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Evolutionary Ecology Research , 1999, 1 : 987–
© 1999 Dan Cohen
Investing for survival of rare severe stresses in
heterogeneous environments
Dan Cohen^1 * and Marc Mangel^2
(^1) Department of Evolution, Systematics and Ecology, The Silberman Institute of Life Sciences,
The Hebrew University of Jerusalem, Jerusalem 91904, Israel and 2 Department of Environmental Studies, University of California, Santa Cruz, CA 95064, USA
ABSTRACT
We model the ESS investment of limiting resources for survival of rare severe stresses, with an emphasis on investment by trees in mechanical strength for survival of storm stresses. The basic model includes the effects of the fitness benefits of increased survival, the cost in reproduction, the effects of the distributions and timing of the stress in the population, and of stress-independent mortality. The ESS investment in survival of unsynchronized stresses increases if stress-independent mortality decreases, and if the cost of the resistance decreases. The ESS investment increases, and then decreases, when the probability of extreme stresses increases. The fitness of each individual increases if the allocation of resources for resisting stress is optimally adapted to its local stress probability distribution. A Bayesian model is constructed for updating the estimate of the local stress probability distribution, which each individual can get from the exposure to sub-lethal stresses during its life. This estimate can then be used for the ESS investment. The results are discussed and applied to a wider class of organisms, stresses and resistance mechanisms.
Keywords : Bayesian updating, ESS survival, exponential distribution, rare stress, storm damage, threshold stress, trees.
INTRODUCTION
The allocation of limiting resources between reproduction and survival is a fundamental problem in the evolution of life-history characteristics of most plants and animals (e.g. Stearns, 1976, 1992; Sarukhan and Dirzo, 1984). Usually, mortality or damage caused by environmental stresses or by predation can be reduced or avoided by greater investment in resistance or defence mechanisms – for example, in chemical and mechanical defences of plants against herbivory (Karban and Baldwin, 1997) – and in mechanical support against wind damage (Coutts and Grace, 1995). However, such investment in resistance usually results in reduced growth rate or reproduction. Long-term evolution is expected, therefore, to lead to an investment allocation strategy that maximizes the net gain in long-term lifetime fitness. In general, therefore, a higher investment in resistance is expected if the cost in reproduction of the resistance decreases.
- Author to whom all correspondence should be addressed. e-mail: dancohen@vms.huji.ac.il
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In addition, a higher stress-independent mortality is expected to decrease the benefit of investing resources in survival against rare stresses, because the benefit will affect a smaller fraction of surviving individuals. The optimal investment in surviving stresses is expected, therefore, to decrease with increasing stress-independent mortality for the same cost of investment. The optimal investment in resisting stress may be a decreasing or an increasing function of the intensity of the stress or the effectiveness of the investment, depending on whether they increase or decrease the marginal benefit of investment in resistance relative to the marginal increase of the cost. This is a general property of optimal allocation models in economics (e.g. Varian, 1984) and in ecology (e.g. Givnish, 1986). As different individuals of the same species may be exposed to different environmental conditions with different probability distributions of stress or of the cost of defence, fitness could be increased considerably if the allocation of resources to resisting stress by each individual were adapted to its local conditions. An environmentally induced phenotypic increase of investment in resistance as a response to stress or to correlated signals has been reported, including for wind (Coutts and Grace, 1995) and herbivory (Karban and Baldwin, 1997). A basic difficulty with the induced allocation of resources to defence or resistance is that it must depend on limited and unreliable information that each individual can have about the probability distribution of stresses in its immediate environment. This is especially true for resistance against rare extreme lethal events that occur infrequently in the lifetime of the individuals, because direct learning by experience is inherently impossible in such cases. The optimal induced investment in resistance has to take into account the inherent uncertainty of this information. In this paper, we construct a simple model for the evolutionarily stable investment in resistance against environmental stresses. We provide an expression for the optimal invest- ment as a function of the probability distribution of the stress and of the reproduction cost function of the resistance. We then analyse and discuss the problem of the optimal use of environmental signals for inducing and regulating the resistance investment, and propose some partial solutions.
THE BASIC MODEL
We consider the investment in mechanical strength by trees against the mechanical stress and damage caused by wind and snow. In this system, both the mechanical investment and the damage can be observed and measured easily, and there are many reported observations and measurements of the effects of wind intensity and other local conditions on wind damage in trees (e.g. Foster, 1988; Quine, 1988, 1995; Foster and Boose, 1995). For example, an average of 0.2–0.4% mean destruction per annum was observed in New Zealand soft wood plantations, with a large variation between sites. The worst damage was caused by sudden increases in exposure (Somerville, 1995). Also, many trees modify their mechanical strength and structure in response to local wind exposure and other environmental factors (Mattheck, 1991; Stokes et al. , 1995; Telewski, 1995). The relative investment in mechanical strength in trees increases, and their wind survival decreases, as trees increase in size and age (Nielsen, 1995; Telewski, 1995). The results of this model are quite general, however, and can be applied to a much wider class of organisms and resistance mechanisms against other rare stresses. Analogous
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A rare type i ( Ni N ) will change its density N (^) i by a constant growth factor ri in any one year. The numbers of the rare type i in year t + 1 are related to those in year t by
Ni ( t + 1) = Ni ( t )(1 − m ) E { S ( I , R (^) i )}
- ( N ( t )[1 − (1 − m ) E { S ( I , R *)}] + N (^) i ( t )[1 − (1 − m ) E { S ( I , R (^) i )}])
Ni ( t )(1 − C ( Ri )) Ni ( t )(1 − C ( Ri )) + N ( t )(1 − C ( R *))
(1a)
where the first term on the right-hand side of (1a) is the mean survival of the rare type and the second term is its mean occupation of vacant sites. Defining the annual growth factor ri = [ N (^) i ( t + 1)]/[ Ni ( t )], and assuming that N (^) i ( t ) N ( t ), we get the simplified expression for the growth of the rare type:
r (^) i = (1 − m ) E { S ( I , Ri )} + [1 − (1 − m ) E { S ( I , R *)}]
1 − C ( Ri ) 1 − C ( R *)
(1b)
To find the conditions for the evolutionarily stable investment in resistance R *, we differ- entiate r (^) i with respect to Ri and set R (^) i = R *, which gives
∂ ri ∂ R (^) i
= (1 − m )
∂ E ( S ( I , Ri )) ∂ Ri
1 − (1 − m ) E ( S ( I , R )) 1 − C ( R *)
d C d Ri
[ R (^) i = R *] (2a)
There is an ESS R * = 0 if the right-hand side of (2a) is negative at R (^) i = 0, that is, when
∂ E ( S ( I , Ri )) ∂ R (^) i
1 − (1 − m ) E ( S ( I , R )) (1 − m )
d C d Ri
1 − C ( R *)
[ R (^) i = R * = 0.0] < (^0) (2b)
For ESS R * > 0, we set the right-hand side of (2a) equal to 0, and R (^) i = R *, to obtain an implicit equation for the ESS R * > 0:
∂ E ( S ( I , R )) ∂ R
1 − (1 − m ) E ( S ( I , R *)) (1 − m )
d C d R
1 − C ( R *)
[ R (^) i = R *] (3)
The general solution for the ESS R * depends on the cost function, the survival function at any level of stress, the probability distribution of the stress, and the non-stress mortality m. Even at this general level of analysis, it can already be seen that R * = 0 becomes more likely and R * > 0 decreases as the mortality coefficient m increases. R * = 0 is also more likely when the derivative of the cost function increases and the derivative of expected survival decreases at R = 0. In the following sections, we derive the ESS R * for several representative simplifying assumptions about the characteristic functions of the model.
A threshold survival function
Let us assume a threshold survival function such that
S ( I , R ) =
1 if I < AR 0 otherwise
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where A is a coefficient of effectiveness of the investment in resisting the stress. A threshold survival function is a reasonable assumption for the damage caused by uprooting or breaking of trees by extreme wind stress. The mean annual expected survival in the population is
E { S ( I , R )} =
AR 0 f ( I )d I = F ( AR ) (5)
where f ( I ) is the annual probability density function of the wind stress I , and F ( I ) is the cumulative probability distribution of I. The implicit equation for the ESS R * > 0 (equation
- then becomes:
(1 − m ) A f ( AR *) =
1 − (1 − m ) F ( AR *) 1 − C ( R *)
d C d R
[ R (^) i = R *] (6)
For simplicity, we replace R * by R and rearrange equation (6) as
d C d R
1 − C ( R )
(1 − m ) A f ( AR ) 1 − (1 − m ) F ( AR )
We call the left-hand side of (7) the cost derivative function and the right-hand side of (7) the benefit derivative function. We can also use equation (7) to determine when R * = 0. In particular, if C (0) = 0, and if d C /d R [ R = 0] exceeds the right-hand side of (7) at R = 0, then R * = 0. Equation (7) is the implicit solution for ESS R * > 0.
The exponential stress function
Next, we specify the probability distribution of the stress by assuming that it is exponen- tially distributed with parameter L = 1/ I (^) m , so that
P r{ i < I < i + di } = L e− Li^ di = 1/ Im e− I / I^ m^ (8)
where L is 1/(mean I = I (^) m ). The exponential probability distribution function of wind stress is a good approximation for the higher end of the observed and measured distributions of wind speed and stress (e.g. Hanna et al. , 1995). The cumulative probability distribution then becomes F ( I ) = 1 − e− I / Im. In this case, the condition (7) for ESS R * > 0 becomes
d C d R
1 − C ( R )
(1 − m )( A / Im ) e−( A / Im ) R m + (1 − m ) e−( A / I^ m ) R^
If C (0) = 0, the condition that ESS R * = 0 is now
d C d R | R = 0
≥ (1 − m ) A / I (^) m (10)
The condition for ESS R * = 0 is satisfied over a smaller range of parameter values when the effectiveness coefficient A increases and the mean stress Im decreases, and when the mortality coefficient m decreases. The benefit derivative function – that is, the right-hand side of (9) – is a decreasing function of R for m > 0, and decreases more strongly as m increases. Thus, R * > 0 always decreases when m increases (Fig. 1c). In equations (9) and (10), and in all the subsequent equations, the effectiveness ratio A / I (^) m can be considered as one variable, which scales the effectiveness coefficient A in units of the mean stress Im.
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function of mean stress if R * is less than R b in different habitats. Thus, the ESS R * > 0 always increases and then decreases as a function of increasing effectiveness ratio A / Im , with an intermediate maximum for any particular cost and mortality parameters. The effects of A and Im on R * are, therefore, always in opposite directions. The crossover point between any two particular levels of effectiveness ratio A / Im , R b2, can be obtained by numerically solving equation (11). Because it is only the ratio A / I (^) m that enters into the equations, we set A = 1. For simplicity we let 1/ Im = L , and for definiteness assume that L 1 > L 2. The two benefit derivative functions then intersect at the value of R b such that
L 1 e−( L^1 AR ) m + (1 − m ) e−( L^1 AR )^
L 2 e−( L^2 AR ) m + (1 − m ) e−( L^2 AR )^
R b2 is an increasing function of the difference L 1 − L 2 for any given L 2 , and is a decreasing function of the mortality coefficient m (see Figs 1a and b).
The cost function
Plausible simple assumptions for the relative cost function C ( R ) are that C (0) = 0, it is monotonically increasing, and becomes 1 at high levels of R = 1. We choose C ( R ) = 1 − (1 − R ) k , where k determines the initial slope and curvature of the cost function. The cost derivative function is
d C d R
1 − C ( R )
k 1 − R
This cost function satisfies the reasonable assumption that, in general, the relative loss of reproductive output is an increasing function of the investment in survival. In this case, the derivative of the relative cost is an increasing function of R and k. Combining this cost derivative function with the ESS condition (9) gives the following solution for the ESS R * > 0:
k 1 − R
(1 − m ) A / Im e− A / Im^ R m + (1 − m ) e− A / ImR^
with ESS R * = 0 if k > (1 − m ) A / Im. Note that if m = 0, the benefit derivative is constant at A / Im , and (13) can be solved analytically for the ESS value of R *, which is
R * = 1 −
k A / Im
In this case, ESS R * = 0 if k > A / I (^) m. This result is a special case of the change of the crossover between increasing and decreasing R * as a function of A / Im and m. With m = 0, R b is infinitely high, so that the ESS investment always increases with increasing A and with decreasing mean intensity of stress I (^) m. We interpret it as follows: When the only source of mortality is the rare but powerful stress, the contribution of stress survival to future reproduction is enhanced, so that it is advantageous to invest even more to ‘assure’ survival when mean stress decreases.
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Numerical results
Figures 1, 2 and 3 provide graphical illustrations of the effects of the parameters of the basic model on the benefit and cost derivatives, and on the resulting solutions for ESS R * and for the average ESS survival of stress. R is the fraction of resources invested in resistance, so the range of R is 0 ≤ R ≤ 1. We chose a representative range of mean wind speed between 0 and 20 m · s−^1 , which is typical of natural mean wind speeds in exposed conditions (e.g. Hanna et al ., 1995). The range of the effectiveness coefficient A was chosen to include values both above and below I (^) m , mostly between 5 and 50. We chose a natural range of stress-independent annual mortality coefficients m between 0.005 and 0.1 per year, with 0.02 as the standard reference. We chose values of the cost coefficient k between 1 and 0.3, to represent varying costs and non-linearities. Figure 1a illustrates the benefit derivative function. As noted above, the benefit derivative function increases with increasing A / Im at low R below the crossover point, and decreases with increasing A / Im above the crossover point, as illustrated in Fig. 1d. This is in contrast to the effect of the mortality coefficient m , which always decreases the benefit derivative function, as shown in Fig. 1c. Numerical examples of the cost derivative function are illustrated in Fig. 1b. The solutions for ESS R * were obtained by solving numerically equation (13), using the FindRoot function in the Mathematica 3.0 program by Wolfram Research. The number of iterations was varied between 50 and 100, and the damping factor between 1 and 0.1, to obtain efficient convergence. ESS R * is an increasing function of mean stress Im at low levels of Im and a decreasing function of I (^) m at high levels of Im (Fig. 2a). The effect of A is similar in the opposite direction, because the effect depends only on the ratio A / I (^) m. The range of k has a large effect on R * (Fig. 2b), in comparison with the effects of the range of m (Fig. 2c). Figures 3a, b and c illustrate the effects of the parameters of the model on the ESS average survival of stress when the ESS R * is invested:
average ESS survival of stress = 1 − exp(− A / I (^) m R *)
Increasing A greatly increases ESS average survival because it scales the mean stress (Fig. 3a). Decreasing the cost coefficient k has a similar large effect (Fig. 3b). Increasing m has a relatively small effect on the ESS average survival, because m has a smaller effect on R * (Fig. 3c). A set of reference curves of unoptimized average stress survival at increasing constant levels of R between 0.1 and 1.0 are plotted in Fig. 3d.
PART II: OPTIMAL INVESTMENT IN A HETEROGENEOUS ENVIRONMENT
We now assume that the environment includes patches of habitats with different probability distributions of stress damage, either because of a different exposure to strong winds, or because the ground support changes the damage probability of any given wind stress. We also assume that the seeds produced in any one-habitat patch are uniformly dispersed over a large enough area that includes all the habitats in their representative proportions. We denote the fraction of the total habitat that is type j by p (^) j and the probability of surviving a stress of intensity I in habitat j when the investment is R by S (^) j ( I , R ).
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In this case, the condition for the ESS R * = 0 is
k > (1 − m ) A [ pL 1 + (1 − p ) L 2 ] (17)
which is determined by the harmonic mean of Im in the two habitats. The ESS R * can be calculated by numerical solution of equation (16) for any levels of mean stress in the habitats and their proportions in the area. We used equation (16) to calculate specific numerical examples of ESS R * and of ESS survival of stress in two habitats as a function of their mean stress and of the proportion p of habitat 1 with the lower mean stress in the environment. The results are plotted in Figs 4a and b. As discussed in the previous section, the ESS R * in each of the two habitats alone may be an increasing or decreasing function of I (^) m. ESS R * decreases with mean stress below the intersections of the benefit derivative functions at the two habitats and increases above the intersections of the benefit derivative functions at the two habitats.
Fig. 3. The average ESS survival of wind stress with R = R *, ESURV, as a function of the mean stress I (^) m. E ( S ( I , R *)) = 1.0 − exp(− A / Im R *). (a) The effect of the effectiveness coefficient A on average ESS survival of stress: A = 5, 10, 20, 50; m = 0.02, k = 1.0. (b) The effect of the cost coefficient k on average ESS survival of stress: k = 0.3, 0.5, 1.0; A = 10, m = 0.02. (c) The effect of the mortality coefficient m on average ESS survival of stress: m = 0.005, 0.02, 0.1; A = 10, k = 1.0. (d) The unoptimized reference curves of the average survival of wind stress as a function of Im , at different constant levels of R = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0; A = 10.
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The informational problem and learning by sampling
Clearly, a better investment strategy for all the trees would be to sense in some way in which habitat they grow, and invest the optimal amount for the characteristic stress probability distribution in each habitat. For simplicity, we consider only two habitats, but our formula- tion can be used in general. Individual trees may obtain information about their habitat type in many different ways. Here we analyse the most direct way of obtaining information: by learning from the experi- ence of previous stresses. It is reasonable to assume that, below the threshold, wind stresses can be perceived by trees during their lifetime, which could then act as signals that convey information about the stress probability distribution at each particular location. These signals are then used to regulate the allocation of resources to investment in mechanical strength. The informational problem is thus for each tree to experience a series of stresses and to estimate, given these data, the probability that the tree is located in habitat 1 or habitat 2. We do this by Bayesian analysis (Hilborn and Mangel, 1997). Assume that the current level of investment is R. The tree can survive any stress less than AR , so we set a threshold stress I th = AR. Consequently,
P r{tree experiences and survives a stress of intensity I } =
L exp[− LI ] 1 − exp[− LI th]
Given that a tree has survived an experienced stress I , we compute the posterior probability that it is in habitat 1 according to:
P r{tree is in habitat 1, given it experiences and survives a stress of intensity I } =
P r{in habitat 1 and observe and survive I } P r{observe and survive I }
Fig. 4. (a) The ESS investment in resistance R * by a mixed population in a heterogeneous environ- ment with two habitats, RSSHet, as a function of changing the fraction p of habitat 1 with the lower mean stress: Im (1) = 2, Im (2) = 8, A = 10, m = 0.02, k = 1.0. (b) The ESS average survival of wind stress, ESURVHet, in habitat 1 (continuous line) and in habitat 2 (dashed line) as a function of the fraction p of habitat 1 in the environment.
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seed production of investment in resistance is not very high for moderate levels of invest- ment (i.e. the relative cost coefficient k is considerably less than 1). The most interesting, but less obvious, results of the model are those that predict an increasing and then decreasing dependence of ESS R * on the ratio between the effectiveness coefficient A and the mean wind speed I (^) m over a range of the other parameters (see Figs 2a, b, c). These results are caused by the effects of A / Im on the benefit derivative function (see Fig. 1d). Although this is a widely known property of optimal allocation models in economics (e.g. Varian, 1984) and in ecology (e.g. Givnish, 1986), we did not expect it to be so pronounced in the optimal investment by trees in wind resistance. The intuitive conventional wisdom of most evolutionary ecologists and of forest ecologists in particular (e.g. Telewski, 1995), predicts an increasing optimal investment in defence or resistance with an increasing probability of higher stress. This would apply in general also to other types of stresses, and not just to investment in wind resistance. Typical
Fig. 5. The effects of experiencing stresses over 20 years on updating the estimates of the probabilities of trees in heterogeneous environments for being in habitat 1 or in habitat 2. The Bayesian updating is implemented by a recursive application of equation (20) to a sequence of 20 random truncated samplings of wind stresses from the distributions in the two habitats. As an example, we consider trees with A = 10, k = 1, m = 0.02, with I (^) m 1 in habitat 1 = 2.0 and I (^) m 2 in habitat 2 = 10. We assume that, a priori , the two habitats occupy the same area; so we set the initial probability for being in habitat 1 = p = 0.5 and find that the corresponding initial ESS investment is R * = 0.38 by solving equation (16). (a) The stresses experienced (dotted line) and estimate of p (solid line) for a tree in habitat 1. (b) The same for a tree in habitat 2. Note the difference in scale. (c) The separation in the levels of optimal investment in resistance for the trees in habitat 1 (dashed line) and in habitat 2 (solid line), as calcu- lated by equation (16) with the updated estimate p of the probability for being in h 1 in each habitat.
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observations and measurements of investment of resources for defence or resistance against stresses report an increased level of investment with increasing levels of wind stress (e.g. Ennos, 1995; Nielsen, 1995; Stokes et al. , 1995; Telewski, 1995) and for herbivory defence (e.g. Karban and Baldwin, 1997). Trees or saplings that are exposed to higher wind stress or are artificially shaken produce more mechanical support tissues, whereas protected or artificially supported trees produce less. The results of our model may in part be explained by the assumption that, above the threshold, lethal wind damage only occurs with a moderate to low probability. On average, this provides long intervals for producing more seeds by trees that invest less in resistance. The predictions of our model on the effects of increasing mean wind stress on the ESS investment in resistance are consistent with the observations over the lower range of mean wind stress. Our model predicts that, as mean wind stress increases, the increased ESS investment in resistance will overcome the increased stress up to some point close to the maximal ESS investment (e.g. compare Fig. 3a with Fig. 3d). This means that the average natural optimal survival of wind stress in trees is predicted to remain high over a wide range of mean wind stress up to some critical level, above which it is predicted to decrease very sharply. This prediction is consistent with the observation that the proportion of trees killed by tornados was similar in locations with very different wind exposures within the forest stands (Dyer and Baird, 1997). Apparently, trees exposed to a higher wind stress for a long time had invested sufficiently in stronger mechanical support to survive to the same extent the higher wind stress. Support for the predictions of the model of decreased optimal survival over the higher range of mean wind stress is provided by the observations that tree species with wind- resistant tall trunks do not grow at sites with very high wind stresses. Depending on the values of the parameters, our model predicts that there is a level of mean wind stress above which the ESS is not to invest in resisting wind stress (i.e. a tree life form is not optimal; see Figs 3a and b). Trees could probably survive and grow in such high wind stress if they invest a sufficiently high fraction of their resources in resistance. However, under these conditions, the marginal benefit presumably becomes less than the marginal cost. Our model is consistent with the observed allocation of mechanical support tissues within a tree to those parts that experience the largest stress caused by deformation or strain (e.g. Ennos, 1995), because this is the expected optimal within-tree allocation for any given total investment. Increased deformation by mechanical stress is known to be the mechanism that causes both the overall and the local adaptive increased production of mechanical support tissue in trees in response to wind stress (Mattheck, 1991; Telewski, 1995). Increased production of the plant hormone ethylene, probably induced by mechano- receptors in cell walls, is probably one of the immediate causes of the physiological responses (Telewski, 1995). Typical growth responses to increased mechanical shaking or wind stress are increased stem diameter, decreased elongation, and decreased elongation of upper branches. These findings are entirely consistent with the predictions of our model. Analogous with this is the increased synthesis of defence chemicals by plants in response to herbivore damage (e.g. Karban and Baldwin, 1997). However, our model predicts in addition that the increased investment in defence would be a cumulative response to a number of exposures to rare stresses over a long time. This prediction is consistent with the finding that trees are generally ‘over-designed’ in response to the mechanical stresses to which they had been exposed in the past, and that rare
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