Exponential Population Growth

Defining a Population •  The change in the size of a population can be

defined with four processes: • Births (B) • Deaths (D) •  Immigration (I) • Emigration (E)

– Thus, • N(t+1) = N(t) + B - D + I – E

•  KHAN ACADEMY RESOURCE: https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/modeling-with-differential-equations/v/modeling-population-with-simple-differential-equation

Open vs. Closed Population

OPEN N(t+1) = N(t) + B - D + I -E

CLOSED N(t+1) = N(t) + B - D

Exponential Population Equation •  exponential population equation

with instantaneous rates:

•  The intrinsic rate of increase (r) = b - d. Thus,

•  r is the per capita rate of change in a population measured as individuals/(individualshtime )

dNdt

b d N= −( )

dNdt

rN=

Assumptions of Model (1) Closed population

(2) Constant b and d •  Unlimited supply of space, food, etc.

(3) No genetic structure/variation

(4) No age or size structure •  i.e. same b and d for all sizes and ages

(5)  continuous growth with no time lags •  Continuous birth, death and instantaneous change

Projecting the Population in Time •  After integration and rearranging the basic

equation we arrived at the projection equation:

N N etrt= 0

Barndoor skate

•  Fecundity (b): 48 eggs per year •  Longevity (Tmax): 50 years •  Age at maturity (Tmat): 12 years •  First year mortality (M1): 2.5 •  Mortality (M): 0.07

Age Structured Models

•  Age structured models are similar to simple exponential models with added age structure.

•  same assumptions •  continuous birth and death rates •  unlimited resources

•  The difference: –  Individuals are classified into discrete age

classes. Thus, these models only approximate continuous growth.

Continuous vs. Discrete Models Discrete Model

- added age/stage structure

Continuous Model - no age/stage structure

The Life Table: Calculating

N N etrt= 0

Fecundity (b)

•  Fecundity: number of offspring born per unit of time (we will use a year) to an individual female of a particular age.

•  We assume a 50/50 sex ratio. Thus, the annual egg production should be halved for the model.

– Example: winter skate fecundity = 24, a value

of 12 would be used in the model.

The Life Table: Calculating

Survivorship (S) •  Survival: the number of individuals in a specific

age class (or cohort) that survive to the next age class.

– Example: If a population produces 1000 newborns (age 0) and 500 live to age class 1, then the survival rate is 500/1000 = 0.5.

•  Next consider l(x), or the probability an individual survives to age class x:

lSSxx

( )( )

( )=

0

The Life Table: Calculating

Net Reproductive Rate (R0) •  Ro: the mean number of offspring produced per

female over a life span.

•  where l(x) is survivorship, b(x) is fecundity and Tmax is longevity. –  R0 is in the units of offspring. –  R0 > 1, population is increasing. –  R0 < 1, population is decreasing.

R l bx xx

T

00

==∑ ( ) ( )max

The Life Table: Calculating

Generation Time (G)

•  G: average age of the parents of all the offspring produced by a single cohort.

•  The units are in time.

Gl x b x x

l x b x

x

T

x

T==

=

∑

∑

( ) ( )

( ) ( )

max

max

0

0

The Life Table: Calculating

Intrinsic Rate of Population Increase (r) •  Using the generation time (G) and net reproductive

rate (R0) an approximation of r can be calculated:

•  However, this is only an approximation. For your models you will use the Euler equation to calculate a more accurate value of r.

rRG

=ln( )0

Euler Equation

•  Since you already know l(x) and b(x), the only parameter that can be changed is r. Doing so until a specific condition is met (in this case until both sides of the equation equal 1) is called iteration. Here you will iterate r until both sides of the equation equal 1.

•  To iterate start with r = ln(R0)/G to get a initial estimate. For the model you will want to use the iteration procedure in EXCEL.

10

= −=∑ e l x b xrxx

T

( ) ( )max

The Life Table: Calculating

Assumptions Reminder

•  Closed population •  Constant b and d •  No genetic structure •  No age or size structure •  No time lags (continuous)

Age Structured Growth

Some Notation

•  Newborn age is considered age 0 not 1.

•  Very easy to confuse!

Survival Probability (g)

•  g is the probability that an individual survives from x to the x+1 age classes:

gl xlx x

( )( )

( )=

+ 1

The Life Table: Calculating

Biological indicators

•  Equilibrium mortality •  Spawning biomass

Equilibrium mortality •  In our life tables equilibrium mortality corresponds to the

point where the intrinsic rate of increase is zero (r = 0). –  We will denote the fishing mortality rate that corresponds to

equilibrium mortality as Feq. – 

Feq = 0.34

•  Equilibrium mortality is a biological threshold. •  Fishing targets should be set below thresholds

•  Nt = N0e-(M+F)

Equilibrium mortality

Spawning Stock Biomass (SSB)

•  A simple accounting exercise. SSB is a measure of the potential production of mature females.

•  It is often used as a reference point. For example, protecting 30% of virgin SSB is a common fishing limit.

SSB m w Na a ai a

a

r

==∑max

Spawning Stock Biomass (SSB)

•  “Several studies have examined the SSB/R among stocks, and the mathematical relationships among SSB/R to determine the most appropriate target and threshold levels” (Deriso, 1987).

•  91 stocks studied by Sissewine & Mace (1993)

%SSB/R Biological reference 38 Target 21 Threshold 19 Threshold

Spawning Stock Biomass (SSB)