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Lecture

BIO120.Lecture (5).docx

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Department
Biology
Course
BIO120H1
Professor
Doug Thomson
Semester
Fall

Description
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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