Class Notes (835,507)
Canada (509,212)
Biology (2,228)
BIO120H1 (1,171)

BIO120.Lecture (5).docx

11 Pages
Unlock Document

Doug Thomson

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
More Less

Related notes for BIO120H1

Log In


Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.