exponential population growth · exponential population equation • exponential population...
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Exponential Population Growth
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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
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Open vs. Closed Population
OPEN N(t+1) = N(t) + B - D + I -E
CLOSED N(t+1) = N(t) + B - D
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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=
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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
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Projecting the Population in Time • After integration and rearranging the basic
equation we arrived at the projection equation:
N N etrt= 0
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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
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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.
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Continuous vs. Discrete Models Discrete Model
- added age/stage structure
Continuous Model - no age/stage structure
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The Life Table: Calculating
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N N etrt= 0
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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.
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The Life Table: Calculating
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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
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The Life Table: Calculating
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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
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The Life Table: Calculating
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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
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The Life Table: Calculating
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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
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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
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The Life Table: Calculating
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Assumptions Reminder
• Closed population • Constant b and d • No genetic structure • No age or size structure • No time lags (continuous)
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Age Structured Growth
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Some Notation
• Newborn age is considered age 0 not 1.
• Very easy to confuse!
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Survival Probability (g)
• g is the probability that an individual survives from x to the x+1 age classes:
gl xlx x
( )( )
( )=
+ 1
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The Life Table: Calculating
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Biological indicators
• Equilibrium mortality • Spawning biomass
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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
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• Equilibrium mortality is a biological threshold. • Fishing targets should be set below thresholds
• Nt = N0e-(M+F)
Equilibrium mortality
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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
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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
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Spawning Stock Biomass (SSB)
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