Transcript - Catalyst

Continuous logistic model
dN
 N
 rN  1  
dt
K

N0Ke rt
Nt 
K  N0  e rt  1
This equation is quite different to the discrete model!
1 Logistic.xlsx – Deterministic continuous
Source: Mangel M (2006) The theoretical ecologist's toolbox, Cambridge University Press, Cambridge
How to decide between discrete
and continuous models
Are the processes discrete or continuous?
In individual-based models discrete processes are
almost always appropriate
For most computer software discrete models are
easier
Differential equations provide analytic solutions for
simple problems
Tradition
The modeling toolbox for
conservation of single populations
Biomass-dynamics models (logistic, Schaefer, Fox,
Pella-Tomlinson)
Generation-to-generation models (Ricker, BevertonHolt)
Delay-difference models (Deriso-Schnute)
Size- and stage-structured models
Age-structured models
Extensions to all
Stochastic or deterministic
Adding environmental impacts
Extending to multiple species
Models with no age structure
(total biomass, total numbers)
Discrete exponential model
Numbers next year = numbers this year + births – deaths +
immigrants – emigrants
D  mNt
E 0
Nt 1  Nt  B  D  I  E
B  bNt
I0
Nt 1  Nt  bNt  mNt  Nt (1  b  m)
2 Exponential model.xlsx: deterministic and discrete
A differential equation version
dN
 (b  m)N
dt
Solution:
Nt  N0 e
 b  m t
2 Exponential model.xslx: deterministic and continuous
Model assumptions
• Population will grow or decline exponentially for
an indefinite period
• Births and deaths are independent
• There is no impact of age structure or sex ratio
• Birth and death rates are constant in time
• There is no environmental variability
Individual-based exponential model
For every year
For every individual
Pick random numbers X1~U[0,1], and
X2~U[0,1] (uniform between 0 and 1)
If X1<b then a new birth
If X2<m then individual dies
2 Exponential indiv based.r
2 Exponential indiv based.r
Binomial shortcut
• Binomial distribution: probability p and N trials,
with mean pN and variance p(1-p)N
• Births: probability b, Nt trials, expected mean bNt,
variance b(1-b)Nt
• Deaths: probability m, Nt trials, expected mean
mNt, variance m(1-m)Nt
2 Exponential indiv binomial.r
2 Exponential indiv binomial.r
Normal approximation
• With large number of trials (individuals) the binomial
approaches the normal distribution
• Randomly sample deviates from a normal distribution
with mean bNt and standard deviation sqrt(b(1-b)Nt)
• mean + X×SD where X is a random normal deviate
• In Excel norminv(rand(),0,1) produces random normal
deviates with mean 0 and SD 1
• Small numbers (<30) this approximation does not work
well, then use binomial draws (R is good at this)
• Repeat for deaths with b replaced by m
2 Exponential model.xlsx: Normal approx to binomial
Why the normal approximation?
• We do not have to calculate the probability of
each individual living or giving birth
• This is very helpful with populations in the
thousands or millions
Types of stochasticity
• Phenotypic: not all individuals are alike
• Demographic: random births and deaths
• Environmental: some years are better than
others, El Nino, hurricanes, deep freeze etc.
• Spatial: not all places are alike
Only demographic stochasticity was
included in previous models
• We often allow for a more general model of
stochasticity:
Nt 1  f (Nt , p,ut )e , wt ~ N(0, s )
wt
2
• i.e. wt is normally distributed with a mean zero
and standard deviation s
• numbers next year depend upon numbers this
year, the parameters p, any forcing function u
(such as harvesting) and random environmental
conditions wt
Lognormal error: a little deeper
• If wt is normally distributed with mean zero, then
exp(wt) is lognormally distributed
• When wt=0, exp(wt) = 1, “average year”
• When wt>0, exp(wt) > 1, “good year”
• When wt<0, exp(wt) < 1, “bad year”
• Since exp(wt) is not symmetric, the expected
value is not 1
• therefore we use a correction factor:

2
s
exp wt 
2

Adding lognormal error to the
exponential model
2
s
 b  m t
Nt  N0 e
exp(wt  w )
2
wt ~ N  0, sw2 
Normal and lognormal error
Non-age-structured models
•
•
•
•
Exponential growth
Logistic (Schaefer) model
Fox model
Pella-Tomlinson model
Schaefer MB (1954) Some aspects of the dynamics of the population important to the management of the commercial marine fisheries. InterAmerican Tropical Tuna Commission Bulletin 1:25-56
Fox WW (1970) An exponential surplus-yield model for optimizing exploited fish populations. Trans. American Fisheries Society 99:80-88
Pella JJ & Tomlinson PK (1969) A generalized stock production model. Inter-American Tropical Tuna Commission Bulletin 13:419-496
Logistic model
Peak catch occurs when B = 0.5K
Catch
 Bt
Bt 1  Bt  rBt  1 
K

Biomass at
time t+1
Intrinsic (maximum)
rate of increase

  Ct

Carrying
capacity
Rate of increase declines as
density approaches K
Rate of increase approaches 0
near K
“Compensation”
• At high densities there will be a shortage of food,
refuge from predators or some critical
requirement
• Birth rates may decline, or mortality rates will
increase
• This is called compensation and can be quantified
as the difference between the rate of increase
when resources are abundant and a rate of
increase of 0.
“Depensation”
• Rates of increase may decline at low
densities
• This is known as depensation (or the Allee
effect)
• It makes local or total extinction much
more likely
• Will be discussed later in course
Offspring produced
Visualizing rates of change with
compensation and depensation
Expected curve
Depensation
Number of spawners
Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. CJFAS 54:1976-1984.
Myers RA et al. (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:1106-1108.
Fox model
Peak catch occurs when B = 0.37K
Catch
Bt 1
Biomass at
time t+1
 ln(Bt ) 
 Bt  rBt 1 

C
t

ln(K ) 

Intrinsic (maximum)
rate of increase
Carrying
capacity
Logistic vs. Fox model
Note: Fox model will have lower SP for the same r value
25
Surplus production
Fox
Logistic
20
15
10
5
0
0
0.2
0.4
0.6
0.8
1
Biomass (fraction of carrying capacity)
2 Non-age models.xlsx: Fox vs logistic
Pella-Tomlinson model
Peak catch occurs anywhere from B = 0 to B = K, depending on n
MSY: maximum
sustainable yield
Catch
n

Bt  Bt  
n
Bt 1  Bt 
m       Ct
n  1  K  K  
n /( n 1)
Biomass at
time t+1
Determines biomass
that yields MSY
Carrying
capacity
Pella-Tomlinson model
These plots all have m = MSY = 500
Becomes the Fox model as n  1
Becomes the logistic model when n = 2
600
n = 0.5
Surplus production
500
n = 0.8
400
n = 1.5
n=2
300
n=5
200
n = 10
100
0
0
0.2
0.4
0.6
0.8
1
Biomass (fraction of carrying capacity)
2 Non-age models.xlsx: Pella-Tomlinson
Advantages and disadvantages
• Logistic, Fox, Pella-Tomlinson offer different
hypotheses about at what biomass level
MSY would be obtained
• The Pella-Tomlinson is more flexible but has
more parameters
Things to do later in course
• What happens if environmental conditions
are not independent in time, but tend to
come in runs of good and bad years more
often than would be expected by chance
• This is called serial autocorrelation
• Results in regime shifts
• Does environment drive surplus production
more than biomass?