Population growth in ideal habitats How does a population grow when - PowerPoint PPT Presentation

Population growth in ideal habitats How does a population grow when colonizing an habitat under ideal physical and biological conditions ?

The muskox Original distribution: North America, Greenland Depleted by hunting from 1700 to 1850 Last individuals in Alaska: 1850-60

Nunivak Island Nunivak Island 31 animals, 1936

Geometric growth Initial population at the Nunivak reserve: 31 individuals Boi almiscarado na Ilha de Nunivak (Alaska) 800 600 Núm indivíduos 400 200 0 Muskox 1930 1940 1950 1960 1970 1980 (Ovibos moschatus) Anos

Measures of variation in N t t+ 1 N t N t+ 1 ∆ N > 0 growth ∆ t ∆ N = 0 no change ∆ N < 0 decline ∆ N = N t+ 1 -N t Absolute variation − ∆ ≡ N N N Mean variation over ∆ t variation time -1 + 1 = t t ∆ ∆ t t ∆ N ≡ 1 Mean relative variation % variation N i ∆ t

Finite rate of increase N = λ + λ t 1 = finite rate of increase N t What happens if λ remains constant ? = λ N N + + t 2 t 1 = λ λ = λ 2 N N N + t 2 t t = λ = λ 3 N N N λ n + + N t+ n = N t t 3 t 2 t ... = λ n N N + t n t

Geometric growth = λ n N N + λ 100 t n t > 1 N t+n λ = 1 50 λ < 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n Do you recognize the muskox here ?

= λ n N N + t n t 100 N t+n 50 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n May λ remain constant ? N λ = + t 1 Contribution of each individual in t, for the population in t+ 1 N t

Biological meaning of λ ? What is the biological meaning of λ ? Are newborns envolved ? deaths ? both ?

Why does N t change ? I mmigrants Natural mortality Births N t Mortality by human activities Closed population Emigrants N t+ 1 = N t -D t + B t Open population

Reproduction timing 11000000 Núm nascimentos -20 0 Tempo 1 Tempo Seasonal breeding Continuos breeding

Census and reproduction in seasonal breeders B t = newborns D t = deaths N t N t+1 ∆ t Pre-breeding census

t B + 1 + t N t N N t+1 = Survival rate t D t − B t + B D t t + N t N B t N t = t S

Number of newborns Number of parents N t+1 t t B N Birth rate b = D t t b t = N t B t

Biological meaning of λ N λ = + t 1 remember t N t Substituting N t+ 1 Using: N ( ) = ∴ = + + t 1 S N S N B + + t t 1 t t t N B t t Birth rate We get: ( ) λ = + S b 1 t t Survival rate

Newborns in Portugal, 1994, INE 1995 Femininos Masculinos 10000 8000 Núm de nados-vivos 6000 4000 2000 0 Jan Fev Mar Abr Mai Jun Jul Ago Set Out Nov Dez meses

Continuous reproduction N changes continuously ! Any time interval ∆ t= [t, t+ 1] will be arbitrary Remember mean variation : N t − ∆ N N N = + 1 t t N t+ 1 ∆ ∆ t t t t+ 1 ∆ N dN = Lim = Instantaneous variation at t ∆ t dt ∆ → t 0

Instantaneous variation dN = − B D Instantaneous variation at time t: t t dt Instantaneous rates, B newborns = = t Birth rate = b t parents N t D deaths = = t d Mortality rate = t population N t

Instantaneous rate of growth ( ) dN = − = − N b N d N b d t t t t t t t dt r Instantaneous rate of growth (Malthusian parameter) dN = rN dt r units: Individuals per individual per unit time Given an initial N t what is N t+ ∆ t ?

Solution dN = Ordinary differential equation of 1 st degree rN dt Assuming r is constant, Solution, by separable variables: Parameter ∆ = r t N N e + ∆ t t t Independent variable For any ∆ t Dependent variable

Exponential growth = r t N N e 100 t 0 r > 0 N r = 0 50 r < 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 tempo

Unregulated growth = λ N N Discrete time: + 1 t t ∆ = r t N N e Continuous: + ∆ t t t If λ applies to the time interval ∆ t= 1, Relationship between instantaneous rate of growth and finite rate of increase = λ r e

Unregulated growth cannot last long 3000 = ∆ r t N N e + ∆ Número indivíduos (N t ) t t t 2400 1800 r =1 r =0.5 1200 600 r =0.25 0 1 3 5 7 9 11 13 15 17 19 21 Tempo r= 1 year -1 N t = 10 initial individuals ∆ = 10 years t N t+ 10 = 10 e 1 x 10 = 220 265 individuals

Survival and reproduction depend upon N t ∆ − ∆ = = r t ( b d ) t N N e N e + ∆ t t t t Survival, birth rate = f (N t )

What good is the unregulated growth model if it does not apply to most populations ? 1. Illustrates the consequence of assuming constant survival and birth rates 2. The model describes the initial stages of population growth, showing the enormous potential of populations to grow 3. It is a good starting point for the introduction of other components that confer greater realism to population growth

Human population 1

Human population 2 Source: Demographic yearbook. Annuaire démographique. New York Dept. of Economic and Social Affairs, Statistical Office, United Nations

b t and d t in an exponential population