Evolutionary distributions: A theoretical framework
for evolutionary ecology
Yosef Cohen
University of Minnesota
1
2
3
August 16, 2006
4
Abstract
5
Adaptive traits are inherited with small mutations. They are subject to natural selection. An adaptive space is made of a set of adaptive
traits. Within this adaptive space live distributions of the density of
phenotypes. The dynamics of this density reflect evolution by natural
selection. We call these distributions Evolutionary Distributions (ED).
The adaptive space with its ED form the evolutionary space. ED are
derived from first principles of population dynamics: birth, death, mutations and selection. This approach leads to a rich behavior of ED
with and without stability. With ED, it is easy to understand how
we may find phenotypes with different fitness values at evolutionary
stability. The theory of ED considers the dynamics of densities of all
possible phenotypes on a set of adaptive traits. Thus, invasions by
mutants represent perturbations to the ED. Stable ED therefore do
not allow invasion of mutants and as such, are akin to the concept
of evolutionary stable strategies (ESS). We apply the theory to three
common coevolutionary interactions: competition, predation and host
pathogen. We also apply the theory to (open) ecosystems and illustrate application in the context of primary producers and primary
consumers ecosystems. We also illustrate applications of coevolution
in the context of global change. The theory is restricted to (not necessarily continuous) adaptive traits that can be characterized by real
numbers.
6
Key words: evolutionary distributions, competition, predator prey,
host parasite, adaptive traits, global change.
1
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Cohen · Evolutionary distributions
1
2
Introduction
30
A large class of population models in ecology and epidemiology boils down
to
z0 (t) = f (z (t)) , z (0) = f0
(1)
where 0 denote derivatives with respect to time, z is a vector of real nonnegative time-dependent functions (species population densities), f is a vector
of real valued smooth functions, t is a nonnegative real number representing
time and f0 is given. All vectors are of dimension n. To fix ideas, the precise
statement of this class of models is comprised of (1) with
z ∈ Rn0+ ,
t ∈ R0+ , f : Rn0+ × R0+ → Rn0+
where ∈ means “a member of”, Rn0+ denotes the n-dimensional space of
nonnegative real numbers, × denotes the Euclidean product (e.g., R × R is
a plane) and → means that f is a function with domain on the left side of
the arrow and range on the right.
31
32
33
34
There are several assumptions implicit in models that belong to this class:
(i) reproduction is by cloning; (ii) z (t) is large; (iii) z (t) represents some
moment of the dynamics (usually the mean) of z; (iv) stochastic effects can
be faithfully represented by z (t); and (v) z are smooth functions of t. Some
of these assumptions were discussed by Dieckmann et al. (1995), Doebeli and
Ruxton (1997), Law et al. (1997) and Geritz and Kisdi (2000). In practice,
conclusions from models of class (1) routinely violate the assumptions or are
stated without regard to the assumptions. For example, some applications of
models of class (1) produce negative population densities; this is particularly
true when oscillations are involved. To circumvent such problems, authors
enlarge this class of models to
f (z (t)) if z (t) ≥ ε
0
z (t) =
, z (0) = f0
(2)
0
otherwise
where ε is a small positive number. This is a class of discontinuous functions
and care must be taken in solving (2)—particularly near the discontinuity.
To cast (1) in the context of evolution by natural selection, one needs to
encapsulate in mathematical models the following minimum set of axioms:
1. Phenotypic traits are inherited with some mutations.
35
36
37
38
39
Cohen · Evolutionary distributions
3
2. These mutations are carried in new individuals in the population.
3. Natural selection acts directly on phenotypic traits; not on genotypic
traits.
These axiom have important consequences to evolution by natural selection:
(i) we detect evolution by natural selection through changes in the relative
densities (distribution) of phenotypes; (ii) without distributions of phenotypes, evolution comes to a halt; (iii) selection results in different mortality
rates of different phenotypes; (iv) mutations occur when new individuals are
formed and therefore can be modeled as part of the birth process; (v) natural
selection is usually modeled as a death process (this is the meaning of “acts”
in Axiom 3); (vi) because mutations are random (not directional), evolution
by natural selection can be observed on asymmetric distributions of phenotypes only. Additional consequences of these axioms will be articulated as
we move on.
40
41
42
43
44
45
46
47
48
49
50
51
52
53
To proceed from (1), one usually identifies a set of adaptive traits and writes
(1) as
z0 (xi , x, t) = f (z (xi , x, t)) , z (0) = f0
(3)
where now xi is the set of adaptive traits of species i. The dimension of xi
is mi . Here x is aP
set of real numbers of all adaptive traits of all species with
dimension m := ni=1 mi . These models can be stated mathematically as
(3) with
z ∈ Rn0+ ,
t ∈ R0+ , f : Rn0+ × R0+ × Rm1 × · · · × Rmn → Rn0+ .
Next, the dynamics of x (called the strategy dynamics) may be isolated by
several methods, each involving additional assumptions (see Murray, 1989;
Weibull, 1995; Fudenburg and Levine, 1998; Hofbauer and Sigmund, 1998;
Samuelson, 1998; Gintis, 2000; Cressman, 2003; Hofbauer and Sigmund,
2003; Vincent and Brown, 2005). For example, in the case of adaptive
dynamics (Dieckmann and Law, 1996; Champagnat et al., 2001), the additional assumptions are: (i) mutations are rare, and at small time intervals
the probability that more than one mutation occur is of order zero; (ii) x is a
moment (usually mean) of some distribution of phenotypes. The nice thing
about the adaptive dynamics approach is that with considerations from measure theory (Doob, 1994), the derivation of the strategy dynamics justifies
the fact that no matter how large the population, it still can be viewed as a
collection of individuals (Dieckmann and Law, 1996). Furthermore, in analyzing predator-prey coevolution, Dieckmann et al. (1995) showed that with
54
55
56
57
58
59
60
61
62
63
64
65
66
67
Cohen · Evolutionary distributions
4
some level of stochasticity, deterministic models (i.e., where z is the mean
path) do represent the mean of the stochastic path.
Other approaches to deriving the strategy dynamics (e.g. Abrams et al.,
1993) involve the assumption that population densities change much faster
than strategy values. Therefore, one assumes that the populations are at
equilibrium and only strategy dynamics need to be considered. This assumption is circumvented by the approaches that Abrams (1992) and Vincent and
Brown (2005) take (see also Cohen et al., 2000), where population dynamics need not be ignored. Vincent and Brown (2005) refer to the strategy
dynamics with the related population dynamics as “Darwinian dynamics”.
68
69
70
71
72
73
74
75
76
77
Regardless of the approach one takes, strategy dynamics usually take on the
form
x0i (t) = gi (v, x, t) , x (0) = x0 , i = 1, . . . , n.
(4)
Here gi is a vector valued function (usually a gradient of some other function)
of dimension mi and v is a vector (of dimension mi ) of the so-called virtual
strategies. To derive (4) from (3), one needs to add the assumptions that
f and g are at least twice differentiable. These assumptions are required
because one needs to examine special points (e.g., maxima and minima) on
the strategy dynamics.
78
79
80
81
82
83
The basic achievement in deriving gi is that it gives invasion-ability criteria.
e, that give negative
Specifically, from gi we derive values of x, call them x
e (with respect to z). Here we have
gradient on gi for all i and for all x 6= x
gi : Rmi × Rm1 × · · · × Rmn × R0+ → Rm .
It should be noted that in the case of Darwinian dynamics, (4) becomes
x0i (t) = gi (v, x, z,t) , x (0) = x0 ,
i = 1, . . . , n
(5)
and the strategy dynamics can no longer be divorced from the population
e on the dynamics of all other values of x
dynamics. Because of the effect of x
e the Evolutionary Stable Strategies (ESS) of the system (4) or the
we call x
system (3) and (5). So far, the strategies are assumed unbounded. Bounded
strategy spaces introduce complications that can be avoided if one assumes
e is in the interior of the strategy space.
that x
At any rate, the same violations or misuse of the assumptions behind (1)
apply to evolutionary models, with one crucial addition: Behind each x hide
distributions of the density of phenotypes of z. In fact, the dynamics of x
84
85
86
87
88
89
90
91
92
Cohen · Evolutionary distributions
5
93
94
95
96
z(x,t)
represent the dynamics of moments of these distributions (usually the mean).
This raises the possibility that we may be following dynamics of phenotypes
whose distribution can hardly be represented by their mean. Worse yet,
we may be following the dynamics of phenotypes that do not exist (Figure
e
1). Furthermore, because ESS is an outcome of a game, the stability of x
x_
^x
_
x
Figure 1: A bounded distribution (between x and x) of phenotypes at time t.
Note that there are no phenotypes at the mean adaptive trait, x
b. The
figure illustrates a snapshot of some evolutionary distribution at t.
97
cannot be determined by traditional stability analysis (except for the case
of Darwinian dynamics). Hence, the proliferation of stability criteria for
ESS (Eshel, 1996; Apaloo, 1997; Taylor and Day, 1997). By examining the
stability of Evolutionary Distributions, (ED1 ) some of the ESS related issues
can be simplified; e.g., stability of ED is similar to ESS. ED also bypass the
potential problems that arise when strategy dynamics are followed by some
distribution moments. Yet, ED do suffer from limitations which will be
pointed out as we proceed.
The roots of the the theory developed here can be found in the following:
Kimura (1965), who was interested in population genetics and characterizing
moments of the distribution of genotypes. Levin and Segel (1985) referred
to the adaptive space as aspect (see also Keshet and Segel, 1984). Slatkin
(1981) used a diffusion model to discuss species selection. Ludwig and Levin
(1991) examined the dispersal of phenotypes in plant communities. A different approach to characterizing evolutionary related distributions is to use
1
I use ED in both singular and plural.
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
Cohen · Evolutionary distributions
6
moments. In fact, some distributions can be characterized by a finite (and
small) number of moments (e.g., see Barton and Keightley, 2002). As we
shall see, in some of the cases we examine, characterizing distributions by
moments is a hopeless task.
So here is the plan. In Section 2 we choose one specific example from smooth
evolutionary games. With this example, the transition from evolutionary
games to ED should be smooth. In Section 3 we introduce ED and define
them formally. The theory is then applied to various ecological systems with
coevolution added: competition (Section 4), predator prey (Section 5) and
host pathogen (Section 6). In Section 7 we see how the ED approach may be
applied to ecosystems, where the inanimate part of the environment must be
included. This leads to potential applications of coevolutionary principles
to global change scenarios. In Section 8 we shall discuss the limitations of
the theory and future developments. Our main interest is to introduce a
theoretical framework. Consequently, each section can serve as a basis for
numerous manuscripts with deeper analysis. We will leave such analyses for
future work.
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
Notation To be unambiguous, we must stick to mathematical notation.
So: := means equal by definition and z ≡ z (t) means that z equals z (t) for
all t. Also,
∂ 2 z (x, t)
dz (x)
, ∂xx z (x, t) :=
,
z 0 :=
dx
∂x2
x ∈ X means that x is a member of the set X and X ⊂ R means that X is a
subset of R where R denotes the real numbers. Rn denotes the n dimensional
space of real numbers, R0+ denotes the set of nonnegative real numbers and
R × R denotes the Cartesian product of two real lines (a plane). f : X →
Y means that f maps elements in X to elements in Y (i.e., f is a function).
Boldface notation indicate vectors; e.g., f (x) is an n dimensional vector
valued function [f1 (x) , . . . , fn (x)] . A is an operator—a rule that applies to
functions; e.g., Az (x, t) = z (x, t) + ∂xx z (x, t). ∆ is a (very) small quantity.
∂X is the boundary of X ; e.g., if X = {1, 2, 3} then ∂X = 1 and 2. X ∪∂X
is the union of the open set X with its boundary ∂X . This closed set is
denoted by X . The notation
∂
G (v, x, z)|v=xi
∂v
says “take the derivative of G with respect to v and then replace v with xi ”.
Boundary conditions sometimes require the so-called unit outward normal.
130
131
Cohen · Evolutionary distributions
7
This means that one needs a directional derivative that is perpendicular to
the tangent to the boundary at a particular point. To fully specify a system
of partial differential equations we need both initial conditions (t = 0) and
boundary conditions. That is, we must specify the system’s behavior at
its boundaries. Both the initial and boundary conditions are called data.
Often we need boundary conditions with no flows across the boundary and
nonuniform initial surface. We achieve this with the sin or cos functions
and with appropriate size of the solution space.
2
Smooth evolutionary games
There are different ways to study evolutionary games. Two distinct approaches are through matrix games and through games that involve smooth
functions (Hofbauer and Sigmund, 1998, 2003). The latter is implemented
with ordinary differential equations (ODE) and the approach to formulating
such games was discussed in Section 1. To illustrate the connection between
smooth evolutionary games and ED, we shall follow the approach taken by
Vincent and Brown (2005) and apply it to coevolution under competition
(see Cohen et al., 2000, 2001, for further details).
The Lotka-Volterra competition model with a single adaptive trait x (see
Vincent et al., 1993) is
zi0 = zi G (xi , x, z) , i = 1, . . . , n.
Here x = [x1 , . . . , xn ] and G (xi , x, z) is the instantaneous individual fitness
function for species i. The explicit form of G is
n
G (·) = r −
r X
α (xi , x) zi
k (xi )
i=1
where k (·) and α (·) are the carrying capacity and competition functions.
The ability of i to adjust to its carrying capacity is a function of the value
of the adaptive trait. The ability of i to compete is a function of the value
of its adaptive trait and the values for all other species. The explicit forms
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
Cohen · Evolutionary distributions
8
of k and α for i are
#
1 xi 2
,
k (xi ) = km exp −
2 σk
"
"
#
#
1 xi − xj + b 2
1 b 2
α (xi , xj ) = 1 + exp −
− exp −
.
2
σα
2 σα
"
(6)
Thus, we obtain the strategy dynamics
x0i = σ
∂
G (v, x, z)|v=xi
∂v
(see Vincent et al., 1993). We use the parameter values
√
km = 100, σk = 12.5, b = 2, σα = 2,
√
σ
=
0.04, r = 0.25
EDFramework-v01.nb
(7)
1
with n = 2. These values lead to ESS (Figure 2).
50
4
45
z2
3
z1
40
x2
2
35
1
30
x1
0
0
5μ10
t
4
10
5
0
5μ104
t
105
Figure 2: Two-species ESS: population dynamics (left) and strategy dynamics
(right).
149
3
Evolutionary distributions
We shall translate the smooth games approach into a single ED with a single
adaptive trait. Later we state the results more generally. We do not identify
z with a species density, but with the distribution of phenotype densities (see
150
Cohen · Evolutionary distributions
9
Figure 1). On the distribution, one might identify species (see discussion in
Cohen, 2003a, 2005). So we write
z 0 = f (z, x, t)
where z ≡ z (x, t). Next, we decompose f into the two components: one
that results in growth in the density of phenotypes and the other in decline:
f (z, x, t) = βe (z, x, t) − µ (z, x, t) .
As Darwin (1859) surmised, when resources are plenty, populations grow
exponentially. In our vernacular, βe is a linear function of z. As resources are
exhausted, β (·) − µ (·) decreases. We usually assume that mutations occur
e and selection on processes that lead to decline in the
at birth (i.e., on β)
density of phenotypes; namely µ (these assumptions are the consequences
of Axioms 1-3). Thus we write
βe (z, x, t) = βz (x, t)
(8)
where β is a rate coefficient. Now assume a mutation rate coefficient η.
Then
1
∂t z (x, t) = (1 − η) βz (x, t) + βη [z (x + ∆) + z (x − ∆)] − µ (z, x, t) .
2
Using Taylor series expansion of z around x, we obtain approximately
1
∂t z = z + ∆2 βη∂xx z − µ (z, x, t) .
2
Now for a single ED with m independent adaptive traits, we obtain
m
X
1
∂t z = z + ∆2 β
ηi ∂xi xi z − µ (z, x, t) .
2
(9)
i=1
For convenience, we define the linear operator
mA
=1+k
m
X
mA
thus
ηi ∂xi xi
(10)
i=1
where k := ∆2 β/2. To emphasize its role, we call
operator.
mA
the m-order mutation
152
For the case of n ED, each with mi independent adaptive traits, we have
∂t zi (xi , t) =
mi Azi (xi , t)
− µi (z, x, t) , i = 1, . . . , n .
151
(11)
Cohen · Evolutionary distributions
10
In (11) we assume that all traits are independent and that mutations are
random. The selective effects on trait values are embodied in µ (·) . Note
that we shall usually assume that the mutation operator depends on the
set of adaptive traits within a specific ED, zi . Thus we write mi Az (xi , t).
Declines in densities of phenotypes on an ED depend on the adaptive traits
of all ED. Therefore, we write µi (z, x, t).
The problems we deal with require specific boundary conditions. In general,
there is no a priori reason to set the boundary conditions to a particular
value, so we shall use the Neumann boundary conditions—any dissipation
across the boundaries disappears from the system. Because we deal with
rectangular boundaries and orthogonal traits, our boundary conditions remain simple. We shall always set the initial conditions such that they are
compatible with the boundary conditions. Also, physical constraints dictate
that x is bounded.
To formally define ED, we need the following. Let zi ∈ R0+ , i = 1, . . ., n
be the distribution of the density of phenotypes with mi adaptive traits xi
and
x := [x1 , . . . , xn ]. Define the bounded open set X ⊂ RM (where M
Plet
n
= i=1 mi ) with boundary ∂X . Then,
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
Definition An ED, zi (x, t), is the solution of the system
∂t zi (x, t) = β
mi Azi (xi , t)
− µi (z, x, t)
(where A is the mutation operator) with the data
zi (x, 0) = zi0 (x)
(where zi0 (x) given) and
∂n zi (x, t)|x=∂X = 0, i = 1, . . . , n
where ∂n denotes the directional (outward normal) derivative on ∂X .
The definition implies that: (i) mutations occur on the processes that lead
to growth in zi ; (ii) growth rate is a linear function of zi ; (iii) selection
occurs on the processes that lead to decline in zi ; (iv) selection depends on
all ED with which zi interacts; and (v) adaptive traits are bounded with
dissipation across boundaries. These are simplifying implications. They can
171
172
173
174
175
176
Cohen · Evolutionary distributions
11
be relaxed at the expense of more elaborate models and there is nothing in
the ED approach that limits us to these simplifications.
The set X = X ∪∂X defines the so-called adaptive space and X ×Rn0+ defines
the evolutionary space. The restriction to linear growth can be relaxed. In
such cases, the Taylor series approximation becomes more involved.
Because we model physical systems, we shall not be concerned with the existence and uniqueness of solutions—if solutions do not exist, then there is
something fundamentally wrong with the model. We do not require uniqueness because there is nothing in the theory of evolution that requires unique
solutions. Next, we translate the continuous game in Section 2 to ED.
4
Competition
178
179
180
181
182
183
184
185
186
187
Here we examine competition in the context of ED. We shall first look at an
example with mutations, but no selection. Then we examine the case with
a single evolutionary trait where selection operates on both the carrying
capacity and competitive ability. Finally, we examine the case where there
are two orthogonal adaptive traits: one that affects fitness with regard to
carrying capacity and the other with regard to competitive ability.
4.1
177
Single adaptive trait without selection
188
189
190
191
192
193
194
Using the Lotka-Volterra competition model in Section 2, we write
∂t z = rAz −
r 2
z .
km
(12)
This is the well known Fisher equation (Fisher, 1930; Smoller, 1982). As in
(7), we use
r = 0.25, km = 100, ∆ = 0.01, η = 0.01.
To completely specify the system, we set the boundaries of x to π/2 and
9π/2 and the data to
z (x, 0) = 20,
∂x z (π/2, t) = ∂x z (9π/2, t) = 0.
This results in a homogeneous stable ED, z (x) = 100 (Figure 3).
195
EDFramework-2d.nb
Cohen · Evolutionary distributions
12
30
t
20
10
0
100
80
z 60
40
20
5
x
10
Figure 3: Single trait ED with no evolution.
4.2
Single adaptive trait with selection
196
To implement adaptation to both competition and carrying capacity, we
must modify (12) appropriately. We assume that phenotypes with x =
5π/2 are most adapted to the environment, but competition takes its toll
and write
"
#!
1 x−ξ 2
α (x, ξ) = kα 1 + k exp −
(13)
2
σα
for competition and
"
k (x) = km
1
1 + exp −
2
x − 5π/2
σk
2 #!
(14)
for carrying capacity. The parameters kα and km scale the influence of
competition and carrying capacity on the ED. The parameters σα and σk
reflect the plasticity of phenotypes with respect to the adaptive traits that
influence competition and carrying capacity, respectively.
197
198
199
200
Cohen · Evolutionary distributions
13
Now (12) becomes
r
∂t z = rAz −
z (x, t)
k(x)
Z
9π/2
α (x, ξ) z (ξ, t) dξ
π/2
with
z (x, 0) = 0.005,
∂x z (π/2, t) = ∂x z (9π/2, t) = 0.
With the parameter values
r = 0.25, σk = 100, kα = 1, σα = 1,
km = 104 , ∆ = 0.01, η = 0.01
the effectEDFramework-2d.nb
of adding selection through both carrying capacity and competition is to depress the phenotype densities in the ED (Figure 4). Mutations
0 0.2
201
t
0.4
0.6
0.8
1
8
6
4 z
2
0
5
10
x
Figure 4: ED with a single adaptive trait and with competition.
202
and selection end up with a concentration of phenotypes around the most
adaptive value of the carrying capacity and at the boundaries. The boundary
phenotypes experience half the competition that phenotypes in the interior
do. Consequently the rise in densities on the boundaries.
203
204
205
206
Cohen · Evolutionary distributions
14
The theory predicts that where competition dominates, we should find the
most fit phenotypes on the boundaries of the adaptive space. For example,
in places where competition among plants for light is strong (e.g., no nutrient limitations and uniform carrying capacity with respect to the values
of the adaptive trait), we should find—at evolutionary stability of ED—
higher density of short and tall plants than intermediate plants: A common
phenomenon in forested plant communities.
4.3
Two adaptive traits with selection
One can hardly expect a single adaptive trait to be involved with both adaptation to carrying capacity and competition. Let us assume that we have
two orthogonal traits, x := [x1 , x2 ]; the first is selected for carrying capacity
and the second for competitive ability. Including covariance between x1 and
x2 with respect to adaptation to carrying capacity is trivial.
207
208
209
210
211
212
213
214
215
216
217
218
219
So now we write
r
z
∂t z = r 2 Az −
k (x1 )
Z
9π/2
α (x2 , ξ) z (x1 , ξ, t) dξ
π/2
with data
z (x, 0) = 20,
∂x1 z (π/2, x2 , t) = ∂x1 z (9π/2, x2 , t) = 0,
∂x2 z (x1 , π/2, t) = ∂x2 z (x1 , 9π/2, t) = 0,
where k (x1 ) and α (x2 , ξ) are defined in (13) and (14) and z ≡ z (x1 , x2 , t).
With the same parameter values as before, we obtain a stable solution shown
in Figure 5. As in all other results, when we say a stable distribution we
mean stable in a numerical sense. The conclusion of stability was reached
after running the numerical solutions for a very long time. With two traits,
the effect of competition on the ED becomes apparent: at both ends of the
adaptive trait that reflects competitive ability (x2 ) we have higher density of
phenotypes on the stable ED. This is so because at the boundaries, phenotypes are faced with half as much competition as phenotypes in the interior
of the adaptive space.
What we see is that the highest fitness (most abundant phenotypes at stable
ED) is with respect to the carrying capacity adaptive trait (x1 ). Among
220
221
222
223
224
225
226
227
228
229
230
231
EDFramework-2d.nb
Cohen · Evolutionary distributions
15
6
4
z
2
5
x2
5
10
10
x1
Figure 5: Stable ED with two adaptive traits.
these phenotypes, the highest fitness is that of phenotypes on the boundaries
of the adaptive trait that reflect competitive ability (x2 ).
4.4
Summary
The models in this section and any of their reasonable modifications result
in parabolic partial differential equation (PDE) for single adaptive traits or
elliptic PDE for multiple adaptive traits. Much is known about the qualitative behavior of such equations and for some, implicit solutions are available
(see for example Evans, 1998). We wish to concentrate on our approach
and therefore bypass deeper mathematical analysis of the equations. The
results here seem to contradict the conclusions by Gyllenberg and Meszéna
(2005) who excluded “coexistence of an infinite set of equidistant strategies
when the total population size is finite.” They then conclude that ecological
coexistence dictates the discreteness of species. We do not address the issue
of species.
232
233
234
235
236
237
238
239
240
241
242
243
244
245
Cohen · Evolutionary distributions
5
16
Predator prey
246
Here we use the logistic model for the prey and add to it mortality due
to predation with saturation. Similar models in the context of ED were
analyzed in Cohen (2003b). The predation is translated to predator birth
adjusted by prey to predator biomass conversion efficiency parameter d:
r
az1
z2 ,
z10 = rz1 − z12 −
k
b + cz1
az1
z20 = d
z2 − µz22 .
b + cz1
(15)
With
r = 0.25, k = 100, a = 1, b = 10,
(16)
c = 1, d = 0.1, µ = 0.01
we obtain limit cycle (Figure 6). Having the tendency to produce evolulimit cycle
z2 t
z
preythin,predatorthick
70
60
50
40
30
20
10
0
0 200 400 600 8001000
t
8
7
6
5
4
3
2
0 10 20 30 40 50 60 70
z1 t
Figure 6: Predator prey limit cycle.
247
tionary cycles, models such as (15) were criticized by Abrams (2000) as not
particularly general. Interestingly, as we shall see in a moment, these cycles,
ubiquitous as they are in point process models, disappear in our case for a
large space of parameters values.
Let x := [x1 , x2 ]. Assume that the prey phenotypes, z1 , evolve on the adaptive trait x1 and the predator phenotypes, z2 , evolve on the adaptive trait
x2 . For example, x1 may represent the running speed of prey phenotypes
and x2 the running speed of predator phenotypes. We also assume that
predation is at its maximum when x1 = x2 with some phenotype plasticity
248
249
250
251
Cohen · Evolutionary distributions
17
(variance) with respect to the adaptive traits, σ. We model the interaction
between the prey adaptive trait x1 and the predator adaptive trait x2 with
"
#
1 x1 − x2 2
α (x) = exp −
.
2
σ
Let the mutation operators of the prey and predator be
1
Az1 := z1 + ∆2 η1 ∂x1 x1 z1
2
and
1
Az2 := z2 + ∆2 η2 ∂x2 x2 z2 .
2
We now have two ED (for prey, z1 and predator, z2 ) with two adaptive traits
r
az1
∂t z1 = rAz1 − z12 − α (x)
z2 ,
k
b + cz1
az1
∂t z2 = dα (x)
Az2 − µz22 .
b + cz1
We choose initial conditions
z1 (x, 0) = 10,
z2 (x, 0) = 1
and boundary conditions
∂x1 z1 (π/2, x2 , t) = ∂x1 z1 (9π/2, x2 , t) = 0,
∂x2 z2 (x1 , π/2, t) = ∂x2 z2 (x1 , 9π/2, t) = 0.
In addition to the parameters in (16) we use
∆ = 0.01, η1 = η2 = 0.01, σ = π/3.
The stable ED are shown in Figure 7. The densities of prey phenotypes along
the positive diagonal (with slight shifts in opposite directions for prey and
predator) of the evolutionary space, x1 = x2 , are depressed. The densities
of predator phenotypes are nearly a mirror image. The shapes of the stable
ED depend on the interplay between the size of the adaptive space and
phenotypic plasticity (Figure 8). For fixed plasticity, increase in the size of
the adaptive space means a decrease in the effective phenotypic plasticity.
Both Figures 7 and 8 indicate that in the evolutionary space, low fitness
252
253
254
255
256
257
258
259
Cohen · Evolutionary distributions
18
EDFramework-2d.nb
1
Prey
Predator
10
100
7.5
99.5
z1
5 z2
2.5
99
5
x2
5
5
10
10
x2
0
5
10
10
x1
x1
Figure 7: Stable ED with phenotype plasticity with respect to the adaptive traits
σ = π/3.
EDFramework-2d.nb
1
Predator
Prey
99.8
8
99.7
z1
99.6
6
z2
99.5
5
x2
99.4
5
10
10
x1
5
x2
5
10
10
x1
Figure 8: Stable ED with phenotype plasticity with respect to the adaptive traits
σ = π.
Cohen · Evolutionary distributions
19
of prey phenotypes is associated with high fitness of predator phenotypes.
The density of low fitness prey phenotypes is maintained (fed) by prey with
high fitness phenotypes. To end with stable ED, it is necessary to have low
fitness phenotypes to “feed” the prey and this is done by the high fitness
prey phenotypes. This is a good example where a spectrum of fitnesses
coexist and in fact is necessary to maintain evolutionary stability.
If x1 and x2 represent the prey and predator running speed, for example,
then we interpret Figure 7 as follows. Fast prey (large x1 ) which coevolve
with fast predators (large x2 ) show low fitness at evolutionary stability. In
these circumstances, predator phenotypes show high fitness. At low values
of x1 and x2 we have the same coevolutionary outcome. This pattern is
maintained by the continuous evolution of prey phenotypes of high fitness
(off the diagonal) into the diagonal region of the adaptive space. Off diagonal prey phenotypes coevolve to be either fast runners associated with slow
predator phenotypes or slow runners associated with fast predator phenotypes. So if you are a slow prey (e.g., hiding), you are fit against fast
(chasing) predators. If you are fast (e.g., running), you are fit against slow
(ambushing) predators. These outcomes are conditioned on the existence of
low fitness prey phenotypes (along the diagonal). The picture for predators
is a mirror image of the picture for the prey. Overall, we find high diversity of prey phenotypes occupying the adaptive space and low diversity of
predator phenotypes—A common feature in Nature.
The predator prey model we use did not result in evolutionary cycles. This
is not to say that they are not possible. The fact that point process evolutionary models are cyclic does not mean that the cycles will remain when
the models are translated to ED.
6
Host pathogen
The evolution of host-pathogen and infectious diseases is of much interest
(Dieckmann et al., 2002); e.g., witness recent news about resistant pathogens
and pathogen mutations that may cause pandemics (such as the bird flu).
Following Anderson and May (1980, 1981), we define the densities of host
population z1 ; infected hosts z2 ; and pathogens z3 . The parameters are the
coefficients of individual host birth rate, a; natural death rate, b; additional
death rate due to infection, α
e; transmission efficiency, ν; recovery rate, γ;
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
Cohen · Evolutionary distributions
20
death rate of infective stages, µ
e; and the rate at which an infected host
produces infective stages of the parasite λ. The model is from Anderson
and May (1980), equations (3), (5) and (6):
z10 = (a − b) z1 − αz2 ,
z20 = νz3 (z1 − z2 ) − (α + b + γ) z2 ,
z30 = λz2 − (µ + νz1 ) z3 .
The parameter values
a = 5.3, b = 5.29, γ
e = 10−20 , α = 5.5,
e = 108 , µ = 0.02, ν = 10−10
λ
EDFramework-v01.nb
1
60
50
40
30
20
10
z2
z1
produce a limit cycle (Figure 9).
0.1
0.05
0
1000 2000 3000 4000 5000
t
0
8 μ 108
8 μ 108
6 μ 108
6 μ 108
z3 HtL
z3
0.25
0.2
0.15
4 μ 108
2 μ 108
1000 2000 3000 4000 5000
t
4 μ 108
2 μ 108
0
0
0
1000 2000 3000 4000 5000
t
10
20
30 40
z1 HtL
50
60
Figure 9: Host-pathogen limit cycle (after Anderson and May, 1980). z1 , z2 and
z3 denote the host, infected and pathogen populations, respectively.
287
Let x1 be the adaptive trait that affects the host death rate coefficient due
to infection. The adaptive trait of the pathogen, x2 , affects the pathogen
death rate of infective stages. Assume that at some value of x1 , say 5π/2,
Cohen · Evolutionary distributions
21
the death rate of hosts due to infection is at its minimum. Thus we write
"
#!
1 x1 − 5π/2 2
.
(17)
α (x1 ) = α
e 1 − 0.1 exp −
2
σα
We also assume that at some value of x2 (= 5π/2) the death rate of pathogen
infective stages is at its maximum
"
#!
1 x2 − 5π/2 2
µ (x2 ) = µ
e 1 + 0.1 exp −
.
(18)
2
σµ
EDFramework-2d.nb
The anticipated effects of α and µ on the host and pathogen ED are shown
in Figure 10.
Host x
10 2
5
288
Pahogen x
10 2
5
x1 10
1
x1 10
5
-5
-5.2
-a
-5.4
5
-0.02
-0.0205
-0.021 -m
-0.0215
-0.022
Figure 10: Anticipated effect of α and µ on the host and pathogen ED in the
adaptive space (x1 , x2 ).
289
Now the ED become
∂t z1 = aAz1 − bz1 − α (x1 ) z2 ,
∂t z2 = νAz3 (z1 − z2 ) − (α (x1 ) + b + γ) z2 ,
∂t z3 = λz2 − (µ (x2 ) + νz1 ) z3 ,
Cohen · Evolutionary distributions
22
with data
z1 (x, 0) = 20,
z2 (x, 0) = 0,
z3 (x, 0) = 2 × 109 ,
zi (π/2, x2 , t) = zi (9π/2, x2 , t) ,
zi (x1 , π/2, t) = zi (x2 , 9π/2, t) ,
i = 1, 2, 3.
The parameters are set to
a = 5.3, b = 5.29, γ = 10−20 ,
kα = 5.5, λ = 108 , kµ = 0.02,
ν = 10−10 , ∆ = 0.01, η1 = η2 = η3 = 0.01,
σa = σµ = π/3
(note the periodic boundary conditions; they were set to produce a desired effect). Figure 11 (and the animation of the dynamics at http:
//yosefcohen.org) show the limit-cycle ED at a particular t. Because
of coevolution, the anticipated ED of the host and pathogen ED (Figure 10)
are different from the evolutionary ED.
Several interesting feature emerge from the coevolution of the host-pathogen
system. Foremost is the fact that we obtain a limit cycle. This cycle can be
observed only via animations (at http://yosefcohen.org). High densities
of phenotypes are pushed to the corners of the adaptive space. These are
separated by zero densities. In the model, reproduction is localized. So
if one chooses to identify species with phenotypes that coproduce, then
the process results in the emergence of four common host “species” and
rare species in between. Phenotypes at the inner boundaries of the high
densities fare well with the pathogen (their densities are the highest). This
is the case because corresponding to these boundaries there are low densities
of pathogen phenotypes (z3 in Figure 11). The infected phenotypes (z2 in
Figure 11) follow the ED of the host (z1 ) with more pronounced densities
of phenotypes along the inner edges—being at the extreme inner edges of
the adaptive space impart extra protection to the infected phenotypes. At
other times of the cycle (not shown), the corners infected phenotypes and
pathogen disappear and a single peak of both appears in the middle.
These results indicate that we may observe cycles in the coevolution of hostpathogen. In these cycles, resistant hosts and virulent pathogens coevolve
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
Cohen · Evolutionary distributions
23
EDFramework-v01.nb
1
10
x1
x2
10
10 x2
5
x1 10
5
5
100
50 z1
0
10
x1
5
10
5
1.5 μ 109
1 μ 109
5 μ 108
0
z3
x2
5
1
0.75
0.5z
2
0.25
0
Figure 11: Limit-cycle ED for host and pathogen in the evolutionary space (compare to Figure 10), at t = 2 × 105 .
Cohen · Evolutionary distributions
24
from single peaks in the middle of the adaptive space to peaks in the corner
of the adaptive space. The approach we take to the coevolution of host
pathogen is different from traditional views (e.g., Levin, 1996). First, the
adaptive traits of the host and the pathogen are independent and certain
parameter values are not fixed; they are functions (given in (17) and (18))
whose values are determined by coevolution. The dependency between the
values of these functions surfaces through mutation and selection. Second,
ours is an infinite dimensional system. Therefore, we do not need to introduce a fixed mutant with a certain parameter value and examine the
outcome of coevolution (as in e.g., Anderson and May, 1982). Finally, and
perhaps most importantly, we do subscribe to the view that natural selection
is a local phenomena (e.g., Levin and Bull, 1994; Levin, 1996). However,
fitness is a distributional phenomena. Thus, we may observe a spectrum of
fitnesses at evolutionary stability (of ED).
313
314
315
316
317
318
319
320
321
322
323
324
325
326
The results rely heavily on the initial conditions. When we use
z1 (x, 0) = 20 + 2 cos (x1 ) ,
z2 (x, 0) = 0,
z3 (x, 0) = 2 × 109 + 2 × 108 cos (x2 )
we obtain a pathogen and infected ED very different from those obtained
from homogeneous initial surfaces (compare Figure 11 to Figure 12). This
is a particularly convincing example of the problems that arise if one follows
the dynamics of the mean of the strategies, as opposed to following the
dynamics of the ED. Following the mean of the strategy dynamics leads one
to believe that one is following dynamics of typical phenotypes when in fact
no phenotypes with adaptive trait values that represent the mean density
of phenotype exist. This application of the theory also illustrates the much
richer behavior of solutions of PDE compared to ODE and thus brings us
closer to the richness of phenomena we see in Nature.
7
ED in ecosystems
So far, we discussed ED from the perspective of coevolutionary interactions
within and between ED. Here, we discuss ED in the context of ecosystems.
Our fundamental tenet is that ecosystems are open. Thus, in addition to
abiotic considerations, we must include input and output to the system.
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
Cohen · Evolutionary distributions
25
EDFramework-v01.nb
1
10
x1
x2
10
10
5
x1
150
5
x2
10
5
1.5 μ 109
5
100
50
1 μ 109
z1
0
10
x1
10
5 μ 108 z3
0
x2
5
1
5
0.75
0.5
z2
0.25
0
Figure 12: Stable infected and pathogen ED.
Cohen · Evolutionary distributions
26
Extending ED to incorporate the abiotic environment opens the door to
applications that include nutrient cycling and global change in the context
of coevolution.
7.1
A simple ecosystem model
loss from the
system
1
herbivory ( )
343
344
345
We shall use a simple system (Figure 13) with nitrogen cycling through three
primary
producers (z1)
342
primary
consumers (z2)
3
mortality ( ) uptake ( )
mortality ( )
soil (z3)
input rate ( )
Figure 13: A simple ecosystem model.
compartments. All units in the model are expressed in N-equivalent units
(e.g., density of N). We model the system with saturation and from Figure
13 derive the model as:
z3
z1
z1 − χ1
z2 − (λ1 + µ1 ) z12 ,
z10 = υ3 υ1
1 + υ 2 z3
1 + χ2 z1
z1
z20 = χ3 χ1
z2 − (λ2 + µ2 ) z22 ,
(19)
1 + χ2 z1
z3
z30 = ι (t) + µ1 z12 + µ2 z22 − υ1
z1 − λ 3 z3
1 + υ 2 z3
where the scaling coefficients υ3 and χ3 are between zero and one. Assume
that the nitrogen input rate, ι (t), oscillates with one cycle per year. So we
write
2πt
ι (t) = A + B sin
.
(20)
12
Cohen · Evolutionary distributions
27
With the parameter values
A = 10,
B = 4,
µ1 = 0.01,
χ1 = 1,
λ1 = 0.01,
µ2 = 0.01,
χ2 = 1,
λ2 = 0.1,
υ1 = 100,
λ3 = 0.01,
υ2 = 1,
υ3 = 1,
(21)
χ3 = 0.4
and initial conditions
z1 (0) = 20,
EDFramework-2c.nb
z2 (0) = 1,
z3 (0) = 0.006
(22)
1
30
25
20
15
10
5
0
z1
z3
z1 , z2
we obtain Figure 14. The time trajectories indicate that the system may be
z2
0
100
200
300
400
0.0075
0.007
0.0065
0.006
0.0055
0.005
0
500
100
200
300
400
500
t
t
Figure 14: Time trajectories of system (19) with parameters values in (21) and
initial conditions (22). z1 , z2 and z3 are the density of nitrogen in
primary producers, primary consumers and soil, respectively.
EDFramework-2c.nb
346
3.51
3.505
3.5
3.495
3.49
3.485
3.48
20
25.8
z1 Ht-t L
z2 HtL
either chaotic or resonant (Figure 15). We shall not pursue this issue any
25.75
25.71
22
24
26
z1 HtL
28
30
25.71
25.75
z1 HtL
25.8
Figure 15: Phase portraits of system (19) with parameters values in (21) and
initial conditions (22). The delay τ = 6. z1 and z2 denote nitrogen
density in primary producers and primary consumers, respectively.
347
further and move on directly to ED for this simplified ecosystem.
348
1
Cohen · Evolutionary distributions
7.2
28
Coevolution in ecosystems
349
From (19) and Figure 13 we conclude that reproduction occurs on the uptake (υ) and consumption (χ) terms. To simplify the system, we shall ignore
competition within ED and assume that x1 is the primary producers adaptive trait that endows them with the ability to resist consumption (e.g.,
tannin and lignin concentrations in plant tissue). Similarly, x2 is the adaptive trait that allows consumers to handle these compounds. Assume that
increase in x1 causes decrease in consumption—the χ related term in Figure
13. The price to pay for increased resistance to consumption is increase
in death rate—the µ1 related term in Figure 13. Increase in resistance to
the primary producers defenses results in decrease in the primary producers
mortality rate—the µ2 related term in Figure 13. The price is decrease in
the consumption efficiency—the χ3 related term in (19); e.g., decrease in
the conversion efficiency from plant to herbivore biomass.
350
351
352
353
354
355
356
357
358
359
360
361
362
These considerations lead us to modify (19) and write
x1
x1
χ1 (x1 ) = χ4 exp −
, µ1 (x1 ) = µ3 1 − exp −
,
σ1
σ2
x2
x2
, χ3 (x2 ) = χ5 exp −
µ2 (x2 ) = µ4 exp −
σ3
σ4
and
z3
z1
Az1 − χ1 (x1 )
z2 − (λ1 + µ1 (x1 )) z1 ,
1 + υ 2 z3
1 + χ2 z1
z1
Az2 − (λ2 + µ2 (x2 )) z2 ,
(23)
∂t z2 = χ3 (x2 ) χ1 (x1 )
1 + χ2 z1
z3
∂t z3 = ι (t) + µ1 (x1 ) z1 + µ2 (x2 ) z2 − υ1
z1 − λ 3 z3
1 + υ 2 z3
∂t z1 = υ3 υ1
where
1
Azi = zi + ∆2 ηi ∂xi xi zi i = 1, 2.
2
We let x := [x1 , x2 ] and zi ≡ zi (x, t), i = 1, 2, 3.
363
We use the following parameter values
A = 10,
B = 4,
µ3 = 0.01,
χ2 = 1,
µ4 = 0.01,
χ4 = 1,
∆ = 0.01,
λ1 = 0.01,
λ2 = 0.1,
υ1 = 100,
χ5 = 0.4,
η1 = η2 = 0.01,
λ3 = 0.01,
υ2 = 1,
υ3 = 1,
σ1 = σ2 = σ3 = σ4 = 106 ,
(24)
Cohen · Evolutionary distributions
29
with the Neumann boundary conditions and initial data
z1 (x, 0) = 20 + 2 cos (x1 + x2 ) ,
z2 (x, 0) = 1 + 0.1 cos (x1 + x2 ) ,
EDFramework-2d.nb
(25)
1
z3 (x, 0) = 0.006
to obtain a stable cyclic solution (Figure 16). For the most part, the stable
10
x1
10
x2
10
5
x1
35
5
10
x2
5
5
3.5
30
3
25 z
1
20
15
2.5 z2
2
Figure 16: Stable cyclic solution of (23) with parameter values (24), Neumann
boundary conditions and initial conditions (25).
364
surface preserves the initial conditions except in the extremes of the adaptive
traits.
The theory provides interesting predictions. The ED of the primary producers illustrates a marked decrease in fitness (density at stability) of those
producer phenotypes that are at the extremes (high or low) of their adaptive traits. These low densities correspond to consumer phenotypes at the
extreme high values of tolerance to the producers resistance to consumption
(high values of x2 ). In other words, if you are a producer coevolving with
a consumer in a resistance-tolerance (to consumption) adaptive space, you
want to be neither devoid of resistance nor too resistant because in either
case your death rate is too high. However, the disadvantage of being in the
extreme of your adaptive trait disappears when you coevolve with consumers
with low tolerance to your defenses—there are no dips in abundance of x1
phenotypes at the extremes of resistance to consumption by consumers with
low tolerance to your defenses (small values of x2 ).
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
Cohen · Evolutionary distributions
7.3
30
Application to global climate change
Here we briefly outline the approach to integrating global change with coevolutionary processes. This is a little investigated area. It is generally
believed that the rate of global warming, for example, is too fast for coevolution to catch up. This, however, is a belief that needs to be investigated
at least theoretically. Furthermore, one can hardly expect global change to
be a short lived (say a few hundred years) phenomenon. In the long run, we
may expect evolutionary changes in response to global change.
380
381
382
383
384
385
386
387
We shall not pursue this topic in detail except to outline, by example, how
the current framework can be applied. Take for example the nutrient input
function (20). Suppose that it represents temperature. Then we may modify
it thus:
2πt
ι (t) = A (t) + B (t) sin
12
where now the average temperature A becomes time dependent; e.g., A (t)
= a1 + b1 t and the variance in temperature changes with time; e.g., B (t)
= a2 + b2 t (this scenario was investigated without reference to evolution in
Cohen and Pastor, 1991).
We can then link ι (t) to a geographical location and then examine the
potential outcome of coevolution due to global change.
8
Discussion
The theory developed here is not a panacea. Albeit applicable to a large
class of population models, it is not applicable to discrete models. Deriving
analytical criteria for stability for a system of nonlinear PDE is difficult (currently, this is an active research area in Mathematics). However, stability
should not be viewed as the holy grail of mathematical biology. The often
made claim that unstable systems are not likely to be observed in Nature
may not be true; particularly if transients are slow. After all, all species are
doomed to extinction sooner or later. Transient phenomena in mathematical models can produce rich results that mimic Nature. Finally, difficult as
it may, stability analysis should not be a reason to abandon any theory that
is based on first principles.
The significant contributions of the theory of ED to evolutionary ecology
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
Cohen · Evolutionary distributions
31
boil down to: (i) ED do not require game theoretic approach to the study
of coevolution in adaptive space; (ii) because ED covers all of the adaptive
space, invasion of mutants to a stable ED is possible for a short time only—
we thus do not need to invoke the ESS concept; (iii) the theory admits that
fitness of phenotypes at stable ED can be positive, negative or zero—the
fitness of phenotypes must be viewed in the context of fitness of the whole
distribution. Thus, as in Nature, ED allow phenotypes of various fitness
values to coexist and resist invasion. In other words, in coevolutionary
dynamic equilibrium, phenotypes are able to maintain high fitness by a
fraction of them evolving to low fitness phenotypes.
All of the stability results here were inferred from numerical solutions of the
ED models. This may raise objections, for inferring stability from numerical
solutions is not a proof of stability. I therefore offer all of the stability results
as conjectures of stability. Much experience with numerical solutions (in
particular of PDE) indicates that unstable solutions surface after a short
time
interval.
In all the results, no numerical instability surfaced for t ∈
0, 106 . For unstable models, numerical instabilities appeared well before t
= 100. Again, these facts are not offered as proofs of stability, but as strong
indications of stability.
As with all differential equation models, ED are limited to large, but finite
populations. There is a common misconception that continuous models in
bounded domains imply infinite population sizes. This is not the case simply because the integrals of phenotype densities over bounded domains are
finite. Furthermore, stochastic effects are ignored. For small populations,
evolutionary ecology systems should be modeled with probabilities. Ignoring
stochastic effects is more difficult to justify. However, we are often interested
in the range of phenomena that deterministic models exhibit. For example,
deterministic models that produce chaotic behavior are interesting precisely
because they produce stochastic-like behavior. In the case of ED, the fact
that entirely different solutions emerge from different initial conditions (Figures 11 and 12) indicates that perturbations (stochastic or otherwise) can
move a system of ED to different regions in the solution space.
One difficulty with ODE models is that they assume smooth functions and
therefore the transition from an individual based approach to the continuous
approach needs to be justified (as in Dieckmann and Law, 1996, for example). In the case of PDE, this problem vanishes because the theory of PDE,
from its foundations, relies on measure theory and on generalized functions
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
Cohen · Evolutionary distributions
32
(Renardy and Rogers, 1993; Evans, 1998). Thus, the extension from points
(individuals) to intervals is natural. Because of the functional space within
which PDE operate, solutions that are not smooth and even not continuous
are acceptable; see for examples Smoller (1982) and Cohen (2005).
The richness of solutions that ED can produce, and the fact that they need
not be unique raise tantalizing possibilities in modeling evolution via ED.
For example, Abrams et al. (1993) raised the possibility of stable ESS where
fitness is at its minimum (see also Cohen et al., 2000, 2001). The results
illustrated in Figure 7 support Abrams et al. (1993) finding. These results
also indicate that phenotypes with a spectrum of fitness values can coexist
in stable ED. The whole issue of maximizing fitness becomes irrelevant when
we deal with stable ED. For example, in the predator prey system (Figure 7),
there are many regions where the slope of the surface (instantaneous fitness)
is zero and others where it is positive or negative. Yet, the surface is stable.
We must therefore revise the notion that stable evolutionary systems should
reflect special points in the fitness space. At this time, I have no alternative
to offer.
We invoke orthogonality of traits in the mutation operator (10). This might
appear as a restrictive assumption. However, many adaptive traits are (or
seem to be) unrelated: it is difficult to see the connection between the
length of an elephants trunk and the color of his skin. Furthermore, if
adaptive traits are not orthogonal, then they can be made so (with rotation
methods akin to Principal Components Analysis). Finally, there is nothing
in the theory of ED that restricts one from using nonorthogonal traits. The
mutation operator will simply need to include mixed partial derivatives.
In developing the framework for ED, we claimed that birth is a linear function (see equation 8). Incorporating nonlinear birth process introduces some
notational difficulties, but no conceptual difficulties. In using the Taylor series expansion, one will need to invoke the chain rule for the Taylor series
approximation to hold.
The fact that solutions on the boundaries of the adaptive space differ from
the interior (where we may find zero densities in some regions of the adaptive space) should come as no surprise. For certain classes of PDE, there
are maximum principles that establish that the maxima of solutions should
always be on the boundaries (see for example Evans, 1998). From an evolutionary perspective, this makes sense—natural selection in a bounded domain is expected to push the most fit organisms to the boundaries of the
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
Cohen · Evolutionary distributions
33
adaptive space. A good example is the body temperature of birds, which
is close to the temperature of protein denaturation—possibly because based
on thermodynamic principles, it is easier to warm up than to cool down.
Another possible example is the tremendous size of the Irish elk antlers that
eventually led to its extinction (Moen et al., 1999).
The application of ED to ecosystem models opens the door to (at least theoretical) investigations of how coevolution responds to inanimate components
of systems in Nature. Such applications offer one way to address the issues
of global change in the context of coevolution.
Finally, we propose the following connection between ED and population
genetics: For a population of organisms with a small number of genes, we
can construct the power set (the set of all possible subsets) of the genes.
Identifying environmental influences on the development of phenotypes, this
set should give us a one-to-one map (a function) between genotypes and
phenotypes. By following the dynamics of ED we can use the inverse of
this map to infer the densities of genotypes. Thus, at least conceptually,
ED may reduce the problem of the dynamics of population genetics to an
algebraic problem. The same approach can be taken with organisms with a
large number genes. Here, however, one has to isolate a subset of genes that
produce adaptive trait values almost independently of other genes. To thus
characterize all of the traits of phenotypes, we need to isolate the influence
of environmental factors on the emergence of trait values—an impossible
task. Thus, the approach we evangelize is limited to those few traits where
it is known that the environment plays a negligible role (e.g., color of skin
and eyes).
References
Abrams, P. A. 1992. Adaptive foraging by predators as a cause of predatorprey cycles. Evolutionary Ecology 6:56–72. 4
Abrams, P. A. 2000. The evolution of predator-prey interactions: Theory
and evidence. Annual Review of Ecology and Systematics 31:79–105. 16
Abrams, P. A., H. Matsuda, and Y. Harada. 1993. Evolutionary unstable fitness maxima and stable fitness minima of continuous traits. Evolutionary
Ecology 7:465–487. 4, 32
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
Cohen · Evolutionary distributions
34
Anderson, R. M., and R. M. May. 1980. Infectious diseases and population
cycles of forest insects. Science 210:658–661. 19, 20
Anderson, R. M., and R. M. May. 1981. The population dynamics of
microparasites and their invertebrate hosts. Philosophical Transactions
of the Royal Society of London. Series B 291:451–524. 19
Anderson, R. M., and R. M. May. 1982. Coevolution of hosts and parasites.
Parasitology 85:411–426. 24
Apaloo, J. 1997. Revisiting strategic models of evolution: the concept of
neighborhoodinvader strategies. Theoretical Population Biology 52:71–
77. 5
Barton, N. H., and P. D. Keightley. 2002. Understanding quantitative
genetic variation. Nature Reviews Genetics 3:11–21. 6
Champagnat, N., R. Ferrière, and G. Ban Arous. 2001. The canonical
equation of adaptive dynamics: A mathematical view. Selection 2:73–83.
3
Cohen, Y. 2003a. Distributed evolutionary games. Evolutionary Ecology
Research 5:1–14. 9
Cohen, Y. 2003b. Distributed predator prey coevolution. Evolutionary
Ecology Research 5:819–834. 16
Cohen, Y. 2005. Evolutionary distributions in adaptive space. Journal of
Applied Mathematics 2005:403–424. 9, 32
Cohen, Y., and J. Pastor. 1991. The responses of a forest model to serial
correlations of global warming. Ecology 72:1161–1165. 30
Cohen, Y., T. L. Vincent, and J. S. Brown. 2000. A G-function approach
to fitness minima, fitness maxima, ESS and adaptive landscapes. Evolutionary Ecology Research 1:923–942. 4, 7, 32
Cohen, Y., T. L. Vincent, and J. S. Brown. 2001. Does the G-function
deserve an F? Evolutionary Ecology Research 3:375–377. 7, 32
Cressman, R. 2003. Evolutionary Dynamics and Extensive Form Games.
MIT Press, Boston. 3
Darwin, C. 1859. The Origins of Species. Avenel Books. 9
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
Cohen · Evolutionary distributions
35
Dieckmann, U., and R. Law. 1996. The dynamical theory of coevolution: A
derivation from stochastic ecological processes. Journal of Mathematical
Biology 34:579–612. 3, 31
Dieckmann, U., P. Marrow, and R. Law. 1995. Evolutionary cycling in
predator-prey interactions: population dynamics and the red queen. Journal of Theoretical Biology 176:91–102. 2, 3
Dieckmann, U., J. A. J. Metz, M. W. Sabelis, and K. Sigmund, editors.
2002. Adaptive Dynamics of Infectious Diseases. Cambridge University
Press. 19
545
546
547
548
549
550
551
552
553
Doebeli, M., and G. D. Ruxton. 1997. Evolution of dispersal rates in
metapopulation models: branching and cyclic dynamics in phenotype
space. Evolution 51:1730–1741. 2
556
Doob, J. L. 1994. Measure Theory. Springer-Verlag, Berlin. 3
557
Eshel, I. 1996. On the changing concept of evolutionary population stability
as a reflection of a changing point of view in the quantitative theory of
evolution. Journal of Mathematical Biology 34.:485–510. 5
Evans, L. C. 1998. Partial Differential Equations. American Mathematical
Society. 15, 32
Fisher, R. A. 1930. The Genetical Theory of Natural Selection. Calderon
Press, Oxford, UK. 11
Fudenburg, D., and D. Levine. 1998. The Theory of Learning in Games.
MIT Press, Boston. 3
Geritz, A. H., and E. Kisdi. 2000. Adaptive dynamics in diploid, sexual
populations and the evolution of reproductive isolation. Proceedings of
the Royal Society, London Series B 267:1671–1678. 2
Gintis, H. 2000. Game Theory Evolving. Princeton University Press, Princeton, NJ. 3
Gyllenberg, M., and G. Meszéna. 2005. On the impossibility of coexistence of
infinitely many strategies. Journal of Mathematical Biology 50:133–160.
15
Hofbauer, J., and K. Sigmund. 1998. Evolutionary Games and Population
Dynamics. Cambridge University Press, Cambridge, UK. 3, 7
554
555
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
Cohen · Evolutionary distributions
36
Hofbauer, J., and K. Sigmund. 2003. Evolutionary game dynamics. Bulletin
of the American Mathematical Society 40:479–519. 3, 7
Keshet, Y., and L. A. Segel, 1984. Pattern formation in aspect. Pages
188–197 in Modelling of Patterns in Space and Time. Lecture Notes in
Biomathematics, Volume 55, Springer-Verlag. 5
Kimura, M. 1965. A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proceedings of the National
Academy of Sciences of the United States of America 54:731–736. 5
Law, R., P. Marrow, and U. Dieckmann. 1997. On evolution under asymmetric competition. Evolutionary Ecology 11:485–501. 2
Levin, B. R. 1996. The evolution and maintenance of virulence in microparasites. Emerging Infectious Diseases 2:93–102. 24
Levin, B. R., and J. J. Bull. 1994. Short-sighted evolution and the virulence
of pathogenic microorganisms. Trends in Microbiology 2:76–81. 24
Levin, S. A., and L. A. Segel. 1985. Pattern generation in space and aspect.
SIAM Review 27:45–67. 5
Ludwig, D., and S. A. Levin. 1991. Evolutionary stability of plant communities and the maintenance of multiple dispersal types. Theoretical
Population Biology 40:285–307. 5
Moen, R., J. Pastor, and Y. Cohen. 1999. Antler growth and extinction of
Irish elk. Evolutionary Ecology Research 1:235–249. 33
Murray, J. D. 1989. Mathematical Biology. Springer-Verlag. 3
Renardy, M., and R. C. Rogers. 1993. An Introduction to Partial Differential
Equations. Springer. 32
Samuelson, L. 1998. Evolutionary Games and Equilibrium Selection. MIT
Press, Boston. 3
Slatkin, M. 1981. A diffusion model of species selection. Paleobioloby
7:421–425. 5
Smoller, J. 1982. Shock Waves and Reaction-Diffusion Equations. SpringerVerlag. 11, 32
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
Cohen · Evolutionary distributions
37
Taylor, P., and T. Day. 1997. Evolutionary stability under the replicator
and the gradient dynamics. Theoretical Population Biology 11:579–590.
5
Vincent, T. L., and J. S. Brown. 2005. Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press,
Cambridge, UK. 3, 4, 7
607
608
609
610
611
612
Vincent, T. L., Y. Cohen, and J. S. Brown. 1993. Evolution via strategy
dynamics. Theoretical Population Biology 44:149–176. 7, 8
614
Weibull, J. 1995. Evolutionary Game Theory. MIT Press, Boston. 3
615
613