discrete variation (e.g. semelparous populations) continuous variation (iteroparous populations)

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1 Basic demographic models N t+1 = N t dn/dt = r N discrete variation (e.g. semelparous populations) continuous variation (iteroparous populations) Where r is the intrinsic per capita rate of increase of the population In this case r > r < r = Positive growth Negative growth Stability only

2 Are basic models of population growth complete? N t+1 = N t discrete dn/dt = r N continuous Are the predictions of these DM coherent with real pops? NOT, or at least NOT ALLWAYS Cricket population N Yeast cells N hrs 2 generations 1 We conclude that the basic DM models are NOT COMPLETE because they are not able to explain demographic homeostasis

3 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS We can imagine two classes of mechanisms: 1) Population growth depends on environmental conditions such as climate r or = f (C) 2) Population growth depends on the availability of resources (food, refugia etc) r or = f (R) We have already considered an implicit for of dependence of on environmental factors (environmental stochasticity in probabilistic Models We can now try to make this explicit by measuring the effect of different conditions levels on the rate of increase of the population

4 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS How to assess = f (C)? Autoecological experiments in the field or under controlled conditions measuring the variation of lambda or its components (mortality and fertility) when conditions do vary 25 Macrosiphum euphorbiae potato aphid Myzus persicae green peach aphid Temperature ( C) 3

5 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Dependance of instantaneous growth rate on two conditions: temperature and relative humidity.8.6 RH 14% Calandra oryzae The rice weevil r.4 RH 11%.2 RH 1% Temperature ( C) 3 35

6 Substrate humidity (g w Kg -1 ) MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Dependance of instantaneous growth rate on two conditions: temperature and substrate (weath) humidity 14 r >.5 r < 12 1 Rhizoperta dominica < r <.5 8 r < r < Temperature ( C)

7 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS In general, this kind of information is important for planning defence strategies against crop pests, or to assess optimal conditions for farming On the euristic side, dependence of rate of increase from environmental conditions can explain fluctuations of natural populations, but does not explain homeostasis As an alternative, dependence of the rate of increase from resorces can be considered This is the Thomas R. Malthus hypothesis: Populations do grow exponentially, but they stop growing under shortage of resources

8 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS The availability of the resorces can depend on fluctuations in habitat productivity (climate, nutrients etc.), but in a constant environment it depends on the number of individuals using them (negative density dependence) So the negative dependence of growth on per-capita resources availability = f (R) becames in fact the negative dependence of growth on population density, due to INTRASPECIFIC COMPETITION = f (N) This is why we speak in general of density dependent models of population dynamics But what about the experimental evidence for density-dependence?

9 Fertility (seeds per plant) MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Negative density dependence of growt rate (or of its components) is a common phenomenon in many different organisms Vulpia fasciculata Poaceae Plant population density (N m -2 )

10 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Negative density dependence of growt rate (or of its components) is a common phenomenon in many different organisms Fraction of seed survived until germination Cakile edentula Brassicaceae Seed density (N dm -2 )

11 Viable eggs per pair MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Negative density dependence of growt rate (or of its components) is a common phenomenon in many different organisms Great tit (Parus major) Number of pairs on a given area

12 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Negative density dependence of growt rate (or of its components) is a common phenomenon in many different organisms B fertility A fertility mortality mortality fertility K density 1 b d K density mortality In a density-dependence scenario there is a density value (K) corresponding to which fertility and mortality rate have exactly balanced values to produce null growth (equilibrium density) C K density

13 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS The population grows with a maximum potential growth rate ( max or r max ) only when population density is so low that competition for resources is null As its size approaches a critical value (K) its growth reduces; when its size reaches K the growth stops; when its size exceeds K, its growth becomes negative K is thus an equilibrium density since, for max > 1 if N t < K then N t+1 /N t > 1 if N t = K then N t+1 /N t = 1 if N t > K then N t+1 /N t < 1 N t+1 N t+1 = N t K N t

14 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS In the continuos growth scenario we have the equivalent idea if N < K then dn/dt > if N = K then dn/dt = if N > K then dn/dt < We can formalize this malthusian idea by the following logistic eqn dn dt = r N K N ( ) K Exponential growth Regulation

15 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Exponential vs. logistic growth Equilibrium density also called Carrying capacity (of that population in a given habitat)

16 RATE OF INCREASE OF POPULATIONS A) Total rate of increase (recruitment) Discrete: N t+1 N t Continuous: dn/dt B) Per capita rate of increase i.e. how the single (average) individual contributes to the population growth Discrete: (N t+1 -N t ) / N t Continuous: dn/n dt

17 GENERAL PREDICTIONS OF THE LOGISTIC MODEL 2) Increase at the level of the population (recruitment) Max growth rate > Instantaneous growth rate (dn/dt) < K/2 Population density (N) K

18 RECRUITMENT ACCORDING TO A LOGISTIC PATTERN Agreement of observational data (pheasant s recruitment in UK) with the general predictions of the logistic model Annual recruitment (N) Pheasant 1 N* Total population (Nx1)

19 GENERAL PREDICTIONS OF THE LOGISTIC MODEL 1) Rate of increase at the level of the single individual r max > Per-capita Growth rate (dn/n dt) < Population density (N) K

20 MECHANISMS ALLOWING DEMOGRAPHYC HOMEOSTASIS Negative density-dependence (competition) in a discrete-growth population The Maynard-Smith and Slatkin eqn. N t 1 1 N t a N t b Potential growth Per-capita effect of competition Variation of the per-capita effect with density 12 8 N t 4 when b=1 Carrying capacity K 1 a 1 b tempo 1

21 PREDICTIONS OF MS & S EQN. WHEN b> N t b = 1 N t b = tempo tempo N t 4 b = 4.5 Simple oscillations 4 b = 5.3 N t tempo tempo Dumping oscillations Double oscillations Intraspecific competition alone is able to generate such different dynamics N t 8 4 b = 5.7 Deterministic chaos tempo

22 PREDICTIONS OF MS & S EQN. What happens when the carrying capacity is exceeded (immigration, exhagerate introduction of specimens, resources fluctuation) N K b<1 Under-regulation b=1 Exact regulation b>1 Overcompensation t t+1 b>>1 Demographic collapse

23 Life table data and DM of structured populations Age specific mortality rate, growth rate and relative fertility determine the dynamics of single classes and of the whole population Annual bluegrass Poa annua p 1 seeds p 2 f 1 f 2 seedlings Younger adults Older adults p 3 p 5 p 7 p 4 p 6

24 DENSITY DEPENDENCE AND HOMEOSTASIS IN STRUCTURED POPs Transition probabilities may depend on density It is enough that some or even only one coefficient do vary with density for producing a density dependent regulation of the whole population Seed destiny Ageing and growing Fertility.2 P 2 1 P 6 2 F P 4 1 F N N N

25 Population size (N) DENSITY DEPENDENCE AND HOMEOSTASIS IN STRUCTURED POPs 1 6 seeds 1 4 seedlings Younger Plants Interm. Plants Older Plants Time (generations)

26 COMPETITION AND MUTUAL FACILITATION facilitation Density dependence can be negative at higher density (competition) but positive at lower density (mutual facilitation) r or competition N* K N

27 COMPLEX DENSITY DEPENDENCE In many cases density-dependence is more complex than expected if it would be generated only by competition Rotifera Brachionus 1 Euchlanis (higher temperature) 5 Euchlanis (lower temperature) N ml -1 Density

28 ALLEE EFFECT Allee effect can be due to many different processes through which fertility is increased and/or mortality is reduced as population density Increases: 1) Probability to find a sexual partner may be difficult at low densitiy and increases as density increases 2) Increased density may facilitate cooperation in searching for resources or for a better manipulation of them 3) Increased density may facilitate interindividual help in avoiding predators, by confusing them or by actively defending against their attacks (mobbing in birds)

29 Reproductive success (%) Allee effect (reproductive success) in Alca torda Density (N m -2 )

30 ALLEE EFFECT AND DEPENSATION Interaction between competition and facilitation generates depensation (e.g. lower growth at lower densities) Weak Alle effect: Small depensation Strong Allee effect: Critical depensation r or N - K N t

31 Annual recruitment (N 1) ALLEE EFFECT AND DEPENSATION Antarctic whales recruitment affected by depensation N* Total population (N 1)

32 Population (N t ) THE EFFECT OF CRITICAL DEPENSATION The Minimum Viable Population (MVP) Due to depensation there is a threshold density (Minimum Viable Population) below which there is not growth even if lambda max is greater than MVP Time units

33 THE EFFECT OF CRITICAL DEPENSATION The importance of knowing the Minimum Viable Population (MVP) for conservation and managing of natural populations N Stochastic oscillations Exploitation Insufficient reintroduction MVP Extinction vortex time

34 DENSITY DEPENDENCE IN r- AND K-SELECTED POPULATIONS In r-selected populations (unstable environments) growth rate is high at lower-medium density (high r), but besides a given density there is a rapid decrease and overcrowding is followed by population crash In K-selected populations (stable environments) growth rate is low at lower density (depensation) but the population remains stable when reaches the carrying capacity N t+1 r-selected Equilibrium line K-selected Stable equilibrium Unstable N t

35 DEMOECOLOGY AND EXPLOITATION POLICIES Exploitation of natural biotic resources (fishing, forest harvesting etc.) require to combine economical and ecological needs: A) To give the maximum short term gain compatible with long term exploitability of the resource (sustainability) A) To guarantee the persistence of population density enough to minimize future risk of extinction (conservation) Empirism does not work The only way to obtain policies able to generate acceptable policies combining these two needs is to know the demography of the exploited resources and to use models for predicting future density under different exploitation pressures

36 DEMOECOLOGY AND EXPLOITATION POLICIES Two basic strategies: fixed quota and constant effort A) Fixed quota Recruitment: number or biomass reproduced by the population in a given interval of time (e.g. one year) N t 1 N t R N t Q Extraction of a fixed quota (number, biomass)

37 DEMOECOLOGY AND EXPLOITATION POLICIES Recruitment curve in case of a density dependent regulation Recruitment (R) Theoretical Maximum Sustainable Yields N* K Population density (N)

38 DEMOECOLOGY AND EXPLOITATION POLICIES Extraction of constant quota of resources (diffent values) Q 3 Extraction (Q) Q 2 Q 1 Population density (N)

39 DEMOECOLOGY AND EXPLOITATION POLICIES Overlapping logistic recruitment and constant quota Q 3 >R max Extraction (Q) Q 2 =R max Q 1 <R max Recruitment (R) Stable equilibrium Unstable N - N* N + K Population density (N)

40 DEMOECOLOGY AND EXPLOITATION POLICIES Problems with fixed quota exploitation: stochasticity of recruitment Extraction (Q) Stable equilibrium Unstable Recruitment (R) N* Population density (N)

41 Relative abundance DEMOECOLOGY AND EXPLOITATION POLICIES Problems with fixed quota exploitation: reduction of whale populations during the whale games 3 2 Fin whale Sei whale 1 Blue whale

42 DEMOECOLOGY AND EXPLOITATION POLICIES B) Constant effort Exploitation effort includes: number of extraction items, their characteristics time of operation etc. (e.g. number of fishing vessels, total length of nets, kind of nets and trawling devices allowed, total number of vessels days) Recruitment: number or biomass reproduced by the population in a given interval of time (e.g. one year) N t 1 N t R N q E N t t Characteristics of extraction devices Number of extraction devices / days operation

43 DEMOECOLOGY AND EXPLOITATION POLICIES With constant effort the amount of resource actually extracted is proportional to the population density (regulated exploitement) Extraction q E Population density (N)

44 DEMOECOLOGY AND EXPLOITATION POLICIES Effects of constant-effort policies and Maxymum Sustainable Yield Unsustainable effort Supraoptimal effort Optimal effort MSY Exploitment (E) Recruitment (R) Suboptimal effort N* K Population density (N)

45 Extraction (Kg 1 6 ) DEMOECOLOGY AND EXPLOITATION POLICIES Under constant effort exploitment, the yield is maximum for a given (optimal) effort value. Greater effort release lower yields Optimal effort Spiny lobster Effort

46 DEMOECOLOGY AND EXPLOITATION POLICIES Constant effort may be dangerous in case of depensation Extraction (E) Recruitment (R) Apparently optimal, but risky effort Safe effort Depensation N - N* K Population density (N)