Extinction risk and the persistence of fragmented populations

Transcription

1 Extinction risk and the persistence of fragmented populations Dipartimento di Elettronica e Informazione Politecnico di Milano Milano, Italy Dick Levins 1

2 The world biodiversity 1.8 million species known Estimate of existing species: 3-30 million 2

3 Extinction rates Average life time of animal species from fossil records: 1-10 million years. From extiction rates recorded in the past century: life time reduced to 10,000 ys From current extinction rates for birds and mammals: life time = ys 3

4 Cited phenomena leading to extinction Inverse density dependence (Allee effect) Sociality Defense against predators Pollination American Loss of genetic variability ginseng Genetic drift and inbreeding Bottlenecks and founder effect Demographic stochasticity Ngorongoro crater lions Synergism with Environmental stochasticity 4 Deterministic chaos Sperm abnormalities

5 The dynamics of extinction 5

6 Habitat fragmentation and destruction Causes of endangerment in the USA. Percentage of species being endangered by each cause. All species Vertebrates Invertebrates Plants Habitat destruction and 85% 92% 87% 81% degradation Exotic species introduction 49% 47% 27% 57% Pollution 24% 46% 45% 7% Overexploitation 17% 27% 23% 10% Diseases 3% 11% 0% 1% Isolated habitat remnants in Western Australia 6

7 The biosphere level: habitat fragmentation Brazil Forests in England 7 Forests in Costarica

8 Destruction can be fast August June July (Landsat 4) 2) 7) 8

9 The soy saga August February June

10 Endangered biodiversity from GCC and habitat loss The Monte Verde golden toad: Extinct Edith s Checkerspot Butterfly: Moved Keel-billed Toucan: Moved Mosquitoes: On the Rise Projection: 25% of the world species extinct by 2050 Thomas et al. (2004), Nature 427:

11 Endangered biodiversity in Europe 11

12 Fragmented habitat in the lab Eotetranychus sexmaculatus Typhlodromus occidentalis 12

13 Populatios are patchy Slobodkin Andrewartha 13

14 Populations are made of subpopulations 14

15 Islands and mainland From island biogeography to 15

16 Populations of populations 16

17 The problem of global extinction in single species metapopulations with discrete individuals Outline The metapopulation paradigm and modeling approaches The importance of being discrete A spatially implicit Markovian model and its moment closure A spatially explicit stochastic model (IPS) with logistic demography The role of random disturbances: habitat loss and environmental catastrophes 17

18 How many variables? The problem of scale + theoretical realistic landscape Describing the habitat... boolean intermediate discrete individuals...and the local demography 18

19 The metapopulation model (Levins, 1969) Similarity with the logistic equation Consider a large number of patches at a given time p = proportion of occupied patches c = colonization rate = the probability that migrants from any given population reach another site e = extinction rate of local populations (demographic and environmental stochasticity) dp cp ( 1 p) ep dt = dp dt = ( c e) p 1 1 p e c The persistence-extinction boundary 19

20 Different modeling approaches Local demography description simple Boolean and implicit (e.g., Levins-like models) Structured and implicit (e.g., Markovian models or PDEs) Habitat description Boolean and explicit (e.g., Incidence Function and SRLM) Structured and explicit (e.g., Coupled maps or IPS) complex 20

21 Various approaches Presence-absence CA Molofsky (1994) Ecology 75:30 Hiebeler (1997) J. theor. Biol. 187:307 + spatially explicit + very flexible boolean Coupled maps + spatially explicit + structured no discrete individuals Hastings (1993) Ecology 74: 1362 Allen et al. (1993) Nature 364:229 Earn et al. (2000) Science 290:

22 Available data implicit-boolean Argiope spp. explicit-boolean Melitaea diamina Schoener and Spiller (1987) Nature 330:474 implicit-structured Bolitotherus cornutus explicit-? Wahlberg et al. (1996) Science 273:1536 Rana lessonae Whitlock (1992) Am. Nat. 139:952 Sjörgen (1994) Ecology 75:

23 The importance of discrete individuals The core of metapopulation concept (and of island biogeography): Each local population is prone to extinction because each habitat patch hosts a finite number of individuals, subject to a sequence of random events In endangered populations (abundance is small in each patch) this number cannot be approximated by a real number, must be integer Correct evaluation of metapopulation persistence 23

24 A spatially-implicit structured model p = probability that a patch is occupied p i = probability that i (integer) individuals be contained in a patch colonization demographic stochasticity birth or arrival death or dispersal no event 24

25 and the intermediate dispersal principle If the expected number of emigrants from a patch that is begun with one individual, and to which subsequent immigration is excluded is greater than unity Chesson (1984) Zeitschrift für Wahrscheinlichkeitstheorie 66:105 Metapopulation persistence is guaranteed E 0 j 1 2 j 1 = a >1 j = 0 µ + D ( µ + D ) L ( µ + D ) j D j ν ν L ν 1 1 j 1 j 1 In a fragmented habitat, an intermediate level of dispersal is generally required for metapopulation persistence Intrinsic rate of increase r p 0 < 1 25 Casagrandi and Gatto (2002a) Theor. Pop. Biol. 61:115

26 The persistence-extinction boundary (for species with the same carrying capacity) Intrinsic rate of increase r Estimating the dispersal rate Deterministic approximation dn dt = birth rate death rate dispersal rate to unsuitable habitat 26

27 Real histograms Pterostichus lepidus P. niger den Boer (1990) J. of Evol. Biol. 3:19 Pterostichus niger Parus montanus Parus cristatus Parus major Haila et al. (1993) Ecology 74:714 27

28 The compact models Extinction risk p 0 = + Probabilities conditioned on non extinction pi δ i = 1 p 0 Moment closure technique Fraction of empty patches Average number of individuals in occupied patches Variance of distribution in occupied patches Advantages Finite (and small!) dimension Persistence-extinction boundaries as simple bifurcation manifolds 28 Casagrandi and Gatto (1999) Nature 400:560

29 Some 2D examples 29

30 and the intermediate dispersal principle 30 Casagrandi and Gatto (2002b) Theor. Pop. Biol. 61:127

31 Robustness of the results Durrett and Levin (1994) Theor. Pop. Biol. 46:363 31

32 Interacting Particle Systems A system of particles (cells, organisms, etc.) that reproduce, move and die on a countable graph (e.g. a lattice) x=space 32 Simulator (winsss)

33 A spatially-explicit model (stochastic cellular automaton or IPS) Dispersal rules Habitat characteristics Patches arranged in square grids (k x k matrix) Typically small grids (6x6 to 30x30) Time steps are discrete (z = 0,1,2,...) Demographic characteristics Abundance is discrete (n xy =0,1,2,...) Local dynamics is logistic Von Neumann (nearest 4 neighbors) Moore (nearest 8 neighbors) Propagule rain (from one patch to any other patch, p=1/k 2 ) 33

34 A spatially-explicit model Conditions at the edge Periodic Absorbing Reflecting Typical pattern (10x10, Von Neumann, periodic, n ij (0)=K/2) 34 local

35 Finding persistence-extinction boundaries in IPS From a theoretical viewpoint The model is a finite Markov chain (ordered strings of k 2 integer numbers) Two equivalence classes: extinction (null string) and all other strings Extinction is recurrent and absorbing lim p () t t x 1 x = 0 = 0 x 0 However, the time to global extinction can be astronomically long! Evidence from the contact process Durrett & Levin (1994) Phil. Trans. R. Soc. B. 343:329, Schinazi (book, 1999) Infinite lattice: the process survives if and only if λ θ c µ > (percolation in space-time) Finite lattice: the expected global extinction time grows as Much larger than the average time to local extinction 2 2 ( ) = LOC ( ) T 1 GLOB : exp ck T exp ck µ 35

36 The contact process Space Time Higher mortality Lower mortality No more than one organism per site 36

37 Finding persistence-extinction boundaries in IPS The local extinction time in the stochastic Verhulst model Nåsell (1996) Adv. Appl. Prob. 28:895; Nåsell (2004), preprint r < 0 If the time to extinction r > 0 scales as T LOC 1 r 1 g K r (, N ) Heuristic definition Persistence-extinction boundaries can be seen as percolation thresholds from extinction to persistence over reasonable time horizons (e.g., much larger than 1/r but not astronomical) Rule of thumb take T = 100/r 37

38 Persistence-extinction boundaries: the computational burden Algorithm for detection (Sequential Probability Ratio Test) Fix two hypotheses to be contrasted (e.g., H 0 = prob. of persistence is 0.8, H 1 = prob. of persistence is 1) Choose the acceptable errors (e.g., type I = 0.05, type II = 0.1) Evaluate trial after trial if the number of simulations performed is enough to discriminate between H 0 and H 1 38

39 The persistence-extinction boundaries in space-explicit metapopulations 30x30, Periodic edge conditions 6x6, Von Neumann dispersal Space-implicit Space-implicit 39

40 From space-implicit to space-explicit Absorbing edge Periodic edge 6x6 grid 40

41 Is there any law of scale with the number of patches? Proportion of species that are lost because the patches are n (and not infinite) Slope is slightly sensitive to patch size (as measured by local carrying capacity K) 41

42 The cost of losing a patch Consider species with the same carrying capacity (e.g., similar in body size and same trophic level) distributed at random in parameter space r-d 0.3 Proportion of species lost Example: The cost of losing a patch when there are 5 patchesisthe loss of a further 4% of the species Number of patches Carrying capacity in each patch = 10 individuals 42

43 The role of disturbances Habitat loss and environmental catastrophes forest fires fragmentation rinderpest floods 43

44 Incorporating disturbances in the Levins model h = proportion of remaining suitable habitat out of the original habitat h - p = fraction of empty suitable patches e + m e + m * h if h > p = c c 0 otherwise m = rate of occurrence of catastrophes wiping out occupied patches dp cp e m dt = + ( h p) ( ) Persistence-extinction boundaries p 44

45 How habitat loss affects the persistence-extinction boundary Via the IPS and the compact space-implicit models 10x10, Von Neumann, Periodic 45

46 Understanding some conflicting experimental results M. oeconomus M. canicaudus In the present experiment, the emigration rate was higher in the permanently fragmented control populations than in the continuous treatment populations before habitat destruction Johannesen and Ims (1996) Ecology 77:1207 Reduced movement of animals among the small fragments was the most obvious effect of habitat fragmentation Wolff et al. (1997) Conservation biology 11:945 The example of Microtus spp. 46

47 How environmental disasters influence the boundary Via the IPS and the compact space-implicit models 47

48 The synergism between disturbances Space-implicit Space-explicit Finite number of patches effect 48

49 Conclusions Persistence-extinction boundaries in metapopulations with discrete individuals can be evaluated And there are a lot of other Analytically (Markovian model) Via bifurcation analysis (Compact models) Heuristically (Spatially explicit models) problems that can be explored and The intermediate dispersal principle is robust to model changes (a fortunate case sensu Durrett and Levin 1994) are left to you! There exists a scaling law that bridges explicit to implicit models; it allows a rough estimation of the cost of destroying a patch Thanks for your attention Frequent dispersers are hit more by random loss of habitat than by random environmental catastrophes and viceversa (but there can be synergism) 49