Resource consumption, sustainability, and cancer
a

a,b

Irina Kareva , Benjamin Morin
a

, Carlos Castillo-Chavez

a,c

Mathematical, Computational Modeling Sciences Center,

PO Box 871904, Arizona State University, Tempe, AZ 85287

b

Global Institute of Sustainability,

Arizona State University, Tempe, AZ 85287

c

School of Human Evolution and Social Changes,
Arizona State University, Tempe, AZ 85287

Abstract
Preserving a system's viability in the presence of diversity erosion is critical if the goal is to sustainably support biodiversity. Reduction in population heterogeneity, whether inter- or intraspecies,
may increase population fragility, either decreasing its ability to adapt eectively to environmental
changes or facilitating the survival and success of ordinarily rare phenotypes. The latter may result in
over-representation of individuals who may participate in resource utilization patterns that can lead
to over-exploitation, exhaustion, and, ultimately, collapse of both the resource and the population
that depends on it. Here, we aim to identify regimes that can signal whether a consumer-resource
system is capable of supporting viable degrees of heterogeneity. The framework used here is an
expansion of a previously introduced consumer-resource type system of a population of individuals
classied by their resource consumption. Application of the Reduction Theorem to the system enables us to evaluate the health of the system through tracking both the mean value of the parameter
of resource (over)consumption, and the population variance, as both change over time. The article
concludes with a discussion that highlights applicability of the proposed system to investigation of
systems that are aected by particularly devastating overly-adapted populations, namely, cancerous
cells. Potential intervention approaches for system management are discussed in the context of cancer
therapies.

1

Introduction
The role of genetic and environmental variation has been well documented within the context of human
genetics and animal/plant breeding [8]. Heritable variation that aects tness makes evolution possible
[9, 16].

The environmental landscape occupied by individuals within a population includes resources

shared by members of cohabiting populations [39, 46, 47, 49] engaged in interference competition, i.e., the
competition that emerges from variation within a population, guild (group of species that exploit the
same resource), or some larger ecological unit. Selective pressures experienced by individuals within a
population are thus directly a function of the group's heterogeneity. These selective forces generate and
sustain feedback loops capable of altering, over dierent temporal, spatial, and multiple organizational
scales, a population's interactions with the landscape in which such dynamics operate. Here, we explore
the consequences of changes in population heterogeneity on overall system dynamics within a

conceptual

consumer-resource model. The heterogeneous population of consumers is assumed to be operating only
under the inuence of interference competition, i.e., the focus is on the dynamics of a

single population.

We study the impact of changes in population heterogeneity on resource viability and, consequently, on
survival of a population that depends on this resource.
The perspective that natural and articially induced reductions in heterogeneity may result in a
population's collapse due either to loss in adaptive capacity or emergence of a critical number of overconsumers, is not new.

It is known that both can lead to species extinction under conditions that

correspond to excessive change, including those that result in the destruction of the species' habitat [51].
The case of an overly-adapted species potentially destroying their own habitat has become known as
evolutionary suicide [24]. In host-parasite interactions, which are an extreme example of a resourceconsumer system, evolutionary suicide is often seen as a natural consequence of tight co-evolutionary
interactions, that is, the fate of the parasite is intimately connected to that of the host [35].

These

systems are not unique to biology and have been documented in a variety of contexts, including socialeconomic systems [42] or in biological systems, where the resource may be dened as an ecological
habitat [23].

Cases of biological control via introduction of invasive species t the class of tight co-

evolutionary interactions that may lead, often by design, to catastrophic events for the host; the case of

Myxomatosis in Australia provides an example of such a system [13].
The specic nature of the immediate habitat is highly relevant.

2

For example, in the case of host-

parasite interactions in diseases such as cancer, the habitat is conned to the body of the host [27].
In this paper, we restrict our attention to a subset of consumer-resource systems, where short-term
adaptive success can lead to long-term system collapse (evolutionary suicide) under the assumption that
the consumer species are unable to utilize alternative resources or nd alternative homes within a viable
time scale; examples include pika range land loss or isolated habitable lands [22, 50].
Managing consumer-resource systems that might be at risk for long-term collapse may demand an
introduction of selective pressures that restrict or possibly eliminate the advantages of over-consumption
(i.e., practices that reduce the advantages of over-consumers), but there are caveats. Management strategies aimed at rapidly increasing mortality of undesirable individuals within a population often have
unintended consequences. One such consequence is the inadvertent selection of individuals highly resistant to control, which may or may not be directly connected to their ability to exploit the resource [13].
In agronomy, it is well documented that administration of toxic substances may result in selection of
resistant individuals [6], while adding high toxicity costs to crops and consumers [12].
The resistance problem posed by pesticide spray regimes in agricultural systems is similar to issues
encountered with treatment of bacterial infections, as administration of antibiotics may result in the
selection for antibiotic-resistant microbes [7, 37].

Consequently, naturally rare superbugs, which are

otherwise suppressed or out-competed by non-resistant microbes, may suddenly thrive following the
elimination of antibiotic-sensitive members of the bacterial population [6, 10]. This may occur via the use
of alternative, often dramatically distinct drug-use protocols that include random distribution or cycling
of antimicrobials at the population level [7,37]. The fact that similar issues are critical to the development
of eective cancer treatments is not surprising, particularly when aggressive treatment protocols are in
place. Standard protocol for administration of chemotherapy, maximal tolerated dose or MTD, involves
the administration of large doses of chemotherapeutic drugs for extended intervals of time. This practice
allows only for the partial recovery of normal tissues (the `under-consumers'), which are also severely
aected by the therapy directed against cancer cells (the `over-consumers').

The goal of MTD is to

generate the highest possible level of cancer cell mortality, but by applying severe selective pressures on
the population fo cancer cells, one predictably selects specically for therapeutically resistant cell clones,
which in turn can lead to metastatic disease and patient death [43].
While interventions are often a necessary evil, the aforementioned observations suggest that the focus
of intervention or management strategies perhaps should not be driven by the goal of completely eradi-

3

cating undesirable members of a population. Instead, the focus should be on nding ways for controlled
sustainable coexistence of the consumer and the resource over time. Intervention strategies that aim
at reducing the likelihood, or altogether preventing, evolutionary suicide, must do so by maintaining some
level of heterogeneity within a population. We highlight cancer because complete elimination of a subpopulation may not be often the best policy. A similar counterintuitive result emerges when evaluating
a population-level policy designed to maintain the viability of a limited supply of antimicrobials. This
class of drugs has not undergone signicant changes or increases over the past three decades, while the
rate of population-level resistance has [7, 37].
Here, we rst provide an overview of the work carried out in the study of a particular consumerresource type system that incorporates the way that consumers interact with the shared resource (specic
consumer level of heterogeneity).

In this context, approaches for evaluation of the success of various

intervention strategies are explored.

We then introduce another metric for mathematically evaluating

predicted health of the system, namely, changes over time in variance, in addition to changes in the mean
values of parameter of (over)consumption, in order to evaluate whether this may help achieve successful
population management while preserving sucient levels of heterogeneity. We conclude with a discussion
of the applicability of the results in the context of existing cancer treatments.

Mathematical Model

Model formulation

First, we evaluate the ecacy of intervention strategies while preserving a population's heterogeneity
with respect to a consumer-resource system. In the proposed system, each individual is characterized by
his or her own value of the intrinsic resource-consumption parameter,

c > 0.

A consumer characterized

by a particular resource-consumption parameter is referred to henceforth as a
Dene

xc (t)

to be the number of

of consumers is given by
whenever

N (t) =

P
A

c-clones
xc (t)

within the population at time

whenever

c

c-clone.
t.

The total population

takes on discrete values, and

N (t) =

´
A

xc (t)dc

c takes on a continuous set of values (A denotes the range of possible values of the parameter c).

In general, application of the theory we apply here is insensitive to whether
We assume that the number of individuals

A

is discrete or continuous.

xc (t) increases according to a logistic growth function, with

the caveat that the carrying capacity is not constant but is a dynamic variable that depends on amount

4

of available resource. Each clone invests in resource restoration at a rate proportional to (1
is considered to be an over-consumer if

− c);

a clone

c > 1.

We start with a consumer-resource type model (see [33]) given by the following system of nonlinear
equations:

P



xc (t)
dxc
A


= rxc (t) c −

 dt
kz(t)
!
X
e
dz



xc (t)(1 − c) − δz(t).
 dt = γ + z(t) + P x (t)
A

c

(1)

A

The dynamics of the resource is determined by the constant natural restoration rate
tionate decay/loss rate

δz(t).

The resource can also be depleted or restored by the

consumer-induced resource depletion or restoration by a specic group of

xc /z
xc
= e(1 − c) z+N
.
e(1 − c) 1+N/z

γ

and the propor-

c−clones.

Specically,

c-clones is captured by the term

In other words, severe interference competition (over-consumption) and

restoration is modeled as an additive, aggregate eect.

Overview of previous work
Full analysis of the parametrically homogeneous model, i.e., the case where the population is composed
of a set of consumers characterized by a single average value of parameter

c,

was carried out in [28]. It

was observed that as the parameter of over-consumption increased, the system went through a series of
transitional regimes prior to collapse. Specically, it was observed that the domain of attraction of the
non-trivial equilibrium, which represents a state of sustainable coexistence of the population of consumers
and their shared resource, shrank from global stability to being encircled by an unstable limit cycle. As
the value of

c

further increased, population size started oscillating, until further increases in

c

eventually

led to a complete collapse of both the population and the resource. From these results we hypothesized
that transition into an oscillatory regime (emergence of oscillations as

c

increases in this example) serves

as a predictor of collapse.
In order to visualize the evolutionary trajectories of an evolving system, where

c

would be able to

change over time as a result of natural selection, we applied the Reduction Theorem, developed by G.
Karev [26]. This technique allows studying the dynamics of a population that is composed of multiple
clones, that is, a population that initially is heterogeneous with respect to

5

c.

The use of the Reduction

Theorem involves the use of a transformation carried out as follows:
Consider the equation for the rate of change of the population size over time:



N
x0c = xc c −
.
kz
Introduce an auxiliary, `keystone', variable

q1 (t)

(2)

dened by

q10 =

N
.
kz

(3)

Then with

x0c
= c − q10 .
xc

(4)

Integrating the above expression leads to the following equation for calculating the size of each subpopulation of clones

xc

at time

t:

xc (t) = xc (0)ect−q1 (t)

(5)

with the size of the total population calculated as:

ˆ
N (t) =

ˆ
xc (t)dc = N (0)e

−q1 (t)

ect Pc (0)dc = N (0)e−q1 (t) M [t],

c

where
and

M [t]

Pc (t)

is

(6)

c

is (by denition) the moment generating function (mgf ) of the initial distribution of clones,

Pc (t) =

xc (t)
N (t)

=

ect
M [t] , in the population. The mean value of

E t [c] = cPc (t) = c

c

ect
M 0 [t]
=
.
M [t]
M [t]

can be simply calculated as

(7)

The dynamics of the parametrically heterogeneous population can now be fully described using the
following system of equations:




 dq1

=



 dz

=γ+

dt

dt

N (t)
kz(t)

(8)

eN (t)
z(t)+N (t) (1

6

− E t [c]) − dz(t),

where

N (t)

E t [c]

is calculated using Equation (6) and

is calculated using Equation (7). Full analysis of

System (8) was carried out in [28].

Persistent selection for clones with the highest possible values of
is observed.

c

in System (8), a predictable result,

Numerical investigation of the transformed parametrically heterogeneous system allows

explicitly plotting evolutionary trajectories that the population follows over time, as the expected value
of

c

steadily increased; we see that trajectories generated as an increasing value of

c

sweeps through

the aforedescribed domains of the parametrically homogeneous system.

Remark.

It is important to note that the initial form of the parametrically homogeneous system

can signicantly inuence the nal form of the parametrically heterogeneous system.

x0c = rcxc 1 −

For instance, if



N

kz , a variation on the form of the rst equation in System (1), also used in [33], leads

after applying the reduction theorem to an auxiliary equation of the form

q20 = (1 −

N
),
kz

from where it follows that

xc (t) = xc (0)ecq2 (t) ,
ˆ
ecq2 (t) Pc (0)dc, and

N (t) = N (0)
c

E t [c] =

M 0 [q2 (t)]
.
M [q2 (t)]

The nal system thus becomes




 dq2
dt



 dz
dt

with the corresponding expressions for

=1−

N (t)
kz(t)

=γ+

eN (t)
z(t)+N (t) (1

N (t)

and

(9)

E t [c]

t

− E [c]) − dz(t),

dened above.

Bifurcation diagrams of the parametrically homogeneous systems for

xc (c −

x0c = rcxc 1 −

N
kz




and

x0c =

N
kz ) are qualitatively the same, up to a constant, in this particular case. However, parametrically

7

heterogeneous systems are dierent, a fact that can be immediately grasped from the nal forms of the
transformed systems.

Conceptually, if

x0c = xc (c −

N
kz ), then the level of resource over-consumption

aects solely the birth rates of each consumer. However, if
self-regulating, since the value of

c

x0c = rcxc 1 −

N
kz , then the system becomes




now aects both the birth and death rates of consumers.

We will focus on the investigation of System (1), since our interest lies in evaluating the ecacy
of intervention strategies in a population of unregulated over-consumers.

Due to the nature of the

particular model given by System (1), the direction of the system's evolution over time can be anticipated.
Specically, the highest value of

c is always favored, by construction.

This property makes this particular

system an excellent conceptual model for the study of the eect of intervention strategies on the overall
long-term health of the system as a whole.

Population management strategies and their ecacy with respect to population composition
The goal of intervention strategies on this type of systems is to prevent population collapse, a potential
outcome given the over-consumer induced resource overexploitation. One possible way of achieving this
goal is to target the over-consumers directly. In [29], we studied the impact of a series of punishmentreward functions that target individuals within the population as a function their

c-value.

Specically, we

investigated the relative ecacy of punishment for over-consumption vs. reward for under-consumption,
and concluded that in a system, where the reproductive success of an individual is determined solely by
their ability to access and utilize shared resources, rewarding under-consumption is insucient to prevent
the tragedy of the commons; only the use of severe enough punishment can achieve this goal. Specically,
each individual can be punished or rewarded according to some function
aects tness of each consumer-producer.
over-consumption if
that

f (c)

c > 1,

f (c) ∈ C 1 (R+ ),

which directly

With this function, each clone is immediately punished for

or rewarded for under-consumption when

c < 1.

(Note, that we only require

be piecewise continuous and right dierentiable in order to be able to apply the Reduction

Theorem.) The system of equations that incorporates a punishment/reward structure becomes:

8

P


b A xc
dxc
= rxc c −
+ xc f (c)
dt
kz
X
dz
e
P
=γ+
xc (1 − c) − dz.
dt
z + c xc

(10)

A

The study of the above system involved tracking not only the changes over time of the size of the
population,

N (t)

the parameter
the eects of

c

and the amount of the shared resource,

z(t),

but also the change of the mean value of

and the overall distribution of clones over time. However, we haven't yet investigated

f (c)

on a relevant metric for population heterogeneity which may be used as one of the

predictors of the upcoming ecosystem crash.

As we argued above, heterogeneity is the key factor in

population adaptability. What is lacking is a specication of an adequate metric. The Reduction Theorem
allows us to track the change in variance of

c

over time, which we use as a stand in for potentially more

sophisticated measures. The following representative examples (see [29]) are highlighted here with the
addition of the change in variance of

c

over time. The values of all the core parameters remain the same

for all three cases unless otherwise specied. Noticeably, the goal of the following investigation is not to
provide a comprehensive investigation of all the cases of various punishment-reward functions and initial
distributions that were analyzed in detail in [29] but to expand the analysis through now additionally
studying the change over time of variance of

c

within the population. This second moment could prove

to be a potentially predictive measure of the health of the overall system.
In Figure 1, the initial distribution is taken to be truncated exponential with parameter
parameters were taken to be
function is
size

N (t),

f (c) = a 1−c
1+c

with

γ = d = e = r = k = 1, N0 = 0.6, z0 = 0.1, c ∈ [0, 2.5],
a = 1.

As one can see, around

decline in the amount of resource

the mean value of

c.

z(t),

t = 20,

µ = 10.

The

and the punishment

we observe a decline in population

rapid increase in over-consumption, as can be seen in

All this is accompanied by a rapid increase in variance, followed by a very sharp

drop, as over-consumers take over the population, leading to its rapid collapse.
In Figure 2, the initial distribution is taken to be Beta, with parameters
function is

f (c) = ρ(1 − cη ),

where

ρ = 1.2, η = 0.5,

and

c ∈ [0, 3].

[α, β] = [2, 5].

This particular punishment function

allows for separation of the eect of punishment for over-consumption (parameter
under-consumption (parameter

ρ).

The punishment

η)

and reward for

In this case, one can observe that the maximal population size is

smaller than was in the previous case, and the rate of increase of

9

E t [c]

is much slower.

However, we

Figure 1. System (10), where initial distribution of clones in the population is taken to be truncated

exponential with parameter

z0 = 0.1, c ∈ [0, 2.5],

µ = 10.

The parameters were taken to be

and the punishment function is

f (c) = a 1−c
1+c

with

γ = d = e = r = k = 1, N0 = 0.6,
a = 1.

once again observe a rapid increase in variance, followed by a rapid decrease, which precedes population
collapse.
Finally, in Figure 3, the initial distribution is taken to be Beta with parameters
punishment function is

f (c) = a(1−c)3 , with a = 0.1 and c ∈ [0, 2.5].

[α, β] = [2, 2].

The

In this particular case, the maximal

value of the population is smaller than in the previous two cases, which implies that the absolute number
of over-consumers is in fact smaller as well. This could provide an explanation for why in this case the
population survives for a longer period of time despite a high mean value of

E t [c]

at the early stages of

population growth. Moreover, as one can see, in this case, where the population survives longer than in
any of the other ones, the variance, while still decreasing, does so without spiking.
The dierences in the shape of the curves of the change in variance of

c

over time could be connected

to an observation, made by Gorban and colleagues [20]. Specically, these authors have observed that

in a number of social, ecological and economical systems, rapid increase in variance, when coupled with
increase in correlation between dierent components in the system, could be indicative of an approaching
system collapse.

Sharp increase in variance is an indication of a rapid change in population composition,

to which the system as a whole needs to adapt, as it is a stressor that comes from inside the system

10

Figure 2. System (10), where initial distribution of clones in the population is taken to be Beta, with

parameters

[α, β] = [2, 5].

The punishment function is

f (c) = ρ(1 − cη ),

where

ρ = 1.2, η = 0.5,

c ∈ [0, 3].

Figure 3. System (10), where initial distribution of clones in the population is taken to be Beta with

parameters

[α, β] = [2, 2].

The punishment function is

11

f (c) = a(1 − c)3 ,

with

a = 0.1, c ∈ [0, 2.5].

as opposed to being generated by external factors. Failure to adapt will mostly likely lead to eventual
system collapse. It is possible that similar mechanisms underlie the dynamics in our model. However,
this question requires further investigation.

The resilience metaphor
A natural question arises as to the

resilience

of systems such as (1). Resilience, dened in [4] as ...the

magnitude of disturbance that can be tolerated before a [social or ecological] system moves to a dierent
set of processes, may be undesirable within certain contexts. For instance, if the resource is desired and
the clones are a nuisance (i.e., in case of management of parasites, or even cancer cells within a body),
then we would wish for the population of clones to be unresilient, and for the resource (the body/host)
to be resilient.
are resilient.

From a parasite's perspective it is probably best that both the population and host

Our punishment/reward structure,

f (c),

is not a shock in the classic sense; rather, it

is a real-time, non-adaptive form of population control, which by construction favors clones with lower
reproduction rates that might instead use their resources to invest in increasing the carrying capacity
of the environment that supports them [44]. Investigations into the robustness of the logistic model as
trade-os between resisting shocks to growth (r ) and shocks to capacity (K ) in Social-Ecological Systems
(SESs) have shown that when a species (in our case the population of clones) is more robust with respect
to one strategy that the other suers [25, 30, 45].
What we wish to do here is to investigate if it is not merely the absolute magnitude of change that the
system has gone through, presumably assessed through
of the disturbance,
of a system.

V art [f (c)] =

´
C

Et [f (c)] =

(t)
(f (c) − Et [f (c)])2 xNc(t)
dc,

´
C

(t)
dc,
f (c) xNc(t)

but also the spread

that may be used to dene the resilience

Secondly, whether the transitions shown in the preceding gures t within the resilience

metaphor of [4], and whether the latter can help elucidate new insights.

By no means are we in a

position to quantify the eect of the spread of the disturbance; however, this possible connection might
provide an interesting avenue of research in both the eld of resilience theory and that of investigation
of parametrically heterogeneous dynamical systems.
First, we will summarize the metaphor introduced in [4].

Carpenter,

et. al.

describe a cycle of

adaptation, either in terms of the biophysical/behavioral properties, of the population, or of adaptive
selection of control strategies for a species, where essentially 4 stages of existence are dened:

12

r, K , Ω, and

α.

The SES rst favors an

transitions to a

r-strategy

where rapid production of a species dominates the system. Then it

K -strategy solution as either equilibrium is attained, or if other species (potentially more

robust to changes in environmental conditions) come to dominate. This robustness does not favor sudden

Ω

shocks (i.e., forest res), and the SES enters the

stage, where collapse, or near collapse, occurs. The

r-stage.

stage is described as renewal or reorganization of the SES preceding another
authors note, the

Ω and α

α

Typically, as the

stages are characterized by very short time scales, which favor the existence of

some sort of SES. The two stages of growth and persistence, r and K , may in contrast occur over much
slower scales. We are now presented with a particular challenge of implementing this theory here, since
we have an intrinsically coupled system (clone-resource), but we nd this framework useful in describing
the outcomes.
In each of the examples in the preceding section we showed how increase in

Et [c]

can be followed by

collapse of the resource, which would then lead to collapse of the population that is dependent on the
resource. The transitional regimes that are observed in various cases studied could in fact be understood
and formalized within the context of 4 stages described above. In Figure 1 our population of clones starts
in the

r-stage

K -stage

(as shown by the overall population growth from time

(the relatively at population growth from time

8

to

19).

t=0

to

t = 8)

and then enters the

Throughout all this time very specic

clones (under-consumers) got selected over time, as can be evaluated by the low value of

V art [c] ≈ 0.

Et [c]

and that

Furthermore, due to the prevalence of highly under-consuming clones, the resource reaches

its carrying capacity (transitioning from its own
spike in both variance and mean of parameter

c

r-stage

to

K -stage

at time

t = 10).

suggest that the system has entered the

The subsequent

Ω-stage,

a rapid

decline in resource availability. Moreover, while trajectory-wise, the initial direction of overall population
growth could be indicative of a new

r-stage,

a closer look at the distribution of

c-clones

indicates that in

fact the under-consumers have undergone a crash (Ω-stage), and the clones have reorganized about a
new phenotype of over-consumers (this reorganization being indicative of a rapid

α-stage).

The growth

of the the now favored over-consumers (r ) leads to exhaustion of the resource, which is followed by an
eventual collapse of the entire population. In the absence of both actors there can be no second renewal,
and the SES is dead.
We could view this all as a 4-tuple of stages for under-consumers, over-consumers, total clone population, and resource. Initially, we are met with a very favorable
absence). The system reaches a seemingly stable

(K, ∅, K, K)

13

(r, ∅, r, r)

conguration (where

∅

denotes

conguration; however, a transition occurs

rapidly where the population of over-consumers undergoes an unintended renewal
ably caused by the abundance of resource.

(K, α, K, K),

presum-

This abundance facilitates temporary coexistence of both

over- and under- consumers within the SES. There is no observable decay in the total population size
because the growth of over-consumers overshadows the decline of under-consumers, and an unobservable
set of transitions occurs:

(K, α, K, K) → (K, α, Ω, K) → (K, α, α, K) → (Ω, r, r, Ω).

These transitions

are characterized by a rapid growth in both the expected value of the parameter of consumption (the
over-consumers entering

r-stage),

and its variance, indicating a transition of the the population as a

whole, leading to the nal set of transitions of the entire system,
eventually leading to evolutionary suicide,

(∅, r, K, Ω) → (∅, r, Ω, Ω) → (∅, K, Ω, Ω),

(∅, Ω, Ω, Ω) → (∅, ∅, ∅, ∅).

Now let us examine another example, given in Figure 3. Here we can see what can be interpreted
as two population regime shifts as the population as a whole undergoes two rapid shifts from

Ω

to

α.

The second regime is characterized by a long period of what can appear as stability. However, a closer
look at the other characteristics of the system reveals a regime shift, selecting against under-consumers

(Ω, r, α, r) → (α, r, r, r) → (∅, r, r, r).
highly over-consuming clones,

This causes the highly varied system to lose diversity and select for

(∅, r, Ω, Ω),

which in turn once again results in eventual system collapse.

Analysis of these 4-tuples of interconnected stages points to several ndings. Predictably, the absence
of under-consumers leads to collapse of the system (i.e.,

(∅, ?, ?, ?)

invariably led to system collapse).

There must always exist a period when the over- and under-consumers coexist in order to allow the
resource to grow to a level where the over-consumptive type may be selected for,
and

Y

(X, Y, α, r),

where

X

may represent any stage within which the clone type is present. Therefore, it appears that any

intervention strategy needs to aim also at preventing an increase in consumptive variance in order to
disallow the total population to undergo a strategy regime renewal, and hence prevent collapse through
evolutionary suicide. One such possible strategy is discussed in the following section.

Intervention through adaptive control
Adaptive control is a method that can be of particular interest within the context of the proposed
problem. Let us introduce a central-planner, whose role it is to decide on the nature of the punishment
function

f (c).

The planner is faced with a vector of choice parameters

punishment function

f (c; ~a).

Parameters in the vector

14

~a

~a

that aect the shape of the

are implicitly dependent on the aspects within

the system that aect dynamics (e.g., state variables, parameters, etc.), and are chosen in such a way
as to maximize the utility (a general measure of benet) observed by the planner over time. This value
could be a function of the outcomes desired by either the central-planner or by each individual.
current time utility can explicitly be written as
benets, and

Ct

Ut (~at ) = Bt (~at ) − Ct (~at ), where Bt

The

represents anticipated

represents the costs that are expected to be incurred depending on both the state of the

system and the choices of the values of parameters in

~a.

Noticeably,

Ut (~at )

reects the discrete nature

of implementation of interventions: the decision about the proposed mode of action is proposed based
on the current state of the system and is executed over a time interval until new data is collected, and a
new decision can be made.

~at

The values of

are chosen in order to maximize

Ut (~at )


Vt = max Ut (~at ) +
{~
a k }T
k=t

where

δ

T
X

over a planning horizon of length

T

via


Ut+j (~at+j )δ j  ,

(11)

j=1

is some discount factor, a measure of how myopic the central-planner is in practice. A

δ

of one

represents that each future utility gain is worth the same if gained in the future as it would be at the
current time; a

δ

of zero indicates a purely myopic valuation of utility, where only the current time utility

is considered. It should be clear that the more myopic the planner, the less likely the intervention strategy
is to avoid future population crashes. In practice, a moderate discount between the two extremes is most
likely.
For our particular system, it appears reasonable to assume the planner should value both the presence
of the common resource

z(t),

as well as the total size of the population of consumers

N (t) =

´

xc (t)dc.

We shall therefore consider a general form of the punishment function with the following specication.
Let

pc = {ci }pi=1

f (c; ~a)

be a partition of the possible consumption levels within the system, i.e.,

class of functions,

f (c; ~a) .

c ∈ [ci , ci + 1).

for

Therefore, as the number of partition points grows,

f

Dene

to be the step function:

f (c; ~a) = ai

that

[0, cM ax ].

p → ∞,

(12)

so does the ability to approximate a large

Noticeably, in order for the Reduction Theorem to be applicable, it is necessary

be at least piecewise continuous and dierentiable as

p → ∞.

Within the context of reward being a subsidy (perhaps monetary) and punishment a ne, we could

15

view them as a cost and benet respectively.
set of relative costs and benets, that

~a

Given this control mechanism it is possible, for a given

could be chosen to follow either of the following three options:

a) greatly favor under-consumers and punish over-consumers, assuming that value placed on persistence
of resource and population are quite large compared to cost of under-consumer subsidy and benet
gained from punishing over-consumers; b) select for a distribution highly skewed toward over-consumers;
particularly myopic planners (either because of a short planning horizon or high discounting on future
returns) could see the large benet gained from punishing over-consumers while not valuing the future
collapse of the system; or c) induce a very small variance (or very slow growth in variance) within the
population regardless of the mean consumption level. While not being valued directly, a small variance
could likely be an unintended consequence for very forward-looking planners who can see the collapse
that a large variance or rapid change therein may induce.

Within the context of our model, it would

appear that some combination of a) and c) would be the most preferable, as one would clearly wish to
favor under-consumers. Additionally, in the case when rapid increases in variance are indeed correlated
with increased likelihood of population collapse, managing variance in addition to population size would

z(t) and N (t) are relatively large when compared to the
´ cM ax
from
xc (t)dc), and when both δ ≈ 1 and T large,
1

also be of importance. Therefore, if the value on
revenue gained from punishment (the benet

then the central-planner would possess the foresight, and place adequate value on persistence, necessary
to avoid population collapse while inadvertently selecting for narrow, slow growing variance.

Managing cancer as a heterogeneous consumer-resource type system
In addition to evident applications in biological systems and resource management [42], the reported
results are also highly applicable in the eld of cancer therapeutics, where inter- and intra-tumor heterogeneity plays a key role in the identication of treatments for cancer [3].
One of the possible ways to view cancer is as a type of a consumer-resource system, where both
normal and cancerous cells rely on the resources provided by the host for survival and reproduction.
Similarly to the proposed conceptual model, the immediate selective advantage of over-consumption of
shared resources, such as glucose [18], glutamine [11], phosphorus [14], etc., by cancer cells selects for
over-consumers at the expense of normal cells (the under-consumers). This in turn can eventually lead

16

to the tumor committing evolutionary suicide via exhaustion of the shared resources, in this case its
host. Noticeably, regardless of the specic reasons for what drives resource over-consumption, the nal
result is exhaustion of the shared resources at the expense of healthy tissues. In the terminal stages
of the disease we indeed often observe nutritionally irreversible decrease in body mass (cachexia), which
precedes death.
Most clinically available treatment strategies, and specically chemotherapy, target the most rapidly
dividing cells at dierent stages of the cell cycle [52]. Such an approach, while often eective in reducing
the overall tumor burden in the short term, in fact highly eciently selects for therapeutically resistant
clones. If the highest rate of resource consumption translates to the highest growth rates, then chemotherapy administration translates directly into targeting individuals, or in this case cells, with the highest
value of the parameter

c.

Unfortunately, in a clinical setting, such interventions are typically highly toxic

to the host, and their long-term eectiveness is highly limited for most cancer types, including sarcomas,
breast, prostate, pancreas and lung cancers [48].
Our results suggest that interventions that maintain heterogeneity within the population are in fact
more eective long term. In cancer therapy, one strategy is administration of chemotherapy at lower doses
but more frequent time intervals, also known as metronomic chemotherapy. In contrast to MTD, which
administers maximally tolerated doses of cytotoxic drugs at largely spaced time intervals, metronomic
regimen allows more gradual reduction in tumor burden, without severe selection for resistant clones, as
is frequently observed with MTD [21, 31, 43].
Moreover, Gatenby and colleagues [17] have evaluated the ecacy of what has been termed adaptive
therapy, where the goal of the therapeutic intervention is not to cause maximal cell mortality but to
maintain a xed tumor burden. Such approach, which conceptually is similar to aforedescribed adaptive
control, has been successful in producing long-term survival in mouse models. Perhaps a combination
of the two approaches (strategies a and c in the previous section), namely, frequent administration of
low dose chemotherapy in combination with adaptive control, would create an adaptive metronomic
chemotherapy that could lead to most benecial results.

17

Perspectives on sustainability within host parasite systems
In this article, the similarities between resource-consumer type systems, in the presence of a particularly
devastating overly-adapted population (e.g., cancerous cells) have been highlighted.

Cancer dynamics

under the selective forces of treatment belong to a class of non-traditional examples, for which implementation of approaches that sustain some level of heterogeneity may be critical even if they seem
counterintuitive. Other systems that can benet from a consumer-resource perspective include disease
transmission systems, particularly systems that are responsible for generation and spread of antibiotic
resistant agents. For instance, management of the limited number of available antimicrobials that are
currently being used to ght nosocomial infections is likely to have dramatic consequences at higher levels
of organization. The need to embrace nontraditional policies in situations, where `common sense' might
dictate otherwise, must be seriously evaluated. The following observations on sustainability and slowing
down the growth antibiotic resistance are based on [5].
The complex dynamics of emergent and re-emergent diseases within landscapes living under the tinkering eects of evolution, adaptive human behaviors, and public health policies that include treatment,
vaccination or both, are used to highlight the dynamics of antibiotic resistance in the context of sustainability. On March 11, 2013, the British Chief Medical Ocer Dame Sally Davies, [38], noted that
the problem of microbes becoming increasingly resistant to the most powerful drugs should be ranked
alongside terrorism and climate change on the list of critical risks to the nation ... Yet while antibiotic
use is rising  not least in agriculture for farmed animals and sh  resistance is steadily growing and the
`pipeline is drying up' of new drugs which can replace those becoming useless. [In fact] No new classes
of antibiotics have been developed since 1987, and none are in the pipeline." Professor Nigel Brown,
president of the Society for General Microbiology, remarks that immediate action by scientists is required
if we are going to identify and mass produce new antibiotics; this kind of eort needed to tackle the problem of antimicrobial resistance and its transmission, particularly in the context of nosocomial (hospital)
infections [38].
Mathematicians use contagion or epidemiological models to study the evolution, dynamics and control
of diseases, including nosocomial or hospital infections, i.e., infections most often responsible for transmission of resistant pathogens in hospitals [7, 37]. The most celebrated epidemic model was introduced
by a medical doctor W.O. Kermack and a statistician A. G. McKendrick [32]. Both researchers modeled

18

disease dynamics under the assumption that transmission depends on the intensity of hosts' interactions (handshakes, kisses, and more) and the frequency of encounters between susceptible and infected
individuals.
In 1994, Malcolm Gladwell, a journalist writing for the New Yorker magazine, re-discovered a plausible
explanation for the growth or decay of a dierent kind of disease, the `disease' responsible for epidemics
of crime in New York City [19]. He observed that the side of the tipping point that a system was `on'
determined whether or not crime was a problem. Hence, the direction of the crossing, as parameters were
varied, could determine the dierence between night and day when it came down to the level of success
of criminal activity. Gladwell saw crime as a contagion process, recognized the existence of a threshold,
and understood and appreciated the signicance of crossing such a threshold. The insights that he gained
from understanding models of contagion led him to the writing of several books. It is precisely within the
classical contagion framework introduced by Kermack and McKendrick that the exploration of the impact
of nosocomial infections in the presence of the selective pressures resistance generated by antimicrobials
that evaluation of policies that manage disease levels, while limiting the spread of resistance disease
strains, have been studied [7, 37].
Kermack and MacKendrick were indeed well aware of the ideas popularized by Gladwell. In fact, both
had gone further by establishing mathematical results for infectious diseases in 1927. Contagion/contact
models are now routinely used in the study of population-level questions in evolutionary biology or
public health. The direct use of the Kermack-McKendrick 1927 model or its modications, in the study
of the dynamics of antibiotic resistance at the population level is therefore natural.

The simplicity of

the Kermack-McKendrick Susceptible-Infected-Recovered or SIR model has increased its popularity and
helped establish the eld of mathematical epidemiology [1, 2].

The following short synopsis using the

transfer diagram in Figure 4 will highlight the essence of the contagion modeling approach. A population
has been divided into three categories: susceptible or S, infected or I (assumed infectious) and recovered
or R (assumed immune); there are no births or deaths.

That is, we have selected a specic (short)

temporal scale to study the dynamics of contagion in order to highlight the contagion modeling approach
in as simple a setting as possible. It is assumed that all individuals are identical and that the population
is large so that the use of dierential equations is acceptable. It is further assumed that deaths from the
disease under consideration are negligible and that there are no births. Hence, the population is constant
and therefore it can be (and has been) normalized to one.

19

It is assumed that infections arise from

S

β IS

I

γ

R

Figure 4. A phenomenological ow diagram for the transmission of a disease with susceptible,

infectious, and recovered/immune stages.

interactions between susceptible and infective individuals (modeled via the mass action law) and that
recovered individuals cannot get this disease again. Several things can be deduced from this particular
model: if the assumptions outlined above are satised, then 1) for an outbreak to occur an infectious
individual must be introduced (within a purely susceptible population); 2) over his/her infectious period

1
γ , individuals that recover gain permanent immunity (moving to R), and 3) the infection success rate is
actually

β,

and when it is applied over the window of opportunity

as the basic reproduction ratio
collapses if

R0 =

1
γ , one can obtain the quantity known

β
γ . An outbreak is possible if

R0 > 1

and the disease eventually

R0 < 1.

It is in variants of the above setting (variants that include births and deaths and disease-induced
mortality) that the issues of the persistence, evolution and the expansion of resistance to antimicrobials
are being studied under the shadow of the fact that the number of drugs is limited since no new ones
have been brought to the market for nearly three decades [2, 7, 37].

The need for identifying ways of

reducing the prevalence of resistant strains while maintaing our antimicrobial supply viable are obvious
in light of the words stated by the British Chief Medical Ocer Dame Sally Davies and a year earlier by
the Director General of the World Health Organization, Dr. Margaret Chan on the pressing criticality
of managing resistance. In fact, the head of the WHO, while addressing a meeting of infectious disease
experts in Copenhagen, noted that we are facing a global crisis in antibiotics, the result of rapidly
evolving resistance among microbes responsible for common infections that threaten to turn them into
untreatable diseases ... every antibiotic ever developed was at risk of becoming useless. `A post-antibiotic
era means, in eect, an end to modern medicine as we know it. Things as common as strep throat or
a child's scratched knee could once again kill ... Antimicrobial resistance is on the rise in Europe, and
elsewhere in the world. We are losing our rst-line antimicrobials.'  [34].
Trying to identify implementable ways of slowing down the evolution of resistance or nding ways of

20

limiting the most powerful outcomes of tight coevolutionary interactions, like evolutionary suicide, are
central to war on diseases like cancer, or tuberculosis, or nosocomial infections. Maintaining, for example,
a sustainable pool of antimicrobials available for treating infections over long-time scales or identifying
drug protocols that allow us to live with low-levels of cancer are but some of the topics that emerge in
the study of models that address the importance of maintaining heterogeneity as a critical foundation for
the development of policies.
Further, complications arise because in the study of sustainability one must account for processes that
interact over multiple temporal and spatial scales as well as several levels of organization. Each particular
system requires us to consider dierent units of selection, since as Lewontin has observed, [36], As seen
by present-day evolutionists, Darwin's scheme embodies three principles ... : 1. Dierent individuals in
a population have dierent morphologies, physiologies, and behaviors (phenotypic variation). 2. Dierent phenotypes have dierent rates of survival and reproduction in dierent environments (dierential
tness). 3. There is a correlation between parents and ospring in the contribution of each to future
generations (tness is heritable)...The generality of the principles of natural selection means that any
entities in nature that have variation, reproduction, and heritability may evolve.

For example, if we

replace the term individual with the term population and then Principles 1, 2, and 3 describe a process
by which one population may increase its proportional representation in the species relative to other populations. Similar reinterpretations of these principles could be made for species rather than populations
and even communities rather than species. Consequently, what may be benecial to an individual may
be detrimental at a higher level of organization. This can become of utmost importance when a population as a whole is forced to respond to environmental shifts, which in turn may result in response as a
unied group, or create a divide within the population [41]. Furthermore, when it comes down to human
behavior, our ability to adapt is often linked to personal interests [15, 40]. These personal interests may
at times conict with and occur on a discordant temporal scale with the public goals. The presence
of diverse and changing classes of individual behaviors present unique challenges to our ability to build
policies capable of managing heterogeneity in the absence of well-formulated and implemented incentives
and rewards within frameworks that adapt to changing (multi-level) landscapes. The implementation of
cancer therapies that keep chemotherapies viable or policies that increase the life-span of antimicrobials
must at the end not only account for the tinkering role of evolution but also for the complexities of human
behavior.

21

Acknowledgments
This work has been partially supported by grants from the National Science Foundation (NSF - Grant
DMPS-0838705), the National Security Agency (NSA - Grant H98230-09-1-0104), the Alfred P. Sloan
Foundation, the Oce of the Provost of Arizona State University and Intramural Research Program of
the NIH, NCBI. This publication was made possible in part by grant number 1R01GM100471-01 from
the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health.

Its

contents are solely the responsibility of the authors and do not necessarily represent the ocial views of
NIGMS.

22

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