Consequences of catastrophic disturbances on population persistence and adaptations
11 Consequences of catastrophic disturbances on population persistence and adaptations Simone Vincenzi a , Michele Bellingeri a a Dipartimento di Scienze Ambientali, Università degli Studi di Parma, Viale G. P. Usberti 33/A, I-43100 Parma, Italy. Corresponding author: Simone Vincenzi, Dipartimento di Scienze Ambientali Università degli Studi di Parma Viale G.P. Usberti 33/A I-43125 Parma Tel.: +39 0521 905696 Fax.: +39 0521 906611 email: [email protected]
23 24
Abstract
25 The intensification and increased frequency of weather extremes is emerging as one 26 of the most important aspects of climate change. We use Monte Carlo simulation to 27 understand and predict the consequences of variations in trends (i.e., directional 28 change) and stochasticity (i.e., increase in variance) of climate variables and 29 consequent selection pressure by using simple models of population dynamics. 30 Higher variance of climate variables increases the probability of weather extremes 31 and consequent catastrophic disturbances. Parameters of the model are selection 32 pressure, mutation, directional and stochastic variation of the environment. We 33 follow the population dynamics and the distribution of a trait that describes the 34 adaptation of the individual to the optimum phenotype defined by the 35 environmental conditions 36 The survival chances of a population depend quite strongly on the selection 37 pressure and decrease with increasing variance of the climate variable. In general, 38 the system is able to track the directional component of the optimum phenotype. 39 Intermediate levels of mutation generally increase the probability of tracking the 40 changing optimum and thus decrease the risk of extinction of a population. With 41 high mutation, the higher probability of maladaptation decreases the survival 42 chances of the populations, even with high variability of the optimum phenotype. 43
Keywords:
Catastrophic disturbance; Population dynamics; Monte Carlo 44 simulations; Mutation; Selection pressure.45
1. Introduction
46 With climate change, many species will experience selection pressures in new 47 directions and at new intensities, and the degree to which species respond 48 adaptively will have an important influence on their capacity to survive over the 49 coming decades and millennia [1]. 50 Changes in the long-term mean state of climate variables (i.e, climate trends) and 51 their consequences on survival, evolution and adaptation of species have been 52 intensively studied for more than 20 years [2]. The intensification of weather 53 extremes is emerging as one of the most important aspects of climate change [3] and 54 the debate is expanding from an analysis of trends to an interest in extreme events 55 and associated catastrophic disturbances, such as periods of heavy rainfall (with 56 associated floods and landslides), fires, droughts, heat waves [4]. Catastrophes are 57 characterized by statistical extremity, timing, and abruptness relative to the life 58 cycles of the organisms affected; they can disrupt ecosystems, communities, or 59 population structure and change resource pools, substrate availability, or physical 60 environment [3, 5, 6]. Many adaptations (in life histories, morphological or 61 behavioral traits) can be associated with catastrophic disturbance events [7]. 62 There is increasing evidence that the frequency and severity of climate extremes and 63 associated catastrophes have already increased in several regions [8, 9]. Hence, there 64 is the urgent research need to meet the challenges posed by extreme events and 65 catastrophic disturbances. Despite this, their evolutionary consequences have 66 largely been unexplored (e.g., [10]), and more attention has been paid by ecologists 67 to exploring adaptations of individuals to changing trends in climate variables (e.g., 68 [11]), abrupt changes in the environment [12] or smooth and periodic changes 69 [13,14]. 70 Recent individual-based quantitative genetic models with stochastic dynamics [15] 71 suggest that evolution may quickly rescue populations after they collapse under 72 abrupt environmental change. Fitness should initially decline after the 73 environmental change, but then recover through adaptation [16]. According to 74 theory, whether populations can be rescued by evolution depends upon several 75 crucial variables, including population size, genetic variation within the population, 76 and the degree of maladaptation to the new environment [17]. Using Monte Carlo 77 simulations, Bena et al. [14] compared the effects of a smooth variation of the 78 optimum phenotype (as determined by environmental conditions) with those 79 emerging from an abrupt change of the environment and found that sufficiently 80 large mutation rates can increase substantially the probability of population 81 persistence in both scenarios. 82 In many systems, selection is not only directional (e.g., higher temperatures, higher 83 rainfall) but fluctuates (e.g., through cycles, stochastically). Climate change models 84 show that the variance of climate variables like temperature or rainfall may change 85 much more dramatically than their means [18] and will thus intensify the stochastic 86 component of selection. Shifts and excursions might cause some populations to 87 perpetually chase (evolutionarily) alternate optimal phenotypic extremes. Such 88 populations would face a demographic cost if evolution during one environmental 89 phase resulted in maladaptation and reduced favorable genetic variation with 90 respect to the next. This could be exacerbated by strong selection pressure and little 91 opportunity for the emergence of new phenotypes (i.e., high hereditability of traits 92 and low mutation rates). With climate change, it is possible that by chance the 93 population will experience a long sequence of particularly extreme environments [3]. 94 This may cause the population growth rate to be negative for a long enough time to 95 cause extinction. In addition, even when extinction does not follow immediately the 96 extreme event, the loss of genetic variability resulting from a low population size 97 can substantially reduce the population's ability to respond to future selective 98 challenges and increases the chances of an extinction vortex [19]. 99 Despite the inherent difficulties of predicting the impact of climate change on 100 species persistence and evolutionary trajectories, the general trends and dynamics at 101 the individual- and population-level are reasonably comprehensible and modeling 102 can provide probabilistic expectations for population dynamics and evolutionary 103 processes. Here, we want to investigate the implications of selection pressure and 104 mutation rates on the behavior of simulated populations living in a habitat subject 105 to changing selective processes. Monte Carlo simulation represents a valid tool to 106 understand and predict the consequences of variations in trends (i.e., directional 107 change) and stochasticity (i.e., increase in variance) of climate variables and 108 consequent selective processes by using simple models of population dynamics. 109
2. Model of population dynamics
110 We consider a population that consists of hermaphrodite individuals living in a 111 spatially-extended habitat modeled as a vector of length K , where K is the carrying 112 capacity of the system (i.e., maximum number of individuals supported). This 113 means that only one individual can occupy each element j =1… K of the vector, and 114 introduces density-dependent population regulation through a ceiling effect, as 115 described below. We assume that individuals cannot move, therefore an individual 116 occupies the same element during the simulation. 117 The populations has discrete generations (i.e., reproduction is discrete in time) and 118 is composed of N( t ) individuals. Generations are overlapping, meaning that parents 119 do not die after reproducing. Each individual is characterized by a single 120 quantitative trait φ with value ranging from 0 to 1. The population lives in a habitat 121 characterized by an optimum phenotype Θ( t) that exhibits temporal change. This is 122 assumed to result from variations in a climate variable, such as rainfall or 123 temperature, selecting for a phenotype. The degree of maladaptation between the 124 optimum phenotype Θ( t) and a single trait φ i defines the fitness of an individual. 125 The time step is one year. 126 In general, the temporal change of the optimum phenotype may be either 127 directional, stochastic or a combination of both. A simple model for this is a 128 optimum phenotype Θ( t) that moves at a constant rate β µ per year, fluctuating 129 randomly about its expected value µ ( t ). We thus introduce a directional and 130 stochastic temporal change of the optimum phenotype (Fig.1a). Θ( t) is randomly 131 drawn at each time step from a normal distribution Ν ( µ ( t ),sd( t )), where µ (t) = µ + 132 β µ t and sd ( t ) = sd + β sd t . 133 In the context of climate variables, extreme weather events causing catastrophic 134 disturbances (i.e., very large deviations of a system’s behavior from the habitual 135 one) in a sequence of independently and identically distributed random variables 136 are either the maximal values in a time window or they are defined by overcoming 137 a predefined threshold (threshold crossing) [20]. In our model, values of Θ( t)
138 outside (0,1) represent an extreme event causing a catastrophic disturbance to which 139 the vast majority of individuals cannot be adapted, thus causing a population 140 collapse (i.e., strong reductions in population size). This may be interpreted as a 141 catastrophic flood following an exceptional rainfall or a heat wave caused by high 142 temperatures.
143 Our model is similar in spirit to the one used by Droz and Pekalsky [21]. The fitness 144 of an individual i with trait value φ i is: 145 f( i )= | φ i – Θ( t ) | (1) 146 The probability p of an individual i with fitness f( i ) survive to next year is: 147 p( i )= exp(-( s * f( i )) ) (2) 148 where s is the selection pressure. With increasing s the habitat is more demanding 149 (for a given fitness f the probability of survival decreases). Since no individual can 150 be perfectly adapted to the moving optimum phenotype Θ, we did not account for a 151 decrease in survival probability with age (in case of constant Θ over simulation 152 time , accounting for it would be necessary to avoid the presence of individuals 153 living forever). 154 Offspring inherit the trait φ from its parents p and p as follows: 155 φ = φ p + φ p2 ) + M ε (3) 156 where φ is the trait value of the offspring, φ p and φ p2 are the trait values of the 157 parents, M represents mutation-segregation-recombination [22] and ε is random 158 number drawn from a uniform distribution bounded between (-1,1). We will refer to 159 M as simply mutation. 160 The Monte Carlo simulation at a time t during the simulation proceeds as follows: 161 1) We draw the optimum phenotype Θ( t) from Ν ( µ ( t ),sd( t )) .
162 2)
We compute the fitness of individuals by applying Eq (1) and calculate their 163 survival probability by applying Eq (2). 164 3)
We define the survival of individuals with Bernoulli trials. 165 4)
We compute the total number of individuals alive N( t ) and check the 166 distribution of trait φ in the population. A population is considered extinct if 167 at any time during the simulation there are less than ten individuals left. 168 5) We pick the first individual alive starting from j = 1. When the individual j is 169 alive, we check if the ( j +1) individual is alive. If yes, the parents j and ( j +1) 170 produce randomly from 1 to 4 offspring (we chose 4 as the maximum 171 number of offspring produced by following a pattern-oriented procedure [23] 172 to allow for a quick rebound of population size after a strong reduction 173 caused by an extreme event). If no, the individual j does not reproduce. This 174 introduces the Allee effect [24], that is a positive density-dependent effect at 175 low densities through higher mating opportunities. Then, we proceed to ( j +2) 176 and repeat the procedure up to j = K . 177 6) As we assume that the optimum phenotype Θ( t) defines the whole time-step, 178 we applied steps (2) and (3) to offspring. Following an example provided 179 above, a heat wave affects the survival of both adults and offspring. This 180 further intensifies the selective consequences of the optimum phenotype. 181 Offspring are placed randomly on the empty elements of the vector to avoid 182 spatial autocorrelation. When all the empty elements have been occupied, the 183 remaining offspring die (density-dependence through a ceiling effect). 184 Offspring at year t become adults at year t +1 and are able to reproduce. 185 Our simulation model has the following control parameters: carrying capacity of the 186 environment K , mutation M , selection pressure s and the parameters which govern 187 directional and stochastic variations of the optimum phenotype, that is µ , β µ , sd
188 and β sd . To simplify the interpretation of results, we set some of the parameters. For 189 each replicate: K = 2000, µ = β µ = 0.001 and sd = 0.1. Simulations were 190 performed for combinations of s (from 2.5 to 3.5), M (from 0 to 0.2) and β sd taking 191 values 0.0005, 0.0010, 0.0015, 0.0020 (scenarios of increasing variability of Θ over 192 simulation time, Fig. 1b). 193 0 Every simulation replicate lasts 100 years and starts with 500 individuals with trait 194 φ drawn from a uniform distribution bounded between (0,1) (Fig.2). We chose a 195 population size of 500 individuals (one fourth of carrying capacity) because we 196 wanted to explore the space of parameters allowing for extinctions in the first years 197 of simulation. 198 We use different quantities to characterize the behavior of the simulated 199 populations. At the level of single replicates, we recorded: 200 i) Extinction or survival (coded as a binary variable equal to 0 for persistence 201 and 1 for extinction). 202 ii)
Time of extinction, obviously recorded only for the populations going extinct 203 during simulation time. 204 iii)
Time-dependent value of the trait φ , and in particular the mean value of 205 φ ( ranging from 0 to 1) at the end of simulation time, only when the 206 population did not go extinct. 207 We did not focus on the number of individuals at the end of simulation time (100 208 years) since it was largely determined by the succession of Θ near the end of 209 simulations (Fig. 2). 210 For an ensemble of realizations (100 replicates for a fixed set of parameters), we 211 computed: 212 a) Frequency of population extinction, computed as the number of replicates in 213 which the population went below ten individuals during simulation time. 214 1 b)
Mean time to extinction (for the populations which went extinct during 215 simulation time). 216 c)
Mean across replicates of the mean value of trait at the end of simulation 217 time, for the replicates in which the populations did not go extinct. 218
3. Results and discussion
219 In Fig. 1b we show the probability of catastrophes with the different scenarios of 220 variability of Θ . The probability of a catastrophe, that is of optimum phenotype 221 Θ( t ) outside (0,1), reaches a maximum of 0.12 at the end of simulation time ( t = 100) 222 for the most variable scenario ( β sd = 0.0020). With the parameters we chose, there is 223 a higher probability of extreme events in the same direction as directional change 224 (more values of Θ > 1 than < 0 are expected), although the probability of both events 225 increases over the simulation time (Fig. 1a). In other words, with increasing 226 temperatures there is a higher probability of heat waves than of cold waves and 227 with increasing rainfall (and thus increasing flows) there is an higher probability of 228 floods than of droughts. 229 The consequences of different values of β sd for the probability of extreme events is 230 clear after the first 40-50 years of simulation while little difference among scenarios 231 in the probability of extreme events can be noted before that time. In Fig. 2 we 232 report examples of replicates for the four scenarios of variability. For all replicates 233 we set s = 3 and M = 0.1 (thus intermediate values for both parameters). With 234 higher values of β sd the population shows repeated collapses. Selection tries to bring 235 2 the average trait close to the instantaneous optimum, while mutation introduces 236 diversity and broadens the distribution of the trait. There is a clear shift in all 237 replicates of the mean value of trait φ toward 0.6 over simulation time - which is the 238 value taken by µ when t = 100 - even in high variability scenarios. The only 239 exception is the scenario with the highest variability, in which at t = 100 there is a 240 mean value of trait φ in the proximity of 0.5. In the specific example provided, a few 241 years of Θ below φ toward lower values than those selected 242 for by the directional component of Θ . 243 As noted by Siepielski et al. [25], the “temporal landscape” of selection across taxa 244 shows that the strength and the direction of selection often vary through time, even 245 in absence of climate change. Especially with strong selection pressure and high 246 variability of the optimum Θ, alternating selection over time might cancel out 247 periods of directional selection such that effective selective (quasi) neutrality of trait 248 variation is maintained over time. However, after a single extreme event or a 249 succession of them, this balancing effect does not occur, leading to directional 250 changes in trait frequency within the population. Apart from the contribution of 251 directional change, the distribution of trait φ is “pulled” toward higher values over 252 simulation time, since there is a higher probability of extreme events in the same 253 direction as directional change (as previously discussed). 254 In Fig. 3 we present a phase diagram of equal probability of extinction in the 255 mutation-selection plane for each scenario of variability of Θ . The survival chances 256 of a population depend quite strongly on the selection pressure and decrease 257 3 substantially with increasing β sd for the same selection-mutation combinations, 258 indicating that populations could rarely adapt to a strong linear increase in variance 259 of Θ . 260 There is a range of the selection pressure values within all scenarios of variability in 261 which populations have some probability to persist (Fig. 3). Outside this range, 262 broadly for s higher than 2.8, the probability of extinction increases in all scenarios. 263 If selection is too strong, then the distance between the average phenotype and the 264 optimum is small at any time during simulation, but the decrease in population size 265 induced by selection may be too high for population persistence. If the selection is 266 weaker, fewer individuals die from ill-adaptation and the population can persist 267 with a greater diversity in trait φ . 268 For β sd = 0.0020 populations can survive only with intermediate levels of mutation 269 and very low selection pressure, while extinction is the inevitable outcome for all 270 other selection-mutation combinations (Fig. 3). The adaptive value of intermediate 271 levels of mutation is clear also for β sd = 0.0005 and for β sd = 0.0010, while for β sd = 272 0.0015 the only clear pattern is along a selection gradient. It appears from Fig. 3 that 273 increasing mutation amplitude is adaptive up to intermediate values, while higher 274 mutation values are not adaptive (they increase the probability of population 275 extinction). 276 Contrary to our results, Bena et al. [14] found that mutation is unfavorable to the 277 survival of a population in a constant environment, since it increases the probability 278 4 of a mismatch of offspring phenotype to the environment optimum, even though 279 the parents might be well-adapted. Therefore, any level of mutation will result in 280 the production of non-optimal trait in a constant environment (given an adapted 281 population), but it will increase the probability of tracking a moving optimum and 282 thus increase the survival chances of a population. According to our results, even in 283 presence of high variability of the optimum phenotype Θ, high mutation increases 284 the probability of losing adaptations in the next generation and thus decreases the 285 probability of population persistence. When mutation is low, the population cannot 286 track the variations of Θ. In conclusion, for both mutation extremes (high or low 287 mutation) there is an increase in the probability of maladaptation, albeit for 288 different reasons, and consequent risk of extinction. 289 The influence of selection, mutation and β sd on the average time to extinction is 290 reported in Fig. 4. For β sd = 0.0005, for the few populations going extinct with 291 intermediate mutations, this happens only in the first years of simulation after an 292 unfavorable succession of alternate Θ (direction of selection varying through time). 293 With intermediate selection pressure, populations go extinct mostly at the end of 294 simulation time, when an increase of occurrence of extreme values is expected for 295 all scenarios of variability (Fig. 1). An increase in selection pressure tends to 296 decrease time of extinction in all scenarios of variability. 297 In general, the system is able to track the directional component of the optimum 298 (Fig. 5). The mean value of trait φ at the end of simulation time does not depend on 299 selection, therefore even for very small selective pressure and in presence of 300 5 sufficient mutation M , the mean value of trait φ follows the directional component 301 of Θ (
Fig . 6) . With no mutation or very low mutation, there is little potential for 302 adaptive shifts and thus the mean value of φ is largely determined by the optimum 303 phenotypes in the first few years (Fig. 6). For β sd = 0.0005 and β sd = 0.0010 the mean 304 value of trait φ in the population increases, and thus tracks the changes in µ (t), also 305 for very high mutation. In contrast, for β sd = 0.0015 and β sd = 0.0020 the mean value 306 of trait φ increases with increasing mutation, but with very high mutation the mean 307 value of trait φ tends to be lower than in scenarios with lower variability . Since in an 308 substantial fraction of replicates with high mutation the population went extinct 309 (Fig. 3), we cannot exclude that for only a particular sequence of Θ near the end of 310 simulation time (resulting in mean value of trait close to 0.5) the populations were 311 able to persist, thus preventing more general insights. 312
4. Conclusions
313 Extreme events occur in all systems with complex dynamics, but the details of the 314 creation of these large fluctuations are still rarely understood, and therefore their 315 prediction, including that of their consequences on natural populations, remains a 316 challenge. However, many significant impacts of climatic change are likely to come 317 about from shifts in the intensity and frequency of extreme weather events and the 318 prediction of their effects on population dynamics and evolution of traits in natural 319 populations call for wide and intense scientific investigations. These events may 320 result in rapid mortality of individuals and extinction of populations or species [26, 321 6 27, 28, 29, 30] and changes in community structure and ecosystem function [31, 32, 322 33]. Variations in disturbance timing, predictability, frequency and severity make 323 difficult to predict sign and strength of selection [10]. In some cases, catastrophic 324 events may be so swift or severe that there is little possibility for adaptive 325 responses, with population extinction being the inevitable result. However, given 326 sufficient evolutionary potential (i.e., genetic variation within a populations), 327 models suggest that species can survive the effects of extreme events [34]. However, 328 if variability of the optimum phenotype is too high, a relevant potential for 329 extinction exists even when populations might possess genetic variation for 330 adaptation. 331 Despite simplifying the life-cycle of a natural population, the model we have 332 presented here provides a useful starting point for the investigation of the potential 333 of the populations to adapt (and survive) to an increase in the variability of 334 environmental conditions. The simulations showed that the probability of survival 335 of populations is dramatically affected by slight increases of the variance of the 336 optimum phenotype. Although not universal across scenarios of variability, 337 intermediate mutation seem to be adaptive, while increasing selection pressure 338 consistently decreases the probability of population persistence. 339
Acknowledgements
340 The authors thank Luca Bolzoni and Kate Richerson for discussion and comments 341 which greatly increased the quality of the manuscript. 342 7
References
343 [1] S. Carrol, Facing change: forms and foundations of contemporary adaptation to 344 biotic invasions, Mol. Ecol. 1 (2008) 361-372. 345 [2] IPCC. Contribution of Working Group I to the Fourth Assessment Report of the 346 Intergovernmental Panel on Climate Change, 2007. 347 [3] A. Jentsch, J. Kreyling, C. Beierkuhnlein, A new generation of climate change 348 experiments: events, not trends, Front. Ecol. Environ. 5 (2007) 315–324. 349 [4] M. Smith, The ecological role of climate extremes: current understanding and 350 future prospects, J. Ecol. 3 (2011) 651–655. 351 [5] P. S. White, M.D. MacKenzie, R.T Busing, Natural disturbance and gap phase 352 dynamic in southern Appalachian spruce-fir forests, Can. J. For. Res. 15(1985)233-353 240. 354 [6] A. Wagner, Risk management in biological evolution, J. Theor. Biol. 225 (2003) 355 45–57 356 [7] C. A. Stockwell, A. P. Hendry, M. T. Kinnison, Contemporary evolution meets 357 conservation biology, Trends Ecol. Evol. 18 (2003) 94-101. 358 [8] T.R. Karl, R.W. Knight, N. Plummer, Trends in high-frequency climate 359 variability in the 20th century, Nature 377 (2005) 217–220. 360 8 [9] C. Schär, P.L. Vidale, D. Luthi, C. Frei, C. Haberli, M.A. Lininger, C. Appenzeller, 361 The role of increasing temperature variability in European summer heatwaves, 362 Nature 427 (2004) 332-336. 363 [10] M.G. Turner, Disturbance and landscape dynamics in a changing world. 364 Ecology, 91 (2010) 2833-2849. 365 [11] L.-M. Chevin, R. Lande, G.M. Mace, Adaptation, Plasticity, and Extinction in a 366 Changing Environment: Towards a Predictive Theory, PLoS Biol. 8 (2010) e1000357. 367 [12] I. De Falco, A. Della Cioppa, E. Tarantino, Effects of extreme environmental 368 changes on population dynamics, Physica A 2 (2006) 619-631. 369 [13] A. Pękalski, M. Ausloos, Risk of population extinction from periodic and abrupt 370 changes of environment, Physica A 11 (2008) 2526-2534. 371 [14] I. Bena, M. Droz, J. Szwabinski, A. Pekalski, Complex population dynamics as a 372 competition between multiple time-scale phenomena, Phys. Rev. E 76 (2007) 011908. 373 [15] R.D. Holt, R. Gomulkiewicz, Conservation implications of niche conservatism 374 and evolution in heterogeneous environments, in: R. Ferriere, U. Dieckmann, D. 375 Couvet (Eds.), Evolutionary Conservation Biology, Cambridge University Press, 376 Cambridge, UK, 2004, pp. 244-264. 377 [16] A. Burt, The evolution of fitness, Evolution 49 (1995) 1-8. 378 [17] G. Bell, A. Gonzalez, Evolutionary rescue can prevent extinction following 379 environmental change, Ecol. Lett. 9 (2009) 942–948. 380 9 [18] V. V. Kharin, F.W. Zwiers, Changes in the extremes in an ensemble of transient 381 climate simulations with a coupled atmosphere-ocean GCM, J. Climate 13 (2000) 382 3760-3788 383 [19] G. Caughley, Directions in conservation biology, J. Anim. Ecol. 63 (1994) 215-384 244. 385 [20] S. Albeverio, V. Jentsch, H. Kantz, Extreme Events in Nature and Society, 386 Springer, Berlin, 2006. 387 [21] M. Droz, A. Pekalski, Population dynamics in heterogeneous conditions, 388 Physica A 362 (2006) 504–512. 389 [22] M. Ridley, Evolution, Wiley-Blackwell, 1996. 390 [23] V. Grimm, E. Revilla, U. Berger, F. Jeltsch, W.M. Mooij, S.F. Railsback, H-H. 391 Thulke et al., Pattern-oriented modeling of agent-based complex systems: Lessons 392 from Ecology, Science 310 (2005) 987-991. 393 [24] P. A. Stephens, W. J. Sutherland, R. P. Freckleton, What is the Allee effect?, 394 Oikos 87 (1999) 185-190. 395 [25] A.M. Siepielski, J.D. Di Battista, S.M. Carlson, It’s about time: the temporal 396 dynamics of phenotypic selection in the wild, Ecol. Lett. 12 (2009) 1261-76. 397 [26] C. Bigler, O. Ulrich Bräker, H. Bugmann, M. Dobbertin, A. Rigling, Drought as 398 an inciting mortality factor in Scots pine stands of the Valais, Switzerland, 399 Ecosystems 9 (2006) 330-343. 400 0 [27] A. R., Gitlin, C. M. Sthultz, M. A. Bowker, S. Stumpf, K. L. Paxton, K. Kennedy, 401 A. Muñoz, J. K. Bailey, T. G. Whitham, Mortality gradients within and among 402 dominant plant populations as barometers of ecosystem change during extreme 403 drought, Conserv. Biol. 20 (2006) 1477–1486. 404 [28] M.N. Miriti, S. Rodriguez-Buritica, S.J. Wright, H.F. Howe, Episodic death 405 across species of desert shrubs, Ecology 88 (2007) 32-36. 406 [29] C. Bigler, D. G. Gavin, C. Gunning, T. T. Veblen, Drought induces lagged tree 407 mortality in a subalpine forest in the Rocky Mountains, Oikos 116 (2007) 1983-1994. 408 [30] K. M. Thibault, J. H. Brown, Impact of an extreme climatic event on community 409 assembly, PNAS 105 (200) 3410-3415. 410 [31] N.M. Haddad, D. Tilman, J.M.H Knops, Long-term oscillations in grassland 411 productivity induced by drought, Ecol. Lett. 5 (2002) 110–120. 412 [32] Ph. Ciais et al., Europe-wide reduction in primary productivity caused by the 413 heat and drought in 2003, Nature 437 (2005) 529-533. 414 [33] R.C. Mueller, C.M. Scudder, M.E. Porter, R. III Talbot Trotter, C.A. Gehring, 415 T.G. Whitham, Differential tree mortality in response to severe drought: evidence 416 for long-term vegetation shifts, J. Ecol. 93 (2005) 1085–1093. 417 [34] R. Burger, C. Krall, Quantitative-genetic models and changing environments, 418 in: R. Ferriere, U. Dieckmann, D. Couvet (Eds.), Evolutionary Conservation Biology, 419 Cambridge University Press, Cambridge, UK, 2004, pp. 171–187. 420 421 1
Figure Captions
422 Fig. 1 –
Weather extremes. (a) Expected increase in the probability of occurrence of 423 extreme weather events with climate change (gray areas) for an hypothetical climate 424 variabile (e.g., rainfall, temperature), as defined in our model. Solid line represent 425 current scenario ( µ = 0.5, sd = 0.1) while dotted line represent a future scenario at 426 the end of simulation time (dotted line, µ = 0.6, sd = 0.25). Jentsch et al. [3] and 427 Smith [4] provided similar representations. (b) Expected probability of optimum 428 phenotype Θ( t) outside (0,1) for different changes in variability during simulation 429 time. Solid line - β sd = 0.0005; short-dashed line - β sd = 0.0010; long-dashed line - β sd
430 = 0.0015; dashed-dotted line - β sd = 0.0020. 431 Fig. 2 – Examples of simulations . Examples of simulation for the four scenarios of 432 variability with selection pressure s = 3 and mutation M = 0.1. The optimum 433 phenotype Θ( t) is randomly drawn at each time step from a normal distribution Ν
434 ( µ ( t ),sd( t )), where µ (t) = µ + β µ t and sd ( t ) = sd + β sd t . The histograms represent the 435 distribution of trait φ at t = 1, 20, 40, 60, 80, 100. The vertical dashed line is set at 0.5. 436 The mean value of trait of the population tracks the directional change ( µ = t = 437 100) in all the examples of simulation except for β sd = 0.0020 at t = 100. The 438 fluctuations in population size tend to increase with increasing β sd, parallel to 439 increase in fluctuations of optimum phenotype Θ( t).
440 Fig. 3 –
Phase diagram for extinction probability.
Phase diagram of equal probability of 441 extinction in the mutation-selection plane for the four scenarios of variability of 442 2 Θ ( β sd = 0.0005, 0.0010, 0.0015, 0.0020). The frequency of population extinction for 443 combinations of selection pressure s and mutation M is computed as the number of 444 replicates in which the population did go below ten individuals during simulation 445 time. 446 Fig. 4 - Phase diagram for mean time to extinction.
Phase diagram of equal mean time to 447 extinction in the mutation-selection plane for the four scenarios of variability of 448 Θ ( β sd = 0.0005, 0.0010, 0.0015, 0.0020). 449 Fig. 5 - Phase diagram for mean value of trait.
Phase diagram of equal mean across 450 replicates of the mean value of trait φ at the end of simulation time in the mutation-451 selection plane for the four scenarios of variability of Θ ( β sd = 0.0005, 0.0010, 0.0015, 452 0.0020). The mean was computed only for the populations which persisted up to the 453 end of simulation time. The white region in the phase diagram for β sd = 0.0020 454 indentifies mutation-selection combinations for which populations had no chances 455 to persist up to end of simulation time (see Fig. 3). 456 Fig. 6 – Distribution of trait for increasing mutation.
Examples of the distribution of 457 trait φ for increasing mutation M at the end of simulation time. All simulations 458 performed with s = 3 and β sd = 0.0010. 459 460 3 Figure 1
461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476
Climate parameter or phenotypic optimum P r obab ili t y o f o cc u rr en c e -0.5 0 0.5 1 1.5 0 20 40 60 80 100 . . . . . . . Year P r obab ili t y o f c a t a s t r ophe a b Mean Variability φ φ φ φ φ φ Year P opu l a t i on s i z e . . . . . . Year O p t i m u m φ φ φ φ φ φ Year P opu l a t i on s i z e . . . . . . Year O p t i m u m Figure 2
477 478 t = 1 20 40 60 80 100 t = 1 20 40 60 80 100 φ φ φ φ φ φ Year P opu l a t i on s i z e . . . . . . Year O p t i m u m t = 1 20 40 60 80 100 φ φ φ φ φ φ Year P opu l a t i on s i z e . . . . . . Year O p t i m u m t = 1 20 40 60 80 100 ββββ sd = 0.0005 ββββ sd = 0.0010 ββββ sd = 0.0015 ββββ sd = 0.0020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S e l e c t i on Mutation
Figure 3
479 480 481 Mutation S e l e c t i on Figure 4
482 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S e l e c t i on Mutation