Ann. Zool. Fennici 40: 331–340 ISSN 0003-455X Helsinki 29 August 2003 © Finnish Zoological and Botanical Publishing Board 2003

A model on the evolution of

K. Ingemar Jönsson* & Johannes Järemo

Department of Theoretical Ecology, Lund University, Ecology Building, SE-223 62 Lund, Sweden (*e-mail: [email protected])

Received 18 Nov. 2002, revised version received 5 Mar. 2003, accepted 6 Mar. 2003

Jönsson, K. I. & Järemo, J. 2003: A model on the evolution of cryptobiosis. — Ann. Zool. Fennici 40: 331–340.

Cryptobiosis is an ametabolic state of life entered by some lower (among metazoans mainly , and ) in response to adverse environ- mental conditions. Despite a long recognition of cryptobiotic organisms, the evolution- ary origin and life history consequences of this biological phenomenon have remained unexplored. We present one of the fi rst theoretical models on the evolution of cryptobi- osis, using a hypothetical population of marine tardigrades that migrates between open sea and the tidal zone as the model framework. Our model analyses the conditions under which investments into anhydrobiotic (cryptobiosis induced by ) functions will evolve, and which factors affect the optimal level of such investments. In particular, we evaluate how the probability of being exposed to adverse conditions (getting stranded) and the consequences for survival of such exposure (getting desic- cated) affects the option for cryptobiosis to evolve. The optimal level of investment into anhydrobiotic traits increases with increasing probability of being stranded as well as with increasing negative survival effects of being stranded. However, our analysis shows that the effect on survival of being stranded is a more important parameter than the probability of stranding for the evolution of anhydrobiosis. The existing, although limited, evidence from empirical studies seems to support some of these predictions.

Introduction otic abilities are often separated into those in which the cryptobiotic state may be entered Cryptobiosis is the collective name for an only within a specifi c (ontogenetic) life stage, ametabolic state of life utilized by some organ- and those that may enter cryptobiosis over the isms to overcome periods of unfavourable entire life cycle (Crowe 1971, Wright et al. environmental conditions (Keilin 1959, Crowe 1992). The fi rst category includes species from 1975, Wright et al. 1992, Kinchin 1994). Cryp- the taxa Arthropoda, Crustacea, Brachiopoda, tobiotic organisms are known from both the Insecta, spores of various fungi and bacteria, plant and animal kingdom, but in animals only and pollen and seeds of some plants, while the among invertebrates (Wright et al. 1992). The second category mainly includes species from main factors inducing the cryptobiotic state are Protozoa, Rotifera, Nematoda, Tardigrada, and desiccation (anhydrobiosis), (cryo- various species of mosses, and algae, biosis) and defi ciency (anoxybiosis, as well as some higher plants (Keilin 1959, Wright et al. 1992). Organisms with cryptobi- Wright et al. 1992). 332 Jönsson & Järemo • ANN. ZOOL. FENNICI Vol. 40

Fig. 1. (a) in a hydrated active state (light microscopy photo by K. I. Jöns- son) and (b) in a desiccated anhydrobiotic state (SEM photo by R. M. Kristensen).

In the cryptobiotic state, several fundamental ably taken in populations living in seashore and biological characteristics are affected. Since all tidal habitats, where great variation in the water metabolic processes have stopped, reproduction, availability created a strong selection for desic- development, and repair are prevented. There- cation tolerance. Some tardigrade populations fore, as a life history component cryptobiosis of the genus Echiniscoides and Archechiniscus is mainly characterized by survival. However, inhabiting littoral marine environments show a to successfully achieve a high survival in the capacity to survive desiccation (Grøngaard et al. cryptobiotic state, the has to make 1990, Wright et al. 1992), whereas other marine biochemical preparations of the body, such as the tardigrades cannot do so. production of carbohydrates to replace the lost In this paper, we present a theoretical model water in anhydrobiotic organisms (e.g., Crowe on the evolution of anhydrobiosis (cryptobiosis 2002). Such processes are energetically costly induced by desiccation). The model describes (Jönsson & Rebecchi 2002), and will withdraw a hypothetical population of an ancient marine resources from potential use in other life history tardigrade inhabiting the littoral zone, where the functions, e.g., reproduction. Thus, most likely risk of exposure to desiccation creates a force there are trade-offs between investment in cryp- of selection for anhydrobiotic ability. The main tobiotic functions and investment in other life purpose of the analysis is to evaluate two general history functions. aspects of anhydrobiotic selection: the prob- Tardigrades (phylum Tardigrada, Fig. 1) are ability of being exposed to a time period of dry well known for their cryptobiotic capacity (e.g., environmental conditions, and the consequences Wright et al. 1992) and occur in a variety of on survival of such exposure. Based on our anal- ecosystems, some of which represent the most ysis, we propose a set of conditions that should extreme natural habitats on earth (e.g., continental favour energy investments in traits that improve Antarctica; Sømme & Meier 1995). They repre- the animalʼs anhydrobiotic capacity. sent an important component of the meiofauna in habitats exposed to desiccation, and may survive for several years in a cryptobiotic state (Jönsson The model & Bertolani 2001, Guidetti & Jönsson 2002). Although the ability of tardigrades to enter General model scenario cryptobiosis has been known for a long time, and has been subject to numerous investigations Our starting point is a non-anhydrobiotic popula- (see Wright et al. 1992, Wright 2001), the evo- tion of ancient marine tardigrades that migrates lutionary background to this ability has not been (actively or passively) between the tidal zone studied much. May (1951) suggested that marine and permanent water. Such migratory behaviour tardigrades were the ancestors of terrestrial and could arise from generally better foraging condi- limnic tardigrades, and that the transition from tions in the tidal zone. Consequently, the organ- marine to terrestrial forms was accompanied by isms forage in the tidal zone during high tide, the acquisition of a tolerance to anoxia and des- and return to the non-tidal zone when the water iccation, and ultimately of cryptobiosis. If this withdraws in order to avoid stranding and sub- scenario is correct, the initial evolutionary steps sequent desiccation. We assume that egg laying towards cryptobiotic tardigrades were prob- in these marine tardigrades takes place in open ANN. ZOOL. FENNICI Vol. 40 • Evolution of cryptobiosis 333 water and that eggs are not able to migrate to the survival parameter, may be distinguished (see tidal zone during high tide (in tardigrades, eggs Jönsson et al. 1995a, 1995b, 1998). A deduction of are either laid freely in the substrate or within the function (Eq. 1) from the Euler–Lotka equation is moulted cuticle (Bertolani 1983)). The reproduc- provided in the Appendix. The point of offspring tive cycle of our model population then begins release, which in the present case is equivalent to with a prebreeding period of foraging in the egg-laying, separates the two phases. tidal zone followed by migration to open water. Now assume the existence of within popula- During the period of foraging and resource accu- tion genetic variation in the ability to survive mulation, eggs are developed. If the population partial or complete desiccation, generating succeeds in migrating back to open water, the continuous variation in traits related to this abil- animals moult and lay their eggs, and in the fol- ity. Such traits may include storage of energy, lowing postbreeding period they migrate back to synthesis of protectant molecules or enzymes, the tidal zone to start a new period of foraging. cuticular structures reducing transpiration, However, the population runs the risk of being aggregation behaviours, or any other trait that exposed to a dry period (low tide) during which increase the stress tolerance of the animal the animals cannot return to open water and (Wright et al. 1992, Jönsson 2001). We call these which exposes them to desiccation stress. The traits “anhydrobiotic traits” and assume that the model assumes no parental care. ability to survive a period of desiccation depends The model scenario corresponds well to a on the amount of resources invested into such marine environment with tidal changes in water traits. The magnitude of these investments is availability, e.g., the supposed environment here denoted by K, which is the phenotypic trait under which the fi rst steps towards anhydro- that we evaluate below. biosis in tardigrades may have been taken (May We assume that anhydrobiotic traits incur 1951). However, the scenario could just as well costs in terms of a reduction in survival under represent the conditions under which anhyd- non-desiccating conditions by withdrawing robiosis evolved in any other metazoan. Also, resources from this life history trait. Survival because the general life history effects should may then be modelled by a function Pi that be similar regardless of the factor (desiccation, depends on K so that freezing, anoxia) inducing a cryptobiotic state, our analysis could well be adopted as a general analysis on the evolution of cryptobiosis. Moreover, investment into anhydrobi- otic traits is assumed to divert resources from Fitness function reproduction, making reproductive output, R, a decreasing function of K so that Let us denote survival over the prebreeding period by P1, survival over the postbreeding period P2, and reproductive output, i.e. the number of surviving offspring at each repro- Inserting the functions of survival and repro- ductive occasion, by R. Given our ecological ductive output into fi tness function (Eq. 1) gives scenario, and assuming that fertility and adult the following fi tness for a migrating animal survival are constant over adult age classes, the l l fi tness ( ) can be approximated by the function = P1(K)[(R(K) + P2(K)] (2)

l = P1(R + P2) (1) However, this is the fi tness for animals that succeed in returning to the non-tidal area after This fi tness function is an extension of the foraging. If they fail in this process, they become one proposed by Charnov and Krebs (1974), and stranded and must wait until the high tide returns is generally applicable for reproductive cycles in before they can migrate to open water. If the which two temporal phases, each with a separate population is stranded, survival decreases sig- 334 Jönsson & Järemo • ANN. ZOOL. FENNICI Vol. 40 nifi cantly. Still, those individuals with anhydro- ends up in stranding for the population. Since we biotic traits will gain an advantage since survival consider a temporal heterogeneity that affects for stranded animals, , increases with K so that the whole population and that spans several reproductive cycles we believe that the geomet- (3) ric mean fi tness is the proper fi tness measure in this type of model (for discussion on geometric Equation 3 describes a survival function that mean fi tness, see Philippi and Seger (1989)). increases asymptotically towards 1 as K increases The use of an arithmetic mean would imply that towards infi nity. Here, c is a parameter between the populations of tardigrades experience spatial 0 and 1 indicating the survival cost for a stranded heterogeneity, i.e. a proportion of the population non-anhydrobiotic animal. A high value of c gets stranded whereas the other individuals do implies that being stranded is devastating for the not. In this model, however, we are only focus- stranded individual, and for c = 1 it will die if no ing on temporal heterogeneity and hence we use investments are made into anhydrobiosis (K = 0). the geometric mean fi tness. The expected fi tness On the other hand, a low value of c implies that in our hypothetical population is then being stranded does not affect survival very much. Note that stranding does not necessarily mean that the animal gets desiccated, only that (6) it cannot return to the moist area. Thus, c refl ects the probability of being desiccated for a stranded non-anhydrobiotic organism. The parameter a Under these circumstances, a phenotype converts K to a dimensionless entity in order to with no anhydrobiotic investments (K = 0) that make survival a probability. We assume in the gets stranded will have a zero fi tness as long as analysis below, for the sake of simplicity, that death is certain to a stranded non-anhydrobiotic a = 1. Note also that we do not take the length of individual, i.e. c = 1. In the long run, it also the period in anhydrobiosis into account, imply- implies that when c = 1, a population of non- ing either that this period is a constant, or that the anhydrobiotic individuals (K = 0) will become length of the period is short enough not to infl u- extinct when there is a risk of being stranded. ence survival. This seems to be a plausible consequence of the When water returns in the tidal area, anhy- system outlined above. drobiotic individuals rehydrate and migrate to Below we analyse the above fi tness function the non-tidal area where they reproduce. The for decreasing functions of Pi(K) and R(K) with survival of an animal that forages in the tidal three curvatures, i.e. linear, concave and convex area, gets stranded, and returns to the area of functions. We also assume that pre- and post- permanent water is then breeding survival is equally affected by K, i.e.

P1(K) = P2(K). Our analysis should constitute a (4) suffi cient basis for drawing general conclusions about the conditions for the evolution of anhy- As a consequence, fi tness of an animal that drobiosis. For linear and concave functions, we becomes stranded is present graphical analyses because analytical analyses become exceedingly complex and dif- (5) fi cult to interpret.

Results Environmental heterogeneity Linear and exponential effects If the probability of being stranded is q, the chance of safe return to open water is (1 – q), Assuming that survival as well as reproduction i.e., a proportion q of all visits to the tidal area decrease linearly with investment of resources in ANN. ZOOL. FENNICI Vol. 40 • Evolution of cryptobiosis 335

2 2.5

1.8 q = 0.3 a b 2 q = 0.3 1.6 q = 0.5 1.4 1.5 1.2 q = 0.5 q = 0.9 l 1 1

0.8 0.5 q = 0.9 0.6

0.4 0 0.2

0 –0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 K K Fig. 2. Fitness (l) plotted against investments in anhydrobiotic structures (K) for three values of the probability of getting stranded (q) when (a) survival and reproduction functions are linearly decreasing (r = 3, a = 1, b = 0.5, g = 1, c = 1), and (b) survival and reproduction functions are concavely decreasing (r = 3, a = 1, b = 0.5, g = 1, z = 2, c = 1). anhydrobiotic structures, the simplest case may and be described by functions of the sort R(K) = r – gKz.

b b g Pi(K) = 1 – K In the equations above, , and r have the and same meaning as in the case of linear effects R(K) = r – gK whereas the parameter z determines the curva- ture of the functions (the higher value of z, the where b (the marginal decrease in survival due to more dramatic are the effects of an increase in investment into anhydrobiosis) converts K into K). As in the linear case, we will let b = 0.5, g = 1 a dimensionless entity, and g is a parameter that while we let z = 2. converts K into the number of surviving young. Under these assumptions the fi tness function In the analysis, we restrict our case to the situa- (Eq. 6) may be plotted against K, which will reveal tion when b = 0.5, i.e. we will not here analyze the optimum investments into anhydrobiosis. variations in the survival cost function. We also Figure 2 shows fi tness as a function of K for three assume that g = 1, i.e. one unit of K represents one values of the probability of getting stranded, q, surviving young less. The parameter r determines and for linear and concave survival/reproduction the reproductive output of an animal with no functions, respectively. The panels indicate that investments into anhydrobiosis whatsoever and optimal investment in anhydrobiotic structures this parameter, together with b and g, determines should increase as q increases. They also suggest how the effects of anhydrobiotic investments that the probability of getting stranded does not affect reproduction in comparison to the effects have to be particularly high for anhydrobiosis to on survival. If b and g are held constant then a evolve, as long as the effects of getting stranded higher value of r means less impact on reproduc- are devastating (c = 1 in the panels). tion from investment into anhydrobiotic struc- In Fig. 3, fi tness is plotted against K for three tures as compared with the impact on survival. values of c. The probability of getting stranded is We also evaluate the possibility that anhyd- in both cases 0.7, i.e. 70% of the visits to the tidal robiotic investments will have an exponentially zone result in stranding. Still, this is not enough increasing negative effect on survival as well as to promote anhydrobiotic investments when on reproduction. The functions will then be con- survival and reproduction decrease linearly with cave and the easiest way of obtaining such cur- K if c is suffi ciently low (Fig. 3a). These effects vature is by adding an exponential to K so that are not so obvious when we assume concave functions. However, in both cases optimum K b z Pi(K) = 1 – K increases as c increases (Fig. 3). 336 Jönsson & Järemo • ANN. ZOOL. FENNICI Vol. 40

3 3 a b c = 0.5 2.5 2.5 c = 0.5

2 2 c = 0.9

l 1.5 c = 0.9 1.5

c = 1 1 1 c = 1

0.5 0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 K K Fig. 3. Fitness (l) plotted against investments in anhydrobiotic structures (K) for three effects on survival if not entering anhydrobiosis when stranded (c) when (a) survival and reproduction functions are linearly decreasing (r = 3, a = 1, b = 0.5, g = 1, q = 0.7), and (b) survival and reproduction functions are concavely decreasing (r = 3, a = 1, b = 0.5, g = 1, z = 2, q = 0.7).

2 2.5 1.8 abr = 5 r = 5 1.6 2 1.4 1.5 1.2 r = 3 1 r = 3 l 1 0.8 0.6 r = 1 0.5 0.4 r = 1

0.2 0 0 –0.2 –0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 K K Fig. 4. Fitness (l) plotted against investments in anhydrobiotic structures (K) for three values of the effects on reproductive output (r) when (a) survival and reproduction functions are linearly decreasing (a = 1, b = 0.5, g = 1, c = 1, q = 0.7), and (b) survival and reproduction functions are concavely decreasing (a = 1, b = 0.5, g = 1, z = 2, c = 1, q = 0.7).

The effects of reproductive output, r, are ever, we are interested in fi nding an analytical presented in Fig. 4. Here there is a tendency for to the problem so instead we chose the increased investment into anhydrobiosis as its following set of functions. relative effects on potential reproductive output decreases (K increases as r increases). Optimal anhydrobiotic investment should thus be higher in phenotypes with high original reproductive output.

In these equations, both b and g convert K to

Diminishing effects a dimensionless entity in order to make Pi and R currencies for survival and reproductive output, Finally we evaluate the case when survival and respectively. As before, r is reproductive output reproduction functions are slightly convex. This when no investments are made into anhydro- could be accomplished by simply setting the biosis. A great advantage with these functions above parameter z to a value below one. How- is that they allow analytical investigation when ANN. ZOOL. FENNICI Vol. 40 • Evolution of cryptobiosis 337 assuming that g = 1 and b = 1. Differentiating Eq. 6 with respect to K and solving for optimum K yields

(7)

Inserting into the second derivative of Eq. 7 gives us the following expression

(8) which is always negative, implying that repre- sents a maximum. Equation 7 provides a number of interesting results. First, if c = 1 and q > 0, i.e. if there is any probability of being stranded and stranding is equivalent to death to a non-anhydrobiotic Fig. 5. Optimum anhydrobiotic investment, , as a organism, anhydrobiotic investment will always function of the probability of being stranded, q, and the reduction in survival when stranded, c. be selected for ( > 0). This result was indicated also for linear and concave functions. Second, increases as the risk of being stranded increases. Third, as the survival cost of being stranded (c) (9) increases, optimal investment into anhydrobiotic traits ( ) increases. Fourth, the probability of being stranded (q) together with the effect of Discussion being stranded (c) must be high enough to satisfy the inequality c(0.5q + 1) > 1 in order for any Given the assumptions of our model, it is clear anhydrobiotic investments to be advantageous. that anhydrobiosis will evolve under two condi- In conclusion, for anhydrobiosis to evolve at all, tions: (i) when both the probability of strand- i.e. > 0, either both q and c have to be rather ing and the survival cost paid by a stranded high, or c = 1 and q > 0. Even with a risk of being individual is relatively high, or (ii) under low stranded equal to 1, i.e. all visits to the tidal zone probability of stranding if the cost of being end up with stranding, anhydrobiosis will only stranded is detrimental. In addition, our analysis be an optimal strategy if the reduction in sur- suggests that the effect, rather than the risk, of vival when being stranded is relatively high, i.e. being stranded is the crucial parameter for the c ≥ 2/3 in the model above. evolution of anhydrobiosis. Formulated more We may plot as a function of both q and c generally, this suggests that cryptobiosis may as in Fig. 5. The largest investment into anhydro- evolve (i) when both the probability of experi- biotic traits is found under circumstances where encing adverse conditions and the survival cost stranding implies death for a non-anhydrobiotic of such conditions are high, or (ii) when survival individual (c = 1) and stranding is inevitable costs are very high despite a low probability of (q = 1). At the other extremes, i.e. when c = 0 adverse conditions. These results emerged from and q = 1, or when c = 1 and q = 0, we expect no the analysis with convex functions of survival anhydrobiotic investments at all (Fig. 5). Another and reproduction, and under rather specifi c important message from Eq. 7 and Fig. 1 is the assumptions (a = 1, b = 1, g = 1). However, confi rmation that c is the major determinant of also in the case of linear functions, in spite of a the magnitude of anhydrobiotic investments. risk of 70% of getting stranded (q = 0.7) even a This result is indicated by the steeper slope of 50% risk of dying if not entering anhydrobiosis as a function of c, as compared with as a func- (c = 0.5) will not suffi ce to promote any invest- tion of q (Fig. 2 and Eq. 9) ments into anhydrobiotic structures (Fig. 3a). 338 Jönsson & Järemo • ANN. ZOOL. FENNICI Vol. 40

When functions are concave, the requirements ing attempts further and further into the dry area for investing in anhydrobiotic traits seem to would select for phenotypes with increasingly be less restrictive. This is not surprising since higher investment into anhydrobiotic traits. concavity implies that minor initial investments In our analysis we have concentrated on the impose low costs on survival and reproduction. initial stage in the evolution of cryptobiosis, The result that the effect, rather than the prob- when anhydrobiotic populations evolved from ability, of experiencing adverse conditions is non-anhydrobiotic populations. Once anhy- the more important parameter in promoting the drobiosis had evolved, the transition towards evolution of cryptobiosis has an important impli- populations inhabiting pure terrestrial habitats cation. It suggests that cryptobiosis may be pro- exposed to occasional or regular periods of des- moted also in environments with relatively infre- iccation may have been relatively rapid. To some quent events of adverse conditions. According to extent, our results should apply also to this phase this prediction anhydrobiosis could, therefore, be in the evolution of anhydrobiotic behaviour, with widespread among organisms even in terrestrial the probability of being stranded turning into habitats that dry out relatively rarely. We know a general risk of dehydration. Our results will of no data evaluating this prediction. then indicate that terrestrial organisms living in A quantitative result of our analysis is that the habitats that always dry out within a reproduc- optimal investment into cryptobiotic traits should tive cycle (q = 1, c = 1) will evolve a high anhy- increase with increasing probability of adverse drobiotic capacity, whereas organisms living conditions as well as with increasing negative in habitats that dry out only occasionally (e.g., survival effects of such conditions. As a predic- q = 0.5, c = 1) will evolve a lower capacity of tion from this result we would expect cryptobiotic anhydrobiosis. In fact, according to our analysis organisms in different environments to allocate we would expect a gradual increase in invest- different relative amounts of energy to crypto- ments to anhydrobiotic traits as the risk of desic- biotic functions. We may also expect differences cation increases (Fig. 5 and Eq. 7). To the extent in cryptobiotic capacity (e.g. in the ability to sur- that anhydrobiotic capacity is directly related vive rapid desiccation) among organisms adapted to energy invested into this trait, this prediction to different environments. In line with these pre- is supported by studies on tardigrades (Wright dictions, Grøngaard et al. (1990) showed a quali- 1991) and nematodes (Williams 1978, Solomon tative difference in anhydrobiotic capacity in the et al. 1999) showing that anhydrobiotic capacity littoral tardigrades Echiniscoides sigismundi and is related to the desiccation conditions in the nat- Echiniscoides hoepneri. Both species are found ural habitat of a species. We may also expect that in the upper littoral zone, but the former species optimal energy investment into anhydrobiotic inhabits more exposed sites (e.g., among barna- traits over a reproductive cycle increases with cle plates) and survives desiccation, while the the frequency of dry periods expected within the latter species inhabits more sheltered sites (e.g. cycle. Thus, in habitats where periods of desic- within barnacles) and cannot survive desicca- cation occur almost on a daily basis, investment tion. The fi rst attempts by marine non-cryptobi- into anhydrobiotic traits (e.g., protectant mol- otic tardigrades to forage in the tidal zone were ecule production; Westh & Ramløv 1991, Wright probably made fairly close to the border of the et al. 1992) should be high. open sea. In this environment, the risk of being In the analysis with linear and concave stranded is low but still exists, whereas the effect survival/reproduction functions, an increase in r of being stranded would be just as devastating as yielded an increase in optimum K (Fig. 4). This further up on the shore. This circumstance would would suggest that an organism with high origi- be enough for a mutant tardigrade with some nal fecundity should invest more in cryptobiotic investment in cryptobiotic traits to gain a selec- functions than an organism with low original tive advantage over non-cryptobiotic individuals. fecundity subject to the same environmental Figure 2 shows that as the probability of being conditions. Hence, the general reproductive stranded increases, there is a gradual increase in strategy (high or low fecundity) should infl uence optimal anhydrobiotic investment. Hence, forag- the optimal allocation to cryptobiotic functions. ANN. ZOOL. FENNICI Vol. 40 • Evolution of cryptobiosis 339

However, this was not the case for the specifi c Acknowledgements convex functions that we used, where r did not appear in the optimality condition (Eq. 7). We thank P. Lundberg, J. Ripa, and two anonymous referees Thus, the curvature of the trade-off functions for comments on the manuscript. K. I. J. was supported by the Swedish Natural Science Council. between cryptobiotic investment and reproduc- tion and survival in the hydrated state may be important for predicting optimal investments References into cryptobiosis. The shape of these trade-offs has to our knowledge never been investigated Bertolani, R. 1983: Tardigrada. — In: Adiyodi, K. G. & Adi- in cryptobiotic organisms, but linear or concave yodi, R. G. (eds.), Reproductive biology of invertebrates, functions may seem more likely to expect than vol. 1: Oogenesis, oviposition and oosorption: 431–441. convex functions. With convex functions, the John Wiley & Sons Ltd., New York. marginal effects of cryptobiotic investment on Charnov, E. L. & Krebs, J. R. 1974: On clutch size and fi t- ness. — Ibis 116: 217–219. reproduction and survival in the hydrated state Crowe, J. H. 1971: Anhydrobiosis: an unsolved problem. will decline the more the organism invests in — Am. Nat. 105: 563–573. cryptobiotic functions, which is probably not Crowe, J. H. 1975: The physiology of cryptobiosis in tardi- a generally expected condition. One could, grades. — Mem. Ist. Ital. Idrobiol. (Suppl.) 32: 37–59. Crowe, L. M. 2002: Lessons from nature: the role of however, envisage a system where the initial in anhydrobiosis. — Comp. Biochem. Physiol. A 131: investment involves forming of new structures 505–513. that have large effects on organism morphology/ Grøngaard, A., Møbjerg Kristensen, N. & Krag Petersen, M. physiology or behaviour. Such initial investments 1990: Tardigradfaunaen på Disko. — In: Andersen, P. F., Düwel, L. & Hansen, O. S. (eds.), Feltkursus i Arktisk could reduce survival and reproduction sig- biologi, Godhavn 1990: 155–179. Zoologisk Museum, nifi cantly. In such a case, additional investments Københavns Universitet. into cryptobiotic capacity would only imply an Guidetti, R. & Jönsson, K. I. 2002: Long-term anhydrobiotic increase in some quantitative trait or adjustments survival in semi-terrestrial micrometazoans. — J. Zool. of the system that may have a less adverse effect 257: 181–187. Jönsson, K. I. 2001: The nature of selection on anhydrobiotic on survival and reproduction. Consequently, capacity in tardigrades. — Zool. Anz. 240: 409–417. convex trade-off functions should not be ruled Jönsson, K. I., Tuomi, J. & Järemo, J. 1995a: Reproductive out completely. A concave relationship between effort tactics balancing pre- and postbreeding costs of K and anhydrobiotic survival seems plausible reproduction. — Oikos 74: 35–44. Jönsson, K. I., Tuomi, J. & Järemo, J. 1995b: On the conse- though, since investment in cryptobiotic func- quences of pre- and postbreeding costs in the evolution tions (e.g. production of protectant molecules) of reproductive effort tactics. — Écoscience 2: 311–320. above a certain level will have smaller and Jönsson, K. I., Tuomi, J. & Järemo, J. 1998: Pre- and postbreed- smaller effects on survival. ing costs of parental investment. — Oikos 83: 424–431. Jönsson, K. I. & Bertolani, R. 2001: Facts and fi ction about From our assumed trade-off between cryp- long-term survival in tardigrades. — J. Zool. 255: tobiotic and reproductive investments we would 121–123. also expect an inverse relationship between cryp- Jönsson, K. I. & Rebecchi, L. 2002: Experimentally induced tobiotic capacity and reproductive investments. anhydrobiosis in the tardigrade Richtersius coronifer: Organisms living in exposed habitats will thus be phenotypic factors affecting survival. — J. Exp. Zool. 293: 578–584. expected to invest a large proportion of available Keilin, D. 1959: The problem of anabiosis or latent life: his- resources into cryptobiotic functions and thus tory and current concept. — Proc. R. Soc. Lond. B 150: have low fecundity as compared to non-cryp- 149–191. tobiotic organisms living in constantly favour- Kinchin, I. M. 1994: The biology of tardigrades. — Portland Press, London. able (moist) environments. At present, there is May, R. M. 1951: Lʼevolution des tardigrades de la vie aqua- no data available to evaluate these predictions. tique à la vie terrestre. — Bull. Franç. Pisc. 168: 93–100. Overall, it would be of great interest to compare Philippi, T. & Seger, J. 1989: Hedging oneʼs evolutionary frequencies of adverse conditions, cryptobiotic bets, revisited. — Trends Evol. Ecol. 4: 41–44. Solomon, A., Paperna, I. & Glazer, I. 1999: Desiccation capacity and life history patterns in cryptobiotic survival of the entomopathogenic Steinernema organisms inhabiting different habitats, to the feltiae: induction of anhydrobiosis. — Nematology 1: results obtained from this study. 61–68. 340 Jönsson & Järemo • ANN. ZOOL. FENNICI Vol. 40

Sømme, L. & Meier, T. 1995: Cold hardiness of Tardigrada London. from Dronning Maud Land, Antarctica. — Polar Biol. Wright, J. C. 1991: The signifi cance of four xeric parameters 15: 221–224. in the ecology of terrestrial Tardigrada. — J. Zool. 224: Westh, P. & Ramløv, H. 1991: accumulation in the 59–77. tardigrade Adorybiotus coronifer during anhydrobiosis. Wright, J. C. 2001: Cryptobiosis 300 years on from van — J. Exp. Zool. 258: 303–311. Leuwenhoek: what have we learned about tardigrades? Williams, T. D. 1978: Cyst nematodes: biology of Heter- — Zool. Anz. 240: 563–582. odera and Globodera. — In: Southey, J. F. (ed.), Plant Wright, J. C., Westh, P. & Ramløv, H. 1992: Cryptobiosis in nematology: 156–171. Her Majestyʼs Stationery Offi ce, tardigrada. — Biol. Rev. 67: 1–29.

Appendix

Assume that survival between two breeding events contains two components, post- and prebreeding survival, and that the point at which offspring survival become independent of parent survival demar- cate these two periods. With the notations P1 and P2 for pre- and postbreeding survival, respectively, the Euler–Lotka equation may be expressed as follows (see also Jönsson et al. 1995)

(A1) where x is age class and m(x) is fecundity at age class x. Let us further assume that our organism reproduces after a juvenile stage plus a prebreeding period so that survival before fi rst breeding is

P10P20P11, and that fecundity is constant over age-classes. The Euler–Lotka equation above then con- verges to

(A2)

Assuming that adult survival is constant over ages, i.e. P11 = P12 =…= P1n and P21 = P22 =…= P2n, equation A2 becomes equivalent to

(A3) which is equal to (A4)

Here the term mP10P20 is the reproductive output at each reproductive event, in our model denoted by R. Equation A4 can be rewritten as

(A5)

The geometric sum of A5 can be simplifi ed as

(A6).

Now let n approach infi nity — recalling that we assume constant age-specifi c survival — and the equation A6 can be simplifi ed and rewritten as

(A7)

l which gives our fi tness function, = P1(R + P2).