# BIO120H1 Lecture Notes - Lecture 18: Net Reproduction Rate, Exponential Growth, Infinitesimal

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BIO120: Population Ecology – Models without Age Structure

Population

-collection of individuals in certain area

-population size (N): number of individuals

-population density = N/area

-population ecology: what influences N?

-population genetics: what genetic variation resides in N?

-population ecology developed by zoologists (often more ambiguity about plant

individuals)

aspen: clones, one seed produces many identical, connected stems (vegetative spread)

larkspur: many unique seeds produce many unique plants

dandelion: no sex, many identical seeds produce many identical, unconnected plants

(one genetic individual spread across many physiological individuals)

Relationships Among Models

1. basic models:

Discrete time steps

(difference equations,

arithmetic)

Infinitesimal steps

(differential equations,

calculus

Density independent

Geometric growth model

Exponential growth model

Density dependent

None

Logistic growth model (as

population gets bigger,

growth rate slows)

2. extensions for greater realism:

Discrete steps

Infinitesimal steps

Density independent

Add age structure

Density dependent

Add Allee effects, add time

lags

Goal of Population Models

-predict the trajectory of population growth through time (N as a function of t)

-how many individuals are in population now (Nt)

time advances one step (t t+1)

-how many individuals are in population one step later (Nt+1)

-general model: Nt+1 = f(Nt)

-challenge: choosing simple but realistic parameters for f

Choice: Time Steps

-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 (e.g. days, years)

use iterated recursion equations, growth is stepwise and bumpy, best suited for

episodic reproduction

-also called “continuous time” and “discrete-time” approaches

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-different organisms might be better fit by one or the other

Some Notational Matters

-time steps and age classes are classically indexed by subscripts or parentheses (e.g. N3 =

N(3))

-by convention, population growth starts at t = 0, so starting population size is N0 or N(0)

Bookkeeping Model

-D = number who die during one time step

-B = number born during one time step

-E = number who emigrate during one time step (same consequences for population as

individuals dying)

-I = number who immigrate during one time step

Nt+1 = Nt – D + B – E + I

-death and emigration are equivalent, birth and immigration are equivalent, can simplify

model by modelling only death and birth

Geometric Growth

-simplify and convert changes to per capita 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 constant factor each time step: Nt+1 = �Nt

-�: multiplicative factor by which population changes over one time unit (net

reproductive rate); a “growth rate”

difference between birth and death rates

if � > 1, births exceed deaths

Exponential Growth

-alternative model 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)

r: intrinsic rate of increase

-differential equation: dN/dt = rN

slope, rate of change of population with respect to time

Solving the 2 Simplest Models of Unlimited Growth

Discrete-time

Continuous-time

Nt+1 = �Nt

dN/dt = rN

-solve for N vs. t: Nt = N0�t

-solve for N vs. t: Nt = N0ert

-t: geometric growth (step function) if � > 1

-t: exponential growth (smooth function) if

r > 0

-r = 0: population steady

-r > 0: population grows

-r < 0: population declines (exponentially approach extinction)

Graphing Exponential Growth: Nt = N0ert

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