A population is agroup of organisms of a single species that live and reproduce in the same area.
Types of variables
State variable: ratio
Rate variable: how fast a state variable changes
Single species model
Starts out with continuous exponential growth.
Exponential growth is the first principle of population dynamics. A population will grow exponentially as long as the environment remains constant.
A continuous model implies rapid feedback, continuous births, and overlapping generations. Uses differential equations.
A discrete model implies births in specific periods, non-overlapping generations, and uses difference equations.
Nt = population size (number of individuals) at a certain time 't'.
N0 = initial population size (at time 0)
B = Total Births
D = Total Deaths
I = Immigration
E = Emigration
Nt = N0+B+I-D-E
Assuming a closed system, there will be no I nor E, thus Nt = N0+B-D
Nt - N0 = B - D
∆N = B-D
b = birth rate per capita
d = death rate per capita
r = intrinsic rate of increase; r is species-specific and can change within a species
r = b-d
∆N = B-D
∆N = bN-dN
∆N = (b-d)N
∂N/∂t = (b-d)N
∂N/∂t = rN
Nt = N0e^(rt)
Doubling time: time it takes to double the population
tdouble = ln(2)/r
Discrete exponential growth
lambda = Nt/N0 (N at a future time/initial population)
lambda = finite rate of increase
lambda > 1.0 population increases
lambda = 0 population constant
lambda < 1.0 population decreases
Best estimate of lambda occurs when the population reaches a Stable Age Distribution (SAD).
SAD: relative proportion of individuals in each class remains constant
Nt = lambdaN0
Two models converge when the time step (interval) in the discrete model becomes shorter. It allows conversion from one model to the other.
lambda = e^r
r = ln(lambda)
Probability of extinction
Pex can be seen as a function of N0
Pex=(d/b)^N0
A lower initial population increases the probability of extinction.
Highlights the importance of population size for persistence of populations.
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