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BIO120H1
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Doug Thomson
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Biology

BIO120H1

Doug Thomson

Fall

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BIO120: Population Ecology: Models Without Age Structure
-defining populations and individuals, with a focus on some unusual cases
-taxonomy of simplest possible mathematical models
exponential and geometric population growth models without density
dependence
incorporating density dependence; effects of crowding, etc.
more complex density dependence: time delays, Allee effects
-population: collection of individuals of the same species in a certain area
-individuals in population interact in various ways which affect reproduction and
mortality
-N=number of individuals in a population
added to reproduction, subtracted from mortality
-population density=N/area
-population ecology: what influences N?
-population genetics: what genetic variation resides in N?
-notion of what is an individual can become complicated
aspen: clones, one seed produces many identical, connected stems
larkspur: many unique seeds produce many unique plants
dandelion: no sex, many identical seeds produce many identical,
unconnected plants The Goal of Most Population Models
-predict the trajectory of population growth through time, i.e., N as a function of t
-how many individuals are in the population now? N t
-time advances one step: t t + 1
-how many individuals are in the population one step later? N t+1
-general model is N t+1= f(N t
-challenge: choosing simple but realistic parameters for f
Choice: What are the Time Steps?
-calculus lets us look at smooth changes where population size is a continuous
function (may be curved but defined at all times)
-when using differential equations, time steps are infinitesimally small: use concept
of limits and calculus; growth is smooth; best suited for species with continuous
reproduction
-when using difference equations, time steps are discrete units (days, years, etc):
use iterated recursion equations; growth is stepwise and bumpy; best suited for
episodic reproduction
-discrete population growth is a good fit for animals in temperate zone that have a
short breeding season once a year, annual pulse of reproduction
-humans have a continuous reproductive pattern
-also called “continuous time” and “discrete-time” approaches
-different organisms might be better fit by one or the other
Notation
-time steps and age classes are classically indexed by subscripts, but parentheses
are seen, also:
-N ts population size at time t, alternatively designated at N(t); N = N33); etc.
-by convention, population growth starts at t = 0, so starting population size is N or 0
N(0)
How can N change from N to N t t+1
-D: number who die during one time step
-B: number born during one time step
-E: number who emigrate during one time step
-I: number who immigrate during one time step
-N t+1= Nt– D + B – E + I
-two ways in which population can change: birth/death, immigration/emigration
-death and emigration are equivalent
-birth and immigration are equivalent
-can simplify the model by modeling only death and birth
Simplify and convert changes to per-capita (per-individual) rates
-assume no immigration or emigration
-treat birth and death during one time step as per-capita rates that are fixed
constants
-population changes by a constant factor each time step: N t+1= λ N t -λ = multiplicative factor by which population changes over one time unit (net
reproductive rate) = a “growth rate”
-if λ > 1, birth exceeds death
-this model is geometric growth
Alternative version with continuous time
-instantaneous, per-capita rates of birth and death fixed (b and d)
-instantaneous, per-capita rate of population change = b – d = r (constant)
-different equation is dN/dt = rN
-this model is exponential growth
Solving
-discrete-time:
Nt+1= λ N t
solve for N vs t: Nt= N 0 t
geometric growth (step function) if λ > 1
-continuous-time:
dN/dt = rN
solve for N vs t: N = N e rt
t 0
exponential growth (smooth function) if r > 0
-step functions, not continuous lines
-steady within each interval, but jumps between intervals -growth of population increases, starts out slowly and then grows quickly, speed
depends on the size of r
-negative r means deaths are exceeding births and the population declines
-regardless of which model is adopted, the important consequence is the same
-if you have a constant, positive growth rate, we get a population size that is not
constant
-in both models, the growth rate is a constant that simply reflects biology
-constant positive growth rate produces a population size that is not constant, but
rather exploding in an exponential way
-all species have the capability of having a positive growth rate
-all species have the possibility of negative population growth
-no species has ever sustained these extremes for a long period
-“explosion” doesn’t happen, mechanisms in nature don’t let population growth stay
exponential for too long
The staggering implications
-simple exponential growth is a bad model of reality over the long term
-some factors must ten

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