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Behavioral Ecology
doi:10.1093/beheco/ari087
Advance Access publication 29 September 2005
Stress, resource allocation, and mortality
J. M. McNamaraa and K. L. Buchananb
Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK, and
b
Cardiff School of Biosciences, Cardiff University, Park Place, Cardiff CF10 3TL, UK
a
We model the optimal allocation of limited resources of an animal during a transient stressful event such as a cold spell or the
presence of a predator. The animal allocates resources between the competing demands of combating the stressor and bodily
maintenance. Increased allocation to combating the stressor decreases the mortality rate from the stressor, but if too few
resources are allocated to maintenance, damage builds up. A second source of mortality is associated with high levels of damage.
Thus, the animal faces a trade-off between the immediate risk of mortality from the stressor and the risk of delayed mortality due
to the build up of damage. We analyze how the optimal allocation of the animal depends on the mean and predictability of the
length of the stressful period, the level of danger of the stressor for a given level of allocation, and the mortality consequences of
damage. We also analyze the resultant levels of mortality from the stressor, from damage during the stressful event, and from
damage during recovery after the stressful event ceases. Our results highlight circumstances in which most mortality occurs after
the removal of the stressor. The results also highlight the importance of the predictability of the duration of the stressor and the
potential importance of small detrimental drops in condition. Surprisingly, making the consequences of damage accumulation
less dangerous can lead to a reallocation that allows damage to build up by so much that the level of mortality caused by damage
build up is increased. Similarly, because of the dependence of allocation on the dangerousness of the stressor, making the
stressor more dangerous for a given level of allocation can decrease the proportion of mortality that it causes, while the proportion of mortality caused by damage to condition increases. These results are discussed in relation to biological phenomena.
Key words: immunocompetence, optimality, resource allocation, stress. [Behav Ecol 16:1008–1017 (2005)]
rganisms often have to survive periods when external
circumstances are adverse. Adaptations that enhance
survival are not only behavioral but also involve physiological
responses to external stresses (Wingfield and Sapolsky,
2003). This concept of adverse environmental conditions
was first defined as ‘‘stress’’ by Selye (1976). Selye considered
the behavioral and physiological responses to adverse environmental conditions to be part of the ‘‘stress response’’ to
a ‘‘stressor.’’ Although considerable controversy has developed
over exact definitions of stress and the stress response (Broom
and Johnson, 2000), Selye’s work emphasizes that even when
exposed to a range of different kinds of adverse environmental circumstances, animals demonstrate a similar array of
physiological responses, designed to maximize survival and
minimize the physiological impact of the stressor. Extreme
examples of physiological responses are seen in a range of
animal taxa, which reduce their energy requirements through
hibernation or transient torpor, during adverse seasonal conditions (Geiser, 2004; Humphries et al., 2003; Lyman, 1982).
On a finer scale many animals exhibit optimal allocation of
resources without such obvious physiological changes. Such
physiological adaptation involves trading off different demands (e.g., immune defense, metabolism, reproduction,
somatic growth) for the available resource levels and optimal
allocation for both short-term welfare and long-term survival
(Cichon et al., 2002; Harshman and Schmid, 1998; Sandland
and Minchella, 2003; Svensson et al., 1998). During periods of
high resource availability all the competing physiological demands can be maintained at closer to the optimal level than
during times of resource restriction. However, in conditions
of limited resource availability, resources should be shunted
between the competing physiological demands, minimizing
O
Address correspondence to J. M. McNamara. E-mail: john.
[email protected].
Received 20 December 2004; revised 24 July 2005; accepted
24 July 2005.
The Author 2005. Published by Oxford University Press on behalf of
the International Society for Behavioral Ecology. All rights reserved.
For permissions, please e-mail: [email protected]
allocation to less essential functions (e.g., Ewenson et al., 2003;
Kilpimaa et al., 2004). This concept of optimal resource allocation under limited resource availability is similar to
the concept of allostasis, where limited resources are allocated
in order to maintain essential homeostasis (McEwen and
Wingfield, 2003). It is suggested that allostatic load, the cumulative result of physiological allocation in relation to environmental stimuli, can lead to pathological conditions when
demands for resources far outstrip the resources available
(McEwen and Wingfield, 2003). We hypothesize that under
natural selection, the processes that allow redistribution of
physiological resources should distribute resources optimally
to maximize fitness, and here we model the optimal strategies
of allocation under conditions of resource limitation.
In poor conditions, a range of linked physiological changes
controlled by the neuroendocrine system occur; these changes
are designed to minimize the impact of stressful events
(Buchanan, 2000; Wingfield et al., 1997). Under chronic stress,
body mass decreases to allow mobilization of energy reserves
and essential stored reserves of amino acids. Such changes in
mass represent the mobilization of resources required for essential metabolic processes, and similar freeing of resources
that is not necessarily mass linked is likely to occur (e.g., calcium bone reserves during egg laying; Graveland and Drent,
1997). Seasonal changes in mass occur under the influence of
photoperiod and in relation to food availability and environmental conditions (Mercer et al., 2000). However, a decrease in
body mass cannot be taken to imply a restriction in resources as
birds have been shown to adaptively regulate their mass both in
relation to food availability and predator threat (Houston
et al., 1997; Rands and Cuthill, 2001).
Under chronic stress there is also a downregulation of the
immune system, thought to occur in part through the action
of the neuroendocrine system (Brown, 1994; Padgett and
Glaser, 2003). There has been considerable discussion of the
reasons underlying this stress-associated immunosuppression
and whether this forms part of an adaptive stress response
(Buchanan, 2000; Råberg et al., 1998). It appears to be the
McNamara and Buchanan • Stress, resource allocation, and mortality
1009
Table 1
A list of examples of potential stressors commonly encountered by wild animals, combined with examples of condition measures that could be
compromised by the stressor
Stressor
Condition measure
Limiting resource
Source of mortality
Cold
Cold
Predator pressure
Territorial aggression
Drought
Seasonal food shortage
Immunocompetence
Body mass
Immunocompetence
Fur/feather quality
Body mass
Body mass/essential amino
acids/micronutrients
Muscle
Calories/amino acids?
Body fat
Calories/amino acids?
Vigilance/time grooming
Energy to find water
Energy to find food/migrate
Cold/disease
Cold/starvation
Predation/disease
Injury/cold
Dehydration/starvation
Starvation
Reserves of essential amino acids
and micronutrients
Energy to find food/defend territory
Carotenoids
Injury/starvation
Competition for foraging resources
Seasonal food shortage
Territorial competition
Territory
Immunocompetence/badges
of status
Starvation
Starvation/disease
Where possible we identify the nature of the resource, which the animal must allocate between maintaining condition and defending itself against
the stressor. The potential sources of mortality from the stressor/poor condition are also given. Different stressors and condition measures can be
combined to form trade-offs under different scenarios, and this list is designed to give examples, not present comprehensive coverage. The top
row represents the example used to motivate the model, whereby the animal encounters a transient period of cold stress and has to allocate the
resources available in the form of body fat stores between maintaining its condition through high levels of immunocompetence and defending
itself against the cold stressor. In this case the animal can die of either cold (stressor) or from disease (reduced condition). The last row represents
a situation whereby an animal has carotenoid-dependent badges of status that determine territory defense and therefore access to foraging
resources. In such a situation, the animal must allocate the available carotenoids between immune function to protect against disease and
ornament quality, which determines territorial defense. In this case the animal can either die from starvation (stressor) or disease (reduced
condition). In each case the forces underlying each of these physiological trade-offs can be considered similar, although modeling each scenario
would need to allow for the specific physiological consequences of each kind of trade-off.
case that the resources allocated to immunocompetence decline when homeostasis is disturbed, through a shortage of
resources (Moret and Schmidt-Hempel, 2000), seasonal environmental changes (Nelson et al., 2002), or in relation to the
challenges associated with reproductive investment (Ardia
et al., 2003; Bonneaud et al., 2003). These changes in immune
allocation suggest that there is a cost to a constantly elevated
immune system. Although such a cost is difficult to quantify
(Sandland and Minchella, 2003), it could be mediated
through energetic reserves or essential limited stocks of minerals, vitamins, or amino acids (Klasing and Calvert, 1999;
McGraw and Ardia, 2003), which are limited through availability in the diet (Nelson et al., 2002).
While the proximate reason for such changes in immune
function may be physiological effects, mediated through neuroendocrine control, the ultimate explanation may involve
long-term resource management (Nelson et al., 2002). There
are seasonal changes in the level of immunocompetence of
both human and nonhuman animals, with elevated levels during the summer months and lower levels of immunocompetence during periods of environmental stress (Nelson and
Demas, 1996). Mortality levels rise during winter months in
relation to reduced immunocompetence and increased levels
of pathogens, both in humans and in other animals (Nelson
and Demas, 1996; Nelson et al., 2002). It seems likely that this
mortality is due in part to the direct effects of disease and
pathogens and in part to resource restriction at this time
(Lochmiller and Deerenberg, 2000; Nelson and Demas,
1996). An optimal resource allocation strategy must involve
protecting the individual from different sources of mortality,
the threat of which varies over different timescales.
Animals are vulnerable to a range of potential stressful conditions including seasonal changes in environmental quality
(heat/cold stress, water shortages) as well as increases in the
occurrence of pathogens, parasites, and predators. It is worth
noting that the duration of some stress events are partially
predictable (e.g., seasonal changes), while the duration of
other stress events (e.g., short, sharp cold spells) may be much
more difficult to gauge. Stress might also be induced by the
removal of resources such as shelter or increases in disturbance levels. The inability to maintain optimal physiological
condition during such periods of environmental stress will
lead the animal to incur a reduction in its physiological state.
This reduction in condition comes from a reallocation of resources in order to deal with the stressor. Such a reduction in
state potentially includes reduced physiological reserves of
essential vitamins, minerals, and amino acids; reduced body
condition and immunocompetence; and reduced fat reserves,
leading to increased mortality and decreased reproductive
potential. Examples of reduced physiological state also include increased levels of oxidative damage, which involves
damage to both somatic and reproductive cells’ cellular systems by oxygen radicals (von Schantz et al., 1999). Examples
of the kinds of commonly encountered stressors, the potential
impacts on condition, the possible limiting resources, and
consequences for survival are given in Table 1. We hypothesize
that under conditions of stress animals must allocate their
limited stocks of these resources between the competing demands of dealing with the stressor and maintaining condition.
The response to chronic stress must therefore involve optimal
resource allocation to minimize the risk from the current
threats, while not compromising long-term survival by incurring too much damage to individual physiological state. We
hypothesize that the physiological mechanisms involved in
this process must have evolved under strong selection to produce optimal strategies for such resource management under
chronic stress.
The model
We consider an animal that faces a time-limited but immediate danger. Examples of such transitory environmental stressors are environmental fluctuations, increases in predator
presence, increases in parasite and pathogen prevalence, or
resource shortages, as outlined above. All these effects could
occur through seasonal modulations that might be of a more
Behavioral Ecology
1010
or less predictable length, but we also envisage that shorter
term changes in their occurrence could also occur. We assume
that the animal can allocate resources between combating the
immediate danger from the stressor and maintenance. Damage builds up if the allocation to maintenance is insufficient.
Damage could represent either reduced physiological reserves
of essential nutrients, minerals or energy, increased levels of
oxidative stress, or reduced condition of protective body covering such as fur or feathers. Such damage results in reduced
physiological condition leading to effects such as impaired
immunocompetence. There are two sources of mortality; one
source is the stressor, the other results from the build up of
damage. The animal thus faces a trade-off; the more the resources allocated to combating the stressor, the less the mortality from this danger but the greater the build up of damage
and its associated mortality. Through natural selection the
animal is expected to maximize the probability that it is not
killed by either source of mortality during the stress period
and the subsequent recovery time needed to repair damage.
In this model we aim to draw general conclusions about the
trade-offs that occur between transient stressors and long-term
condition. However, for the ease of interpretation, we suggest
that the stressor can be viewed as a period of environmental
cold stress of variable duration that has an impact on the
animal’s condition by reducing immunocompetence and
rendering the animal more susceptible to disease. The animal
can therefore be killed either by the cold stress or by disease.
We find the animal’s optimal strategy, and the level of mortality from the two sources under this strategy. Because the
animal must recover after the stressor disappears, the level of
mortality from damage during this recovery phase is of particular interest. We analyze the effects of various parameters: the
dangerousness of the stressor (represented by the parameter
m0 in Equation 1), the danger of allowing damage to build up
(represented by the parameter K in Equation 5), the mean
duration of the stressful period, and the predictability of this
duration.
The stressor appears at time t ¼ 0. It then is present for
a period of time whose duration is a random variable with
mean s. The stressor then disappears and does not reappear
again. Allowing the time at which the stressor disappears to be
a random variable allows us to model the situation in which
the stressor is present for an unpredictable duration. Although the animal is not able to predict the exact duration
of the stress period, we assume that it is adapted to the probability distribution of the duration. In other words it behaves
as if it knew this probability distribution.
The animal must allocate resources between combating the
immediate danger from the stressor and bodily maintenance.
Maintenance should be regarded as maintaining body condition through basic metabolic processes including antioxidant
activity, immune function, and maintaining defenses against
cold stress, such as fur or feather condition. We denote the
proportion of resources allocated to combating the stressor by
u. Here 0 u 1: The remaining resources (proportion 1 u)
are devoted to maintenance.
If the stressor is present and the animal allocates u to combating it, then the animal has instantaneous mortality rate
M(u) from this danger. Here the function M(u) decreases
with increasing u. Our baseline model assumes the function
M ðuÞ ¼ m0 ð1 uÞ;
ð1Þ
where m0 is a positive constant that we refer to as the dangerousness of the stressor. The function M(u) has a maximum of
M ¼ m0 when u ¼ 0 and a minimum of M ¼ 0 when u ¼ 1.
Once the stressor disappears this source of mortality also
disappears.
The variable x measures the damage brought about by the
lack of resources allocated to bodily maintenance. As x increases, the condition of the animal decreases. Damage has
a minimum value x ¼ 0, corresponding to the animal being in
its best condition. The rate of change of damage at a given
time depends on the allocation u at that time. When u is low,
most of the available resource is allocated to maintenance and
damage decreases. When u is high, little resource is allocated
to maintenance and damage increases. We represent this by
xðtÞ
_ ¼ f ðu; xÞ for x . 0;
ð2Þ
where xðtÞ
_
denotes the rate of change of x. Here f is an
increasing function of u satisfying f(0, x) , 0 and f(1, x) . 0
for all x. Results presented are based on the function
f ðuÞ ¼ 4u 2 1:
ð3Þ
For this function the break-even point at which x remains
constant is u ¼ 0.5. Equation 2 needs to be modified at
x ¼ 0 because x cannot be negative. We do so by setting
xðtÞ
_ ¼ maxf f ðu; 0Þ; 0g for x ¼ 0:
ð4Þ
There is a second source of mortality associated with the
animal being in poor condition. For definiteness we might
think of this source of mortality as death from disease. If
the animal has damage x, its instantaneous rate of mortality
as a direct result of this damage is D(x). We assume that
D(0) ¼ 0 and that D(x) increases with x. In our baseline model
we take
DðxÞ ¼ Kx;
ð5Þ
where K is a positive constant that controls the damageinduced mortality rate at a given level of damage. The total
instantaneous mortality rate while the danger is still present is
the sum of M and D.
After the stressor disappears, the animal allocates all resources to maintenance (u ¼ 0). It is assumed that a second
stressor does not arrive during the period of recovery from
the focal stressor. Thus, after the focal stressor disappears, the
level of damage decreases until it reaches x ¼ 0. Once this
value of x is reached, the animal stays at this level. Because
D(0) ¼ 0, there is no further risk of mortality once x ¼ 0 is
reached.
For this scenario a strategy specifies the allocation of resources to combating the stressor while it is still present. The
optimal strategy maximizes the probability that the animal
survives both the period when the stressor is present (the
stress period) and the subsequent recovery period. A method
by which this strategy can be found is given in the Appendix.
Under the optimal strategy the allocation to combating the
stressor may depend on both the current level of damage x
and the time t that the stressor has been present. We denote
this allocation by u*(x, t). Of course the level of damage at
a given time, and hence the animal’s allocation at this time,
will depend on the initial damage level. In presenting results
on the consequences of following a strategy, we always assume
that the animal has damage level x ¼ 0 at time t ¼ 0.
The baseline case
To investigate the effects in our model we first analyze a baseline case. The effects of deviations from baseline assumptions
are then investigated. In our baseline case, M(u) and D(x) are
given by Equations 1 and 5, respectively. We also assume that
the stress period has an exponential duration with mean
s ¼ 10. Then, because the exponential distribution has the
McNamara and Buchanan • Stress, resource allocation, and mortality
1011
‘‘lack of memory’’ property, given the current level of damage
x, the time since the stressor appeared is irrelevant and the
optimal allocation u*(x) only depends on x. Let the allocation
uc satisfy f ðuc Þ ¼ 0 (so uc ¼ 0:5 when f is given by Equation 3).
Then by Equation 2 the level of damage remains constant over
time at this allocation. For the functions that we use, u*(x)
decreases as x increases, with u . uc when x is small and u , uc
when x is large (Figure 1). Thus, there is a level of damage xc
such that u*(xc) ¼ uc (Figure 1a). An animal following the
optimal strategy starts with damage x ¼ 0 at time t ¼ 0. Its level
of damage then increases over time (while the stressor is still
present), tending to the limiting value xc. We refer to xc as the
maximum acceptable damage level.
Investigation of the costs parameters
We first analyze the effect of the form of the functions M(u)
and D(x).
Effect of K in the baseline case
The parameter K (Equation 5) controls the cost of a given
level of damage. We analyze the effect of this parameter in the
baseline case.
The greater the value of K, the greater the increase in the
rate of damage-induced mortality that is produced by a further
increase in damage. Thus, as K increases it is optimal to devote
less resources to combating the stressor and more to maintaining condition. Thus, the maximum acceptable damage
level xc decreases (Figure 1a). Consequently, the greater the
value of K, the more slowly damage accumulates and the lower
its asymptotic level.
Because the level of damage increases over time and u(x) is
a decreasing function of x, the proportion of resources devoted to combating the stressor decreases over time. Thus,
while the stressor is still present, rates of mortality from both
the stressor and damage increase over time (Figure 2). Increasing K has two opposing effects: the rate of damagerelated mortality at a given level of damage increases, but
the change in strategy means that the rate of damage accumulation is less. The net result is little change in damagerelated mortality experienced over time (Figure 2). As K increases the allocation to combating the stressor when it first
appears is decreased because it is more important to prevent
the build up of damage. Thus, as K increases, the rate of
mortality from the stressor rises sooner (Figure 2). The asymptotic level of mortality from the stressor is, however, the same,
and equals M ðuc Þ for all K. Thus, asymptotic levels of mortality
from the two sources are highly robust under changes in K.
The main effect of this parameter is in determining how
quickly mortality from the stressor reaches its asymptotic level.
In the baseline case the mean time the stressor is present is
s ¼ 10. This is the time the stress period is expected to last on
average, but the actual duration of the stress period can vary
about this mean. Figure 3 illustrates the probability of death,
conditional on the duration of the stress period. Sources of
mortality are broken down into death from damage while the
stressor is still present, death from damage after the stressor
disappears, and death from the stressor itself. As can be seen,
the overall probability of death is a nonlinear function of the
duration of the stress period; for example, the mortality when
the duration is 10 time units is more than twice the mortality
when the duration is 5 units. As the duration increases, the
proportion of the mortality that is due to the stressor increases
and the proportion due to damage after the disappearance of
the stressor decreases. As K increases, the proportion of mortality due to the stressor increases at all durations.
The effect of K on average mortality, averaged over the
length of the stress period, is shown in Figure 4a–c. A striking
Figure 1
The optimal allocation to combating the immediate danger from
the stressor, u*(x), as a function of the level of damage, x. For f(u)
given by Equation 3 the break-even point is uc ¼ 0.5. Thus, the
intersection of the u*(x) curve with the dashed line at u ¼ 0.5 occurs
at x ¼ xc (indicated by the arrows in (a)). This value xc represents the
asymptotic level of damage if the stressor fails to disappear and can
be thought of as the maximum acceptable level of damage. (a) The
effect of K in the baseline case. m0 ¼ 0.1. (b) The effect of the mean
duration s when the duration of the stress period is exponential.
m0 ¼ 0.1, K ¼ 0.001.
feature is that as a factor (here parametrized by K or m0) gets
less dangerous, the proportion of the total mortality that is
due to this factor increases. Moreover, as K decreases, so that
damage-related mortality is reduced for a given level of damage, the greater build up of damage results in the absolute
level of mortality from damage increasing in part of the range.
As can also be seen, as K decreases the proportion of total
mortality caused by damage after the stressor disappears
increases until it predominates.
Effect of m0 in the baseline case
The parameter m0 is the mortality rate from the stressor if no
resources are allocated to combat this danger (Equation 1).
Thus, m0 is a measure of how dangerous the stressor is. Increasing the dangerousness of the stressor results in more
allocation to combating the stressor and less to preventing
an increase in damage. The main effect is that when the
stressor appears, the rate of mortality from this source rises
more slowly initially, although asymptotic levels of mortality
from both the stressor and damage are higher. Thus, increasing m0 has a similar effect on the shape of the mortality rate
functions to decreasing K, although absolute levels of mortality increase. As Figure 4a–c illustrates, the increase in allocation to combating the stressor as it is made more dangerous
1012
Behavioral Ecology
Figure 2
Rates of mortality from the stressor and damage while the stressor
is still present. Baseline case with m0 ¼ 0.1. (a) K ¼ 0.0002,
(b) K ¼ 0.0005, and (c) K ¼ 0.001.
Figure 3
The dependence of the sources of mortality on the actual duration,
d, of the stress period. Mortality can be the result of damage after
the stressor disappears, damage while the stressor is still present, or
due to the direct effect of the stressor. Baseline case with m0 ¼ 0.1.
(a) K ¼ 0.00005, (b) K ¼ 0.0002, and (c) K ¼ 0.001.
can mean that average mortality from the stressor decreases,
with the effect particularly strong when K is small. Average
mortality from damage, both while the stressor is present
and after, increases as the stressor becomes more dangerous.
overall mortality, mortality due to damage after the stressor
has disappeared increases the most rapidly.
Combined effect of K and m0
We have examined the effect of increasing both K and m0
while keeping their ratio constant. The main effect is an increase in overall average mortality. Of the components of this
Robustness of the effects of K and m0
In the baseline case DðxÞ ¼ Kx: It seems possible, however,
that the mortality rate from damage might increase at an
accelerating rate with the level of damage. We have thus examined the case DðxÞ ¼ Kx a ; where a . 1. The qualitative
effect of K and m0 are very similar in this case to the baseline
McNamara and Buchanan • Stress, resource allocation, and mortality
1013
Figure 4
The effect of m0 and K on average mortality, where the average is over the possible durations of the stress period. Sources of mortality are
as in Figure 3. (a) Baseline case with m0 ¼ 0.05, (b) baseline case with m0 ¼ 0.1, (c) baseline case with m0 ¼ 0.2, and (d) stress period of
predictable length s ¼ 10, m0 ¼ 0.1.
case. For the range of parameters considered in Figure 4a–c,
the effect of increasing a is qualitatively similar to that of
increasing K.
We have also examined mortality functions of the form
M ðuÞ ¼ m0 ð1 u b Þ: In the baseline case b ¼ 1; so that
M ðuÞdecreases linearly with increasing u. When b , 1; M ðuÞ
is convex; that is, it decreases rapidly with increasing u for
small u but decreases less rapidly for larger u. When b . 1;
M ðuÞ is a concave function of u that decreases slowly with
increasing u for small u but decreases more rapidly for larger
u. Changing K and m0 produce very similar qualitative effects
regardless of the value of b. The main effect of increasing the
parameter b is to increase the allocation to combating the
stressor when the level of damage is small. Profiles of mortality
over time are thus changed in a similar way to the change
induced by decreasing K in the baseline case (Figure 2).
In the baseline case the stressor is present for an exponentially distributed time with mean duration s ¼ 10. The qualitative effects on average mortalities of varying K are robust to
changes in both the mean duration of the stressor and the
shape of this distribution. In particular, effects are robust
when the duration of the stressor is constant, so that the
duration is predictable rather than exponential when it is
unpredictable (see, e.g., Figure 4d).
Effect of the duration of the stress period
We now examine the effect of the mean duration s when the
stressor is present for an exponentially distributed amount of
time with this mean. Figure 1b shows that as s increases u*(x)
decreases. This can be understood as follows. When the stress
period is short, there is not sufficient time for damage to build
up to high levels even if all resources are allocated to combating the stressor. Conversely, when the stress period is long,
damage will build up to high levels unless resources are put
into maintenance. Thus, for a given level of damage, the longer the stress period is expected to last, the greater the allocation to maintenance rather than combating the stressor.
The consequences for the mortality rates over time are as
illustrated in Figure 5a. As can be seen, when the mean duration s is short, the rate of mortality from the stressor is less
(especially initially) and the rate of mortality from damage is
greater (particularly at larger times) than when s is longer.
Average mortalities reflect this phenomenon. As Figure 6a
shows, increasing s increases the proportion of total mortality
that is due to the stressor at the expense of the mortality from
damage (both during and after the presence of the stressor).
This effect is robust regardless of the values of K and m0.
In all the above cases the duration of the stress period has
an exponential distribution. Under this distribution the end
of the stress period is very unpredictable. In contrast, we now
take the stress period to have a fixed duration, so that its end
is completely predictable. Comparing Figure 5a with 5b shows
that replacing an exponential duration with a predictable
duration of the same mean has a dramatic effect on how
the rates of mortality change over time. When the duration
has an exponential distribution, the mode is less than the
mean and most durations are shorter than the mean. The
organism can thus take a chance on the fact that the duration
will be short and hence initially allocate most of its resources
to combating the stressor, only reducing allocation if the stress
period does not end soon. In contrast, when the duration is
predictable this duration will not be shorter than the mean,
and the above strategy would allow too much damage to accumulate. Thus, there is initially a larger allocation to maintenance. Only when there is little time left before the stressor
disappears can the organism afford to divert most of its resources to combating it. Figure 6a and 6b compare average
mortalities under exponential and predictable durations. As
can be seen, the proportion of the total mortality due to the
1014
Figure 5
Rates of mortality from the stressor and damage while the stressor is
still present for different probability distributions on the duration
of the stress period. (a) Exponential duration with mean s.
(b) Predictable duration equal to s. m0 ¼ 0.1, K ¼ 0.0002 throughout.
stressor tends to be less for a predictable duration when s is
short and greater when s is long. It can also be seen that
replacing an unpredictable duration with a predictable duration increases the ratio of mortality from damage after the
stressor disappears to mortality from damage before it disappears. This is because the resources put into maintenance are
initially high and then decrease when the duration is predictable, whereas they vary in the opposite manner over time
when the duration is unpredictable. Figure 4b and 4d compare the effect of K under predictable and exponential durations, respectively. Again the poststress mortality from damage
is greater in the predictable case.
DISCUSSION
The results from this model confirm that in a situation of
resource restriction, the optimum strategy for resource allocation to combating an immediate physiological threat varies,
depending on (1) the cost to individual condition and (2) the
threat and duration of the stress period. The model assumes
that there is a cost associated with dealing with the stressor,
through the allocation of resources that can be divided between different physiological demands. In a situation where
there is no cost to combating a stressor, an animal would
allocate all available resources to this stressor, without compromising its own physiological condition. In this situation, an
animal would continue to allocate all the resources to protecting itself from the stressor until eventually a cost is incurred.
As our model shows, even if the cost of poor condition is small
Behavioral Ecology
Figure 6
The effect of the probability distribution on the duration of stress
period on average mortality, where the average is over possible
durations of the stress period. Sources of mortality are as in Figure 3.
(a) Exponential duration with mean s. (b) Predictable duration
equal to s. m0 ¼ 0.1, K ¼ 0.0002 throughout.
(small K) its effect can be significant; it is precisely when it is
small that most of the mortality is due to poor condition
(Figure 4). We would suggest therefore that even in conditions where the initial cost of protection appears be small, this
model is biologically relevant.
Second, our model assumes that resources can be shunted
between different physiological activities in order to either
maintain condition or to protect against a stressor. This assumption is likely to hold more or less true, depending on the
nature of the resource. For example, it is currently unclear
how the cost of mounting an immune response is mediated
and how biologically meaningful this cost is (Bonneaud et al.,
2003; Sandland and Minchella, 2003). Indeed, it has been
suggested that the cost of maintaining basic level immune
function that repels constant low-level pathogen attack may
be negligible (Derting and Compton, 2003). In contrast,
many recent studies suggest a cost associated with elevated
immune function (Ardia et al., 2003; Kilpimaa et al., 2004;
Moret and Schmidt-Hempel, 2000). Although caloric intake
plays an important role in determining the energy available to
combat disease (Klasing, 1998; Kramer et al., 1997; Nelson
et al., 2002), certain types of stressors may require specialist
resources (e.g., essential amino acids; Klasing and Calvert,
1999). Moreover, as increased dietary fat has been shown to
decrease immunocompetence, it seems unlikely that caloric
intake per se restricts immune function (Nelson et al., 2002).
Our model predicts that optimal resource allocation strategies are affected by the costs associated with incurring damage
to condition. At small values of K, the costs of sustaining
a given level of damage are low, and mortality increases very
McNamara and Buchanan • Stress, resource allocation, and mortality
1015
slowly with the accumulation of damage. Any marginal increase in damage therefore has very little impact on mortality.
For this reason at low K, it is beneficial to let damage build up
(Figure 1). This does not mean that the costs of damage are
necessarily unimportant, just because they are low, as the
cumulative damage could have significant effects on mortality
rates in the longer term, depending on the duration of the
stressor. We suggest this finding could have important implications for how animals deal with situations where incurring
a small decrease in condition does not lead to a large increase
in mortality risk. If there is a sufficient build up of damage to
individual condition, as a result of continued small amounts
of investment in protection against a current stressor, in the
long term there could be a considerable cost to individual
condition, resulting in increased mortality levels. We suggest
that an example of this can be seen in the behavioral changes
seen in predator-exposed prey. Here, prey respond to predators by changing their foraging strategies to reduce predation
risk and hence decrease food intake. They can afford to do
this because a small decrease in body reserves has small cost,
but the result of many such small decreases is a large decrease
in reserves, with a large associated cost. Although our model
only deals with a single stressful event, the above reasoning
may well also apply to repeated short stressful events. Predictions are, however, likely to depend on the length and predictability of occurrence of these episodes, so that this topic
needs further study.
Increasing the rate of mortality due to reduced condition
(increasing K) has little effect on mortality due to poor condition, during the period when the stressor exists (Figure 2)
because of a reallocation of the available resources. Under
such circumstances, individuals protect themselves against
reduced condition during the period of the stressor because
of the increased risk. This may be analogous to the response
of some organisms to cold periods in temperate regions. During these periods the risk of disease increases, for a given level
of the immune response, and resources are reallocated resulting in enhancement of the immune system (Nelson et al.,
2002). Results from both field and laboratory studies suggest
that these resource reallocation effects (Mercer et al., 2000)
appear to be in part stimulated by photoperiodic changes and
melatonin levels (Nelson, 2004), which are likely to act as
physiological cues to changing environmental conditions
(Nelson et al., 2002).
For the given expected length of the stress period, if the
actual duration of the period is short most of the mortality
may be due to poor condition, both during the stress period
and after (Figure 3a,b). As the actual duration increases, the
probability of death from both poor condition and the
stressor increase at an accelerating rate, with the stressor becoming proportionately more important as a threat of mortality (Figure 3). This is because the longer the stress period
lasts, the more resources are allocated toward maintaining
condition.
When the expected duration of the stress period is short an
animal can afford to devote all resources to combating the
stressor, allowing damage to accumulate during the short period. This strategy would be disastrous over a long period because damage would build up too much. Thus, the longer the
expected duration of the stress period, the less the allocation to
combating the stressor (and the greater the allocation to maintaining condition), even at the start of the stress period (Figure
5). Averaging over possible durations of the stress event we find
that for short expected duration most of the mortality is from
poor condition (Figure 6). In particular, most may be from
poor condition after the stressor is removed, especially when
stressor is present for a predictable duration (compare Figure
6a with 6b). Increasing the expected duration of the stress
period does not necessarily increase the probability of death
from poor condition (Figure 6). However, the probability of
death from the stressor rises rapidly (Figure 6). Thus, for those
environmental stressors that are liable to be prolonged, we
predict that almost all the mortality will be from the stressor
as opposed to the effects of poor condition.
It is worth noting that at a low risk of mortality from a given
level of condition (low K), condition may deteriorate to such
an extent that the proportion of individuals dying from poor
condition, particularly after the stressor has been removed, is
substantial (Figures 3a and 4). We would suggest that in such
circumstances where animals can sustain some damage with
lower mortality risk (e.g., when pathogen levels are low and
immunosuppression is less risky), resources are mainly aimed
at combating the stressor, so that the main risk to mortality
comes from poor condition both during the stressor and after
its disappearance. We suggest that these results may be analogous to the stress-induced changes in the mortality levels of
snowshoe hares (Lepus americanus) in the presence of predatory lynx (Lynx canadensis) (Boonstra et al., 1998). Such effects
have been implicated in mediating the population changes
seen during 10-year population cycles on exposure to variations in predator pressure (Boonstra et al., 1998). Sublethal
physiological effects within the prey populations (condition,
immunocompetence) are suggested to contribute to the population lag times when prey numbers continue to decline, even
in the face of declining predator pressure. During the decline
phase of the hare populations, hares showed increased cortisol
levels, reduced leucocyte counts, increased glucose mobilization, and higher body mass loss during winter compared to
hares from stable, low-density populations (Boonstra et al.,
1998). Increased mortality rates were seen in declining populations as a result of physiological stress rather than as a direct
result of predation and did not decline until the predation risk
declined (Boonstra et al., 1998). Our model may thus be entirely realistic in suggesting that under certain conditions a substantial proportion of the mortality can occur after the removal
of the stressor. Where high levels of damage are incurred in the
prey population, substantial declines in individual condition
occur with consequential high mortality rates, even after the
predatory threat (stressor) has declined.
Comparisons within Figures 5 and 6 (a and b) demonstrate
the fundamental importance of the degree of predictability of
the length of the stress period for the optimal allocation strategy. Where the duration is broadly predictable the organism
can predict its resource demands and make appropriate budgeting decisions regarding allocation. Resources allocated to
maintaining condition decline over time. As a result mortality
rates due to the stressor are initially high but decline over time
(Figure 5b). Mortality rates from reduced condition are initially low but increase at an increasing rate (Figure 5b). We
have illustrated the case where the duration is more unpredictable by an exponential distribution of stressor duration.
This distribution has the property that the modal duration is
less than the mean. For this case, at the start of the stressful
period the animal takes the risk that the duration of the stress
period will be substantially less than the mean duration, initially protecting against the stressor and allowing condition to
deteriorate. This results in an initial increase in mortality associated with poor condition. If the stress period does not
turn out to be short, the animal can no longer allow condition
to deteriorate so fast. Consequently, although the rise in mortality due to the stressor is initially delayed, it then rises faster
than that due to poor condition (Figure 5a). Some stressors
that are seasonal may be more predictable in duration, and we
presume that, in this case, physiological systems have evolved
to exploit photoperiodic cues regarding the duration of such
seasonal stressors (Nelson, 2004). But there are a considerable
Behavioral Ecology
1016
number of stressors whose durations are largely unpredictable
(e.g., extreme snowfalls, increased predator pressure). Our
model suggests that the optimal allocation strategy for dealing
with these types of stressors would be entirely different and
that the predictability of the duration has important consequences for the mortality patterns seen both during the stress
period and afterward.
Animals encounter many transient stressors, presumably
with consequences for resource trade-offs, and we hypothesize
that selection has favored the optimal physiological and behavioral responses to such events. In our model, we have considered a situation where an animal is exposed to a transient
stressor, for example a drop in temperature, and investment
in protection against this stressor has implications for the
maintenance of condition. During such a period of cold
stress, detrimental effects occur because under limited resource availability, the energy generated from body fat metabolism can be used either to produce body heat or to maintain
condition. The stressor causes a trade-off in resource allocation, and how the animal responds to the stressor determines
its probability of surviving the stress event, as well as the cause
of mortality (i.e., due to poor condition or the stressor, in this
case cold).
Here, we have suggested one particular situation, where the
animal is exposed to cold stress, which causes a reallocation of
resources from maintaining condition, making the animal
more susceptible to disease from reduced immunocompetence. In this case, as in the first row in Table 1, the sources
of mortality are cold (stressor) and disease (effect of poor
condition). It is important to emphasize however, that our
model should apply to animals facing many different types
of stressors (e.g., cold, heat, drought, predator pressure, food
shortages, disease, or disturbance) (e.g., Table 1) and that
these have a range of potential ways of affecting individual
condition through different limiting resources. Potential
limiting resources include fat reserves, muscle mass, protein
levels, micronutrient reserves (vitamins, minerals), or essential
amino acids. Under stress, the trade-offs in allocation of these
limited resources can affect measures of condition such as
immunocompetence, dominance status, or fur/feather condition. Examples of such hypothetical scenarios are given in
Table 1. We hypothesize, for example, that the sort of tradeoff proposed in our model occurs during the immunosuppression seen during chronic stress events (Nelson et al., 2002;
Råberg et al., 1998), as essential resources are reallocated away
from immune function to maintaining more essential metabolic functions during the stress challenge. This suggests that
such a response to chronic stress may be adaptive, despite the
fact that many stressed individuals appear to suffer increased
levels of disease and reduced condition both during and after
the stress event (Nelson et al., 2002). If this physiological
response is adaptive, despite the obvious costs of long-term
immunosuppression, we need not invoke functional explanations, such as the dangers of autoimmune disorders to explain
the evolution of immunosuppression as a response to chronic
stress (Råberg et al., 1998).
The concept of stress-mediated resource allocation has
been presented previously as the integration of physiological
systems that allow the minimization of allostatic load
(McEwen and Wingfield, 2003). Allostatic load is understood
to be cumulative physiological effects of stress that move the
animal away from its ideal state. Here we envisage our model
to be applicable to Type 1 allostatic load, where energy demands exceed supply and the animal moves toward an emergency life-history stage. In our model the animal does not
have the option to escape from the stressor. There is often
such an option in the wild, such as changing location to avoid
the effects of a storm (McEwen and Wingfield, 2003).
Recent work has demonstrated adaptive shifts in life-history
traits in relation to allocation of resources to immune challenges (Moret and Schmidt-Hempel, 2000). Such work confirms the resource-based trade-off that exists between survival
or reproduction and immunity (Ardia et al., 2003; Bonneaud
et al., 2003; Kilpimaa et al., 2004). Given such trade-offs, we
suggest that our model is realistic in suggesting that the optimal life-history strategy is based on the allocation of resources
between competing activities, including defense against stressors and the risk of disease (Moret, 2003). It is interesting to
note that such trade-offs are only seen under conditions of
limited resource availability (Moret, 2003). Thus, such tradeoffs are biologically meaningful under the stressful conditions
that are found in the wild, even if they are not under many
laboratory conditions. At the population level, stress effects on
individual condition could generate a range of delayed lifehistory effects, with considerable consequences for population
dynamics (Beckerman et al., 2002; Boonstra et al., 1998). As
a result, we suggest that such resource-based trade-offs could
have fundamental implications not only for individual lifehistory strategies but also for population dynamics.
APPENDIX
The dynamic programing equations
We are concerned whether an animal survives both the period
that a stressor is present and the subsequent recovery time. We
describe this as survival to full recovery. Let V ðx; tÞ be the
maximum probability of survival to full recovery given that
the stressor is present at time t and the level of damage at this
time is x. We formulate the dynamic programing equations
for V. To do so we first need two preliminary results.
The hazard function
We consider a stress period that lasts for a time that is a nonnegative random variable with probability density function
hðtÞ: Then given the stressor has not disappeared by time t,
the probability it will disappear between times t and t 1 Dt is
H ðtÞDt for small Dt; where the hazard function H(t) is given by
H ðtÞ ¼ R N
t
hðtÞ
:
hðvÞdv
Survival probability on disappearance of the stressor
Suppose that the stressor disappears when the level of damage
is x0 : After the stressor disappears the animal allocates all
resources to maintenance (u ¼ 0), so that its rate of change
of damage is xðtÞ
_ ¼ f ð0; xÞ: (note that xðtÞ
_
is negative.) The
probability the animal manages to reduce the damage to level
0 before damage kills it is thus
Rðx0 Þ ¼
Z
x
0
DðxÞ
dx:
f ð0; xÞ
Now suppose that the stressor is still present at time t and
that the animal has level of damage x at this time. We consider
the optimal allocation to combating the stressor over a short
time interval of length Dt: Suppose that during this interval
the animal allocates a proportion u of resources to combating
the stressor. The animal then survives until time t 1 Dt with
probability Su ðx; tÞ ¼ ð1 M ðuÞDtÞð1 DðxÞDtÞ: If the animal
survives, its level of damage at this time is x 1 f ðu; xÞDt: At this
time the stressor is still present with probability 1 H ðtÞDt
and has disappeared with probability H ðtÞDt: Thus, given
McNamara and Buchanan • Stress, resource allocation, and mortality
1017
the allocation after time t 1 Dt is optimal, the animal survives
to full recovery with probability
Klasing K, Calvert CC, 1999. The care and feeding of an immune
system: an analysis of lysine needs. In: Protein metabolism and
nutrition (Lobley GE, White A, MacRae J, eds). Wageningen,
The Netherlands: Wageningen Press; 253–264.
Kramer T, Moore R, Shippee R, Friedl K, Martinez-Lopez L, Chan M,
Askew E, 1997. Effects of food restriction in military training on
T-lymphocyte responses. Int J Sports Med 18:S84–S90.
Lochmiller R, Deerenberg C, 2000. Trade-offs in evolutionary immunology: just what is the cost of immunity? Oikos 88:87–98.
Lyman CP, 1982. Why bother to hibernate? In: Hibernation and
torpor in mammals and birds (Lyman C, Willis J, Malan A, Wang
L, eds). San Diego, California: Academic Press; 1–11.
McEwen B, Wingfield JC, 2003. The concept of allostasis in biology
and biomedicine. Horm Behav 43:2–15.
McGraw KJ, Ardia DR, 2003. Carotenoids, immunocompetence, and
the information content of sexual colors: an experimental test.
Am Nat 162:704–712.
Mercer JG, Adam CL, Morgan PJ, 2000. Towards an understanding of
physiological body mass regulation: seasonal animal models. Nutr
Neurosci 3:307–320.
Moret Y, 2003. Explaining variable costs of the immune response:
selection for specific versus non-specific immunity and facultative
life history change. Oikos 102:213—216.
Moret Y, Schmidt-Hempel P, 2000. Survival for immunity: the price of
immune system activation for bumblebee workers. Science
290:1166–1168.
Nelson R, 2004. Seasonal immune function and sickness responses.
Trends Immunol 25:187–192.
Nelson R, Demas G, 1996. Seasonal changes in immune function.
Q Rev Biol 71:511–548.
Nelson R, Demas G, Klein S, Kriegsfeld L, 2002. Seasonal patterns
of stress, immune function and disease. Cambridge: Cambridge
University Press.
Padgett D, Glaser R, 2003. How stress influences the immune response. Trends Immunol 24:444–448.
Råberg L, Grahn M, Hasselquist D, Svensson E, 1998. On the adaptive
significance of stress-induced immunsuppression. Proc R Soc Lond
B 265:1637–1641.
Rands SA, Cuthill IC, 2001. Separating the effects of predation risk
and interrupted foraging upon mass changes in the blue tit Parus
caeruleus. Proc R Soc Lond B 268:1783–1790.
Sandland GS, Minchella DJ, 2003. Costs of immune defense: an
enigma wrapped in an environmental cloak? Trends Parasitol
19:571–574.
Selye H, 1976. The stress of life. New York: McGraw-Hill Book
Company.
Svensson E, Raberg L, Koch C, Hasselquist D, 1998. Energetic stress,
immunosuppression and the costs of an antibody response. Funct
Ecol 12:912–919.
von Schantz T, Bensch S, Grahn M, Hasselquist D, Wittzell H, 1999.
Good genes, oxidative stress and condition-dependent sexual signals. Proc R Soc Lond B 266:1–12.
Wingfield JC, Jacobs J, Hillgarth N, 1997. Ecological constraints and
the evolution of hormone-behavior interrelationships. In: The
integrative neurobiology of affiliation (Carter CS, Lederhendler
II, Kirkpatrick B, eds). New York; Academy of Sciences; 22–41.
Wingfield JC, Sapolsky RM, 2003. Reproduction and resistance to
stress: when and how. J Neuroendocrinol 15:711–724.
Wu ðx; tÞ ¼ Su ðx; tÞfð1 H ðtÞDtÞV ðx 1 f ðu; xÞDt; t 1 DtÞ
1 H ðtÞDtRðx 1 f ðu; xÞDtÞg:
Finally,
V ðx; tÞ ¼ max Wu ðx; tÞ;
where the maximum is over u in the range 0 u 1: The
value of u achieving the maximum is u*(x, t).
This problem has an infinite time horizon. In computation
we take a finite time horizon, T, and set V ðx; T Þ ¼ 0 for all x,
but choose T sufficiently large so that increasing it further
does not alter numerical results.
To compute u*(x, t) we solve the above equations, using
a fine grid of t values as decision epochs. Damage is modeled
on a fine grid of x values, and linear interpolation between
grid points is used in updating changes in damage.
We thank Mark Viney, John Wingfield, and Ron Ydenberg for comments on a previous version of this manuscript. J.M.M. was supported
by a Fellowship from the Leverhulme Trust.
REFERENCES
Ardia DR, Schat KA, Winkler DW, 2003. Reproductive effort reduces
long-term immune function in breeding tree swallows (Tachycineta
bicolor). Proc R Soc Lond B 270:1679–1683.
Beckerman A, Benton TG, Ranta E, Kaitala V, Lundberg P, 2002.
Population dynamic consequences of delayed life-history effects.
Trends Ecol Evol 17:263–269.
Bonneaud C, Mazuc J, Gonzalez G, Haussy C, Chastel O, Faivre B,
Sorci G, 2003. Assessing the cost of mounting an immune response.
Am Nat 161:367–379.
Boonstra R, Hik D, Singleton GR, Tinnikov A, 1998. The impact of
predator-induced stress on the snowshoe hare cycle. Ecol Monogr
79:371–394.
Broom D, Johnson K, 2000. Stress and animal welfare. Dordrecht:
Kluwer Academic Publishers.
Brown R, 1994. An introduction to neuroendocrinology. Cambridge:
Cambridge University Press.
Buchanan KL, 2000. Stress and the evolution of condition-dependent
signals. Trends Ecol Evol 15:156–160.
Cichon M, Chadzinska M, Ksiazek A, Konarzewski M, 2002. Delayed
effects of cold stress on immune response in laboratory mice. Proc
R Soc Lond B 269:1493–1497.
Derting T, Compton S, 2003. Immune response, not immune maintenance, is energetically costly in wild white-footed mice (Peromyscus
leucopus). Physiol Biochem Zool 76:744–752.
Ewenson E, Zann R, Flannery G, 2003. PHA immune response assay in
captive zebra finches is modulated by activity prior to testing. Anim
Behav 66:797–800.
Geiser F, 2004. Metabolic rate and body temperature reduction during
hibernation and daily torpor. Annu Rev Physiol 66:239–274.
Graveland J, Drent RH, 1997. Calcium availability limits breeding
success of passerines on poor soils. J Anim Ecol 66:279–288.
Harshman LG, Schmid JL, 1998. Evolution of starvation resistance in
Drosophila melanogaster: aspects of metabolism and counter-impact
selection. Evolution 52:1679–1685.
Houston AI, Welton NJ, McNamara JM, 1997. Acquisition and maintenance costs in the long-term regulation of avian fat reserves.
Oikos 78:331–340.
Humphries MM, Thomas DW, Kramer DL, 2003. The role of energy
availability in mammalian hibernation: a cost-benefit approach.
Physiol Biochem Zool 76:165–179.
Kilpimaa J, Alatalo RV, Siitari H, 2004. Trade-offs between sexual
advertisement and immune function in the pied flycatcher (Ficedula
hypoleuca). Proc R Soc Lond B 271:245–250.
Klasing K, 1998. Nutritional modulation of resistance to infectious
diseases. Poult Sci 77:1119–1125.