Life history evolution

Life history evolution Key concepts ˆ Cole s paradox ˆ Tradeoffs ad the evolution of iteroparity ˆ Bet hedging in random environments Life history theory The life history of a species consists of its life cycle together with the vital rates, including age- or stage-specific reproduction and survival. Life histories are quantitatively characterized by the schedules of reproduction and survival. Previously we described a range of different life histories (for instance, different survivorship curves) and how these affected population dynamics. Life history theory typically asks two kinds of questions: ˆ What schedules of reproduction and survival are most adaptive given the constraints provided by life cycle and environment? ˆ How do life cycles evolve given the constraints of history (for instance canalised developmental pathways) and environment? In what follows we will assume that adaptation acts to maximize fitness, defined either as lifetime reproductive success or population growth rate (which will often be equivalent). Such fitness is typically constrained by material constraints (e.g., resources) resulting in trade-offs, such as whether to invest resources in reproduction or survival. The problems of 1

2 LIFE HISTORY EVOLUTION life history theory all derive from considering the kinds of constraints that species face and their effects on fitness. Evolution of iteroparity Semelparity refers to life histories in which an organism reproduces on just one occasion during the lifespan. Annual plants are semelparous. By contrast, iteroparity is a life history in which an organism reproduces multiple times during the lifespan. Iteroparous species are perennial. Intuitively, one expects iteroparity to come at a cost: the metabolic costs of overwintering will be much lower for a small propagule than a mature, reproductive individual. For example, to accomplish perenniality deciduous trees cease photosynthesis and have evolved highly complicated adaptive responses for food storage, not to mention structures that can withstand the weight of snow and ice and water loss due to dessication. Similarly, many mammals have evolved to hibernate and birds have evolved to migrate, each of which is energetically costly. Indeed, iteroparity is a very common life history across many taxonomic groups. For iteroparity to be common, one expects that it provides a significant fitness benefit. That fitness benefit should be measured in terms of the cost of producing additional individuals in each reproductive season. That is, to calculate this benefit we should compare the costs of investing resources in overwintering (yielding the benefit of a second entire season of reproduction) with the alternative of immediately investing in additional reproduction. A little reflection shows that if both new offspring and overwintering adults exhibit the same reproductive output then the fitness gain from iteroparity is the same as that which would be obtained by producing merely a single additional propagule in the first year. The two-stage life cycle is indifferent to whether an individual overwinters as a seed or as a fully growth adult. This is Cole s paradox (L.C. Cole, 1954). We can formalize this idea using the stage-structured demographic theory we developed previously. For our purposes, we will refer to individuals reproducing for the first time as juveniles and individuals that have reproduced previously and overwintered as adults. Assuming there is no density dependence, then this two stage life history has four parameters (fecundity of juveniles, f j ; fecundity of adults, f a ; survivorship of juveniles to become adults, s j ; and survivorship of adults to reproduce again, s a ). Population dynamics will be given by the system of equations n t+1 = Ln t (1)

EVOLUTION OF ITEROPARITY 3 with projection matrix ( ) fj f L = a. (2) s j s a We know from our study of stage-structured populations that the asymptotic growth rate of the population is given by the dominant eigenvalue. In the case of a 2 2 matrix the eigenvalues are given by the quadratic formula λ = f j 2 + s a 2 ± fj 2 2f js a + s 2 a + 4f a s j. (3) 2 Provided that the parameters are all non-negative (as they must be in this case), the dominant eigenvalue will the solution obtained by adding the final term. Thus, we now have a formula for the fitness of the two-stage life history (i.e., the forumla in equation 3), that may take semelparity as a special case (when s j =0 then λ = f j ). We can verify Cole s Paradox by examining some special cases. For instance, we might assume an annual life history with 11 offspring. In this case we have (from the fundamental equation) semelparous fitness λ s = 11 or the projection matrix from the stage-structured theory, L s = ( ) 11 0, (4) 0 0 for which we can determine (either from the equation above or numerically) the dominant eigenvalue, λ s = 10. In contrast, the dominant eigenvalue of the iteroparous projection matrix, L i = ( ) 10 10, (5) 1 1 is λ i = 11. Of course, the question still remains whether or not the cost of overwintering is indeed greater than the cost of producing another propagule. To investigate the effect of resource allocation to survival and reproduction, we will assume that there is a fixed quantity of resource (i.e., energy) that may be either invested in reproduction or in survival, for instance in the production of secondary compounds that contribute to defense against natural enemies. Let φ, 0 φ 1, be the fraction of resource invested in reproduction and

4 LIFE HISTORY EVOLUTION the remaining fraction, 1 φ, allocated to survival. Then, a more general projection matrix is L φ = ( ) fj φ f a φ. (6) s j (1 φ) s a (1 φ) This model is more general than either of the other two because it allows us both to toggle between life history strategies (it corresponds to L s when φ = 1 and L i otherwise), and to very the strength of iteroparity as φ is increased (i.e., as a greater fraction of resources are allocated to survival). We already know that if the reproductive output of juveniles and adults are equal, then only in the extreme case where survival is 100% and at no equivalent cost to reproduction can the iteroparous life history be equivalent to investment in reproducing a single additional individual. Although it is not easy to see analytically, it is straightforward to verify numerically that in the case of resource allocation as represented in equation 6 the optimal fitness is obtained when 100% of resources are invested in reproduction (Figure 1). Thus, so far it appears that resource allocation does little to explain the evolution of iteroparity. There is, however, one further possibility that is worth investigating. Importantly, it is necessarily true that the total available resources must sum to 100%. That is, that if a fraction of resources φ is devoted to reproduction at most only 1 φ fraction of resources in available to invest in survival. However, it is not necessarily true that changes in reproductive outout or survival are necessarily proportional to the resources invested. In general, tradeoffs between reproduction and survival might be nonlinear. Once we admit nonlinear equations into our conceptual framework the set of possible models is virtually infinite. How are we to choose how to proceed? We will start by making the approximating assumption that the energetic costs of reproduction are investment in biomass. Thus, perhaps it is not unreasonably that the differential of reproduction with respect to investment is constant, with endpoints limited by some maximum (at 100% investment in reproduction) and zero (at 0% investment in reproduction). Accordingly, our earlier assumption that juvenile and adult reproductive output are given by f j φ and f a φ, respectively, should be appropriate. Survival, by constrast, depends on a wide range if disparate factors including, for instance, foraging for food (which may be depleted as it is consumed), withstanding harsh environmental conditions, outwitting predators or investing in toxic defensive compounds, avoiding parasites, etc. For these cases, it is much easier to imagine that there will be nonlinearities, including both positive differentials (small investments in structure or fat reserves may yield large returns in survival) and negative differentials (from diminishing marginal returns). Thus, to investigate nonlinear life history tradeoffs we will

EVOLUTION OF ITEROPARITY 5 Fitness 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 φ Figure 1: Population growth rate for an interoparous species with φ fractional investment of resources into reproduction. Other parameters are f j = f a = 10 and s j = s a = 1.

6 LIFE HISTORY EVOLUTION Survival 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 Reproduction 0.0 0.2 0.4 0.6 0.8 1.0 φ Figure 2: Survival (solid line) and reproduction (dashed line) for an interoparous species with φ fractional investment of resources into reproduction and nonlinear tradeoff between survival and reproduction. Other parameters are z = 30, f j = f a = 10 and s j = s a = 1. introduce the nonlinearity into survivorship. To achieve both realism and a certain level of tractability, we will assume this nonlinear is reasonably represented by an exponent, a tuning variable z that governs the strength of the nonlinearity. (When z = 0 the linear tradoff is retrieved.) L φ = ( ) fj φ f a φ s j (1 φ z ) s a (1 φ z. (7) ) To illustrate, we first plot survival (assumed the same for both juveniles and adults) and reproduction against z (Figure 2). Now we calcuate the growth rate as a function of φ (Figure 3). This figure confirms the intution. If the tradeoff between reproduction and survival is strong enough so that relatively small diversions of resources from reproduction

BET HEDGING IN VARIABLE ENVIRONMENTS AND THE EVOLUTION OF DORMANCY7 Fitness 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 φ Figure 3: Population growth rate for an interoparous species with φ fractional investment of resources into reproduction and nonlinear tradeoff between survival and reproduction. Other parameters are z = 30, f j = f a = 10 and s j = s a = 1. to survival may all but guarantee reproduction in the following year then there will be an intermediate optimum favoring iteroparity. Bet hedging in variable environments and the evolution of dormancy The preceding section provided one explanation for the evolution of iteroparity. However, this explanation requires a very specific relationship between survival and its cost that may not obtain in many situations, but rather only in those very special circumstances where maturity might expected to confer a fitness advantage. Is there, in constrast, any situation in which

8 LIFE HISTORY EVOLUTION a species with a very annual-like life history might benefit from iteroparity? The answer is yes, when fitness is dependent on environmental conditions and these are unpredictable. The main constraints assumed by this theory are (i) that reproductive output depends on an environmental condition that varies temporally, and (ii) that there exists a cost of survival in an unpredictable environment. For simplicty, we assume a simple annual life history with a dormant propagule stage such as a seed or resting egg. Germination or hatching of the propagule occurs with probability g. Following Cohen (1966) we assume that the environmental condition in each year may be classified as good, occurring with probability p, and bad, occurring with probability q = 1 p. Each propagule germinating in a good year develops and produce f offspring (where f incorporates both absolute fecundity and intermediate transitions such as juvenile survival and mate encounter); propagules germinating in a bad year fail to produce any offspring. The cost of survival is expressed in the mortality of non-germinating propagules, d. This scenario allows for bet-hedging: in an environment where bad years may occur, possibly even predominate, propagules are not all required to germinate immediately but may distribute their germination over many years. But, there is a cost in that the longer the time over which germination is distributed (the lower the value of g) the greater the total cost paid through the loss of propagules from the propagule bank. Denoting the number of propagules by n, we have the following discrete time model for change in population size in good years: n t+1 = n t n t g d(n t n t g) + n t gf = n t ((1 g)(1 d) + gf). (8) Taken in order, the terms of this model may be interpreted as (i) the size of the seed bank at the prior time, (ii) loss from the seed bank due to germination, (iii) loss of ungerminated seeds due to mortality, and (iv) gain due to reproduction. Dividing both sides by n t we have the growth rate good years λ good = n t+1 n t = (1 g)(1 d) + gf. (9) The model for bad years is the same except the last term is left off because there is no reproduction, with growth rate n t+1 = n t n t g d(n t n t g), (10)

BET HEDGING IN VARIABLE ENVIRONMENTS AND THE EVOLUTION OF DORMANCY9 λ bad = n t+1 n t = (1 g)(1 d). (11) We know from our study of population dynamics in variable environments that when the annual growth rates are not all identical the long run growth rate is given by the product of the realized growth rates. That is for sequence [good, good, bad, good, bad] the realized growth rate will be λ g = (λ good λ good λ bad λ good λ bad ) 1/5. Since multiplication is commutative we can rearrange these in any order we want, say λ g = (λ good λ good λ good λ bad λ bad ) 1/5. Over a long time t the number of good years will be tp and the number of good years will be (1 p)t. Thus, the geometric mean growth rate will be is is, λ g = ([(1 g)(1 d)] (1 p)t [(1 g)(1 d) + gf] pt ) 1/t (12) = [(1 g)(1 d)] (1 p) [(1 g)(1 d) + gf] p (13) For our purposes, we assume that d, p, and f are fixed and ask what gemination rate g maximizes λ g. To find the germination rate that maximizes fitness we need only identify the maximum of these curves. Since the maximum of a function coincides with the maximum of its logarithm we first log-transform equation 12 yielding ln λ g = (1 p) ln[(1 g)(1 d)] + p ln[(1 g)(1 d) + gf] (14) Differentiating equation 14 with respect to g we obtain (Appendix 1): ln λ g = 1 p g 1 + p d + d 1 (1 g)(1 d) + fg. (15) Setting to zero and solving for g, now designated g opt, we find that the optimal germination rate is given by (Appendix 2): g opt = fp+d 1 f+d 1. (16)

10 LIFE HISTORY EVOLUTION Geometric mean growth rate 1 2 5 0.0 0.2 0.4 0.6 0.8 1.0 Germination rate (g) Figure 4: Geometric mean growth rate for a bet-hedging species over a range of germination rate (g). Other parameters are d = 0.2 (black) or d = 0.8 (blue) and p = 0.6 (solid lines) or p = 0.4 (dashed lines); f = 100 for all curves.

APPENDIX 1: DERIVATIVE OF LN λ(g) 11 In some important paradigm cases (e.g., annual plants) f will be large (say, hundreds) relatively to d (which must be between 0 and 1) and p (which also must be between zero and one). In this case, the numerator is dominated by fp and the denominator is denominated by p, yielding the approximation g opt fp f = p. (17) This result is interpreted to imply that in the presence of an abundance of propagules the theoretically optimal germination rate is approximately equal to p, the probability (or proportion) of good years. Appendix 1: Derivative of ln λ(g) For convenience, we define and h = (1 g)(1 d) (18) and re-label some of the terms in equation 14 as follows. k = (1 g)(1 d) + gf (19) f 1 {}}{{}}{ ln λ g = (1 p) ln[(1 g)(1 d)] +p ln[(1 g)(1 d) + gf] (20) By use of the sum rule together with the constant rule we have f 2 ln λ g = (1 p) f 1 + p f 2. (21) By the chain rule we determine that

12 LIFE HISTORY EVOLUTION f 1 = ln h h h = 1 h h = (1 g)(1 d) (1 g)(1 d). (22) Factoring out the (1 d) in the numerator and canceling with the denominator, we have (1 g)(1 d) (1 g)(1 d) = (1 d) (1 g) (1 g)(1 d) Finally, evaluating the derivative in the numerator leaves = (1 g). (23) 1 g (1 g) 1 g = (1) (g) 1 g We proceed similarly with f 2. First, by the chain rule, = 1 1 g = 1 g 1. (24) f 2 = ln k k k = 1 k h Expanding the numerator yields = (1 g)(1 d) + fg (1 g)(1 d) + fg. (25) 1 d g + dg + fg (1 g)(1 d) + fg Inserting these results into equation 21 we have = f + d 1 (1 g)(1 d) + fg. (26) ln λ g = 1 p g 1 + p d + d 1 (1 g)(1 d) + fg. (27)

APPENDIX 2: DERIVING G OP T 13 Appendix 2: Deriving g opt Here we begin with the solution from Appendix 1, set ln λ g to zero, and solve for g. Thus, ln λ g = 1 p g 1 + p d + d 1 (1 g)(1 d) + fg = 0. (28) First, note that (1 g)(1 d) + fg = dg d g + fg + 1 and bring the two terms together using the common denominator (g 1)(dg d g + fg + 1): 1 p g 1 + p(d + d 1) (1 g)(1 d) + fg = (1 p)(dg d g + fg + 1) (g 1)(dg d g + fg + 1) + p(d + d 1)(g 1) (g 1)(dg d g + fg + 1).(29) Expanding the expressions in the numerators and adding the two fractions yields fgp + dgp gp fp dp + p + 1 d g + dg + fg p + dp + gp dgp fgp (g 1)(dg d g + fg + 1) = 0. (30) Lots of terms cancel, yielding 1 d g + dg + fg fp (g 1)(dg d g + fg + 1) Multiplying both sides by the denominator, we have = 0. (31) 1 d g + dg + fg fp = 0. (32) Collecting terms in g and moving these to the right hand side 1 d fp = g(1 d f). (33)

14 LIFE HISTORY EVOLUTION Finally, we isolate g, now designated g opt, and multiply numerator and denominator by 1 to obtain the optimal germination rate: Homework exercises g opt = 1 d fp 1 d f = fp + d 1 f + d 1. (34) 1. Our discussion of the benefits of iteroparity vs. semelparity focused on energy allogation. But, often it is the case that larger or older individuals produce more offspring because they are able to secture more resources. (This happens in plants, fish, and numerous invertebrates). Suppose larger/older individuals are more productive so that f j < f a. Does this provide a mechanism whereby iteroparity is adaptive? Modify the model to show this.

Bibliography Cohen, D. (1966). Optimizing reproduction in a randomly varying environment. Journal of Theoretical Biology, 12, 119 129. L.C. Cole (1954). The population consequences of life history phenomena. Quarterly Review of Biology, 29, 103 137. 15