat.
Parasitism is a relationship in which one species, the parasite, is always benefited at the cost of the
other species, the host. A parasite derives nourishment from the host and, in many cases, it finds pro-
tection and living space on or inside the host. Usually, the host is bigger than the parasite. Bacteria and
fungi are the most important parasites of both plants and animals. Humans use antibiotics to defeat
several bacterial parasites. Plants themselves may be parasites either on other plants or animals.
Amensalism is a relationship wherein an organism is inhibited or destroyed whereas the other is
unaffected. The classical example is that of the bread mold Penicillium that secretes penicillin and
destructs bacteria. Many other examples are offered by plants that secrete chemicals to cause harm
to other plants.
Neutralism describes a relationship between two species where the health of one species has no
effect whatsoever on that of the other. Neutralism is challenging to detect and prove and is often
used to describe situations wherein interplays are insignificant.
In commensalism, one organism draws benefits whereas the other receives neither benefits nor
harms. Examples of commensalism are offered by epiphytes. All epiphytes use trees only for attach-
ment and derive moisture and nutrients from the air, rain, and sometimes from debris accumulating
around them. Sessile invertebrates that grow on plants or other animals represent many commensals.
Finally, a relationship is defined mutualistic when it is favorable to both partners. Mutualism may
be facultative when the species involved can exist independently. Otherwise, it may be obligatory,
when the relationship is imperative to the existence of one or both the individuals. For example, each
lichen is a mutualistic association between a fungus and an alga. The alga photosynthesizes food for
itself as well as for the fungus. The fungus, in turn, furnishes water and carbon dioxide. Pollination
of flowers by insects is another manifestation of mutualism. Humans engage in mutualisms with
other species, including their gut flora without which they would not be able to digest food efficiently.
TABLE 5.1
List of Possible Relationships between Two Species
within an Ecosystem
Effect on Species 1
Effect on Species 2
Relationship
−
−
Antagonism
−
+
Parasitism
−
0
Amensalism
0
0
Neutralism
+
0
Commensalism
+
+
Mutualism
9 Symbiosis derives from the Greek συv, i.e., together, and βıoσıς, i.e., life. In this book, it is considered in its broadest sense, referring to a close and prolonged association between two or more organisms of different or the same species. In some text, you may find the use of the term symbiosis only for the mutualistic relationship.
130
Untangling Complex Systems
5.6 MATHEMATICAL MODELING OF SYMBIOTIC RELATIONSHIPS
The general model to describe any symbiotic relationship between two species A and B, where F is the available food, can be the following one:
k
1 →
A + F ←
2 A
k−1
k
2 →
B + F ←
2 B
[5.33]
k−2
A + B k
12→
nAA+ B
A + B k
21→
nBB + A
Note that in the third process, B acts as a catalyst, being present among both the reagents and the
products. The same is true for A in the fourth process. The type of relationship that mechanism
[5.33] represents depends on the value of the coefficients n and n . All the possible combinations
A
B
are reported in Table 5.2.
5.6.1 anTagonism
In an antagonistic relationship between species A and B, the coefficients n and n appearing in A
B
[5.33] are both equal to 0. Therefore, mechanism [5.33] becomes:
k
1 →
A + F ←
2 A
k−1
k
2 →
B + F ←
2 B [5.34]
k−2
A + B k
12→
B
A + B k
21→
A
TABLE 5.2
List of All Possible Combinations of the Coefficients
nA and nB Appearing in the Scheme [5.33]
nA
nB
Relationship
0
0
Antagonism
0
2
Parasitism
0
1
Amensalism
1
1
Neutralism
2
1
Commensalism
2
2
Mutualism
The Emergence of Temporal Order in the Ecosystems
131
Both species, A and B, have logistic growth in the absence of the other (Murray 2002). According to
[5.34], the differential equations describing how the number of elements of species A and B change
over time are:
dA
2
A
k
=
12
k 1 FA − k−1 A − k 12 AB = k 1 FA 1−
−
B
dt
KA k−1 K
A
[5.35]
dB =
2
B
k 21
k 2 FB − k 2 B − k 21 AB = k 2 FB 1−
−
A
dt
−
KB k−2 KB
In [5.35], K
2
/ −2 )
1 / −1 )
A = ( k F k
and KB = ( k F k . To make easier the treatment of the system [5.35],
we nondimensionalize it, introducing three dimensionless variables: a = AC A; b = BC B and τ = tCt.
Inserting these new variables within [5.35], we obtain:
da
k 1 F
a
k 12
b
=
a 1−
−
dτ
t
C
KAAC k−1 KA B
C
. [5.36]
db
k 2 F
b
k
=
a
b 1−
−
21
dτ
t
C
KBBC k−2 KB AC
The next step is to choose the non-dimensionalizing constants. If we fix A
1/
)
1/
)
C = ( K A , BC = ( K B
and tC = k 1 F , we obtain
da
k 12 K
=
a 1− a
B
−
b
dτ
k−1 K
A
[5.37]
db
k 2 F
k 21 K
=
b 1− b
A
−
a
dτ
k
.
1 F
k−2 KB
Finally, fixing ρ = ( k
/ −
)
/ −
)
2 k
/ 1), r 12 = ( k 12 KB k 1 KA , and r 21 = ( k 21 KA k 2 KB , the two differential equations in dimensionless form become:
da = a(1− a− r 12 b) = f
τ
d
[5.38]
db = ρ b(1− b− r a
21 ) = g
dτ
The possible steady states are ( a
) 0 0) (1 0) (0 )1 ((1
( r r ))
12 ) (1
12 21 ) (1
21 )
ss , bss = ( ,
; , ; , ;
− r / − r r , − r / 1− 12 21 .
Their stability can be inferred by linear analysis. The Jacobian is
f
∂
f
∂
a
∂
b
∂
1− 2 ass − 12
r bss
− 12
r ass
J
ss
ss
=
=
. [5.39]
g
∂
g
∂
−ρ 21
r bss
ρ − 2ρ bss − ρ r a
21 ss
a
∂
b
∂
ss
ss
The steady state (0,0) is unstable. In fact,
1
0
J =
, tr ( J ) = 1+ ρ > 0, det ( J ) = ρ > 0 [5.40]
0
ρ
132
Untangling Complex Systems
For the steady state 1
( ,0), we have
−1
− r 12
J =
, tr ( J ) = ρ (1− r 21) −1, det ( J ) = −ρ (1− r 21)
[5.41]
0
ρ −
ρ r 21
If r
( )
( )
( )
21 > 1, det J
> 0, tr J < 0. Hence, the steady state is stable. If r 21 < 1, det J < 0; therefore, the steady state is a saddle point.
For the steady state (0,1), we have
1− r 12
0
J =
, tr ( J ) = 1− r 12 − ρ, det ( J ) = −ρ (1− r 12 ) [5.42]
−
ρ r 21
−ρ
If r
( )
( )
( )
12 > 1, det J
> 0, tr J < 0. Hence, the steady state is stable. If r 12 < 1, det J < 0; therefore, the steady state is a saddle point.
For the last steady state,
r
1
12 −1
r 12 ( r 12 − )
(1− r 12 r 21)
(1− r 12 r 21)
J =
ρ r
(
)
21 ( r 21 − )
1
ρ r 21 −1
(1− r
(
)
12 r 21 )
1− r 12 r 21
( r
)
(
)
12 −1 + ρ r
−1
tr ( J
21
) =
[5.43]
1
( − r )
112 21
r
ρ 1
( −
)(
)(
)
12
r 21
r
21
r −1 12
r −1
det ( J ) =
2
1
( −
)
12
r 21
r
We distinguish four situations: the first (i) is when both r 12 and r 21 are smaller than 1; the second (ii) is when both r 12 and r 21 are larger than 1; the third (iii) when r 12 > 1 and r 21 < 1; the fourth (iv) when r 12 < 1 and r 21 > 1. They are graphically represented in Figure 5.8.
In case (i), the steady states (1,0) and (0,1) are saddle points, whereas the steady state represented
by the intersection point between the two nullclines, f = 0 and g = 0, is stable because tr ( J ) < 0, whereas det ( J ) > 0. In the latter steady state, both species coexist. Such situation can occur when
two species, having practically the same carrying capacities ( KA ≈ KB ),10 show small interspecific competition parameters, that is ( k
)
(
)
12 k
/ 1
−
< 1 and k 21 k
/ 2
−
< 1.
In situation (ii), the steady states (1,0) and (0,1) are stable, whereas the steady state, represented
by the intersection point between the two nullclines, f = 0 and g = 0, is a saddle point since it has det ( J ) < 0 and tr ( J ) < 0. In such situation, if the two species have similar carrying capacity, then they have also large ratios ( k
)
(
)
12 k
/ 1
− and k 21 k
/ 2
−
. The competition is strong. It is difficult to predict
who ultimately wins out. It depends crucially on the starting condition. Each of the two stable steady
states, (1,0) and (0,1), has a domain of attraction. The eigenvector representing the stability direc-
tion of the saddle point divides the a−b space into regions: R1 and R2. If the initial condition lies
on R1, then eventually species B dies out, and a becomes equal to 1: species A reaches its carrying capacity. On the other hand, if the initial condition lies on R2, then species A will become extinct,
and B will reach its carrying capacity K . We expect to see the extinction of one species even if the B
10 More information about the carrying capacity is presented in Chapter 10, within the paragraph dedicated to the Logistic Map.
The Emergence of Temporal Order in the Ecosystems
133
1/ r 12
f
1
g
R2
b
b
1
g
1/ r 12
f
R1
(i)
(ii)
a
1
1/ r
a
1/ r 21
1
