Download Population Size N 1

BMED 3510
Population Models
Book Chapter 10
General Questions
How does the size of a population change over time?
Speed of growth
Final population size; carrying capacity; steady state
Stability
Dynamics if size is above carrying capacity
How does the composition of a population change over time?
Age structure
Dynamic changes in sizes of interacting subpopulations
SIR systems
Predator-prey systems
Populations competing for the same resources
Behavior of individuals making up a population
Deterministic vs. stochastic
Homogeneous vs. spatial
Dynamics of a Single Population
Uncounted growth functions; most grow ~exponentially and then taper off
All functions are very simple; how can they possibly fit something as complicated
as the human population?
Possible answer*:
Growth is dominated by a few processes at the “right” time scale
Faster processes (e.g., signaling, biochemistry) are in steady state
Slower processes (changes in climate) are essentially constant
Do some math: The few processes at the right scale can lead to simple trends
ODEs with one or two equations ~correct at a coarse level
*Savageau, MA. PNAS 1979
Growth of a Single Population
Typical growth functions
Exponential growth
Logistic growth (Chapter 4):
N  r N 
r 2
N
K
Growth of a Single Population
Many growth functions may be interpreted as exponential growth with
time- (or population-size) dependent growth rates
Exponential growth
N  r N
Logistic growth:
N  r N 
r 2
(K  N)
N  r
N
K
K
“growth rate is modulated by distance from K”
Gompertz growth:
NG  r  ln(K / NG )  NG
“growth rate is modulated by ratio K/NG”
Richards growth:
NR  r  [1  (NR / K ) ]  NR
“growth rate is modulated by function of NR”
Growth of a Single Population
More growth functions may be generated as generalizations of the logistic function
Logistic growth:
N  r N 
r 2
N
K
Generalized growth function:
N    Ng    Nh
Savageau, MA. Math. Biosci. 1980
“One-variable S-system”
Extrapolation
Growth functions are difficult
to extrapolate to final sizes
Example: Logistic growth (N1)
versus Gompertz growth (N2)
PLAS Code
0
0
10
20
growth rate
decreases
exponentially
r1 = .325
K = 1000
r2 = .1
2400
N1 = 2.2
N2 = 0.1
R=1
1200
t0 = 0
tf = 20
hr = .1
N2
300
N1' = r1 N1 - r1/K N1^2
N2'= R N2
R' = - r2 R
N1
600
N1
N2
0
0
50
100
Extrapolation
Very important and
very difficult to do
Pertinent Example:
World Population
Source: see Book Fig. 3, Chapter 10
Reality is More Complicated
Conditions change (habitat reduction; climate; …)
Stochastic effects
New predators; competitors, invasive species, new/fewer food sources; diseases …
Example: Commercial fishing
(1) Fish a certain percentage
r
2
N 1  r  N1   N1  p  N1
K
(2) Fish at a constant rate
r
2

N2  r  N2   N2  P
K
Reality is More Complicated
Conditions change (habitat reduction; climate; …)
Stochastic effects
New predators; competitors, invasive species, new food sources; diseases …
Population Size
Example: Commercial fishing; additional perturbation at time 2.5 (e.g., disease)
(1) Fish a certain percentage
(2) Fish at a constant rate
50
N2
N1
N1
25
N2
0
0
2.5
Time
5
0
2.5
Time
5
Reality is More Complicated
Populations do not have to be macroscopic
Cell populations
Normal physiology
Erythropoiesis
Replacement of skin and gut cells
Cancer
Organismal growth and differentiation
Molecular populations
Dynamics of Subpopulations
SIR models: groups are human subpopulations
Important case: age stratification
Recall Leslie model with constant parameters
For four age classes Pi:
 1  2  3  4   P1 
 P1 

  
 
  1 0 0 0   P2 
 P2 

 
0 

P 
0 0
P
2

  3
 3
P 

  
 4 t   0 0  3 0   P4 t
proliferation rates 
survival rates 
Leslie matrix
Time- or age-dependent parameters make the model more realistic
Interacting Populations
Start with logistic models for two independent populations:
N1  r1
(K1  N1 )
 N1
K1
(K2  N2 )
N2  r2
 N2
K2
N1  r1
(K1  N1  aN2 )
 N1
K1
N2  r2
(K2  N2  bN1 )
 N2
K2
Add interactions:
Minus sign: inhibitive effect
Gause’s exclusion principle (Gause, G.F., The Struggle for Existence, 1934):
Two species with exactly the same needs cannot exist in the same
constant environment. Usually the needs are slightly different.
Questions: Is coexistence possible? Will one species go extinct?
Do the two species always coexist?
What happens if one goes extinct?
Interacting Populations
(K  N1  aN2 )
N1  r1 1
 N1
K1
N2  r2
(K2  N2  bN1 )
 N2
K2
Numerical Example:
Alternative representation:
Phase-plane plot
Assess Complete Repertoire of Scenarios
As usually, start with steady state:
N1  r1
(K1  N1  aN2 )
 N1  0
K1
N2  r2
(K2  N2  bN1 )
 N2  0
K2
Recognize that trivial solutions exist; divide by N1 and N2, respectively:
𝑁1 = 0
𝑁2 =
𝐾1 − 𝑁1
𝑎
Both “nullclines” are straight lines of N2 versus N1.
𝑁2 = 0
𝑁2 = 𝐾2 − 𝑏𝑁1
Assess Entire Repertoire of Scenarios
(K1  N1  aN2 )
 N1  0
K1
N2  r2
(K2  N2  bN1 )
 N2  0
K2
𝑁1 = 0
𝑁2 =
𝐾1 − 𝑁1
𝑎
𝑁2 = 0 𝑁2 = 𝐾2 − 𝑏𝑁1
K1/a
Population Size N2
Nullclines:
N1  r1
Four steady states!
.
N1 = 0
K2
.
N1 = 0
.
N2 = 0
.
0
0
N2 = 0
K1
Population Size N1
K2/b
Assess Entire Repertoire of Scenarios
Steady-state lines (“nullclines”) dissect the space in different model responses:
.
N
.
1
K1/a
>0
.
N
.
Population Size N2
N2 < 0
1
<0
N2 < 0
.
N1 = 0
K2
.
N
.
1
.
>0
N2 = 0
.
.
N1 < 0
N2 > 0
N2 > 0
0
0
K1
Population Size N1
K2/b
Assess Entire Repertoire of Scenarios
Dissection allows us to predict dynamics (and stability of steady states):
.
.
N1 > 0
Population Size N2
K1/a
N2 < 0
.
N
.
1
<0
N2 < 0
.
N1 = 0
K2
.
.
N
.
0
.
.
>0
N1 < 0
N2 > 0
N2 > 0
1
0
N2 = 0
K1
Population Size N1
K2/b
Assess Entire Repertoire of Scenarios
Dynamics exactly on the nullclines:
.
.
N1 > 0
Population Size N2
K1/a
N2 < 0
.
N
.
1
<0
N2 < 0
.
N1 = 0
K2
.
.
N
.
0
.
.
>0
N1 < 0
N2 > 0
N2 > 0
1
0
N2 = 0
K1
Population Size N1
K2/b
Assess Entire Repertoire of Scenarios
Stitch together dynamics for the entire space (qualitatively):
.
N
.
1
>0
N2 < 0
K1/a
.
.
Population Size N2
N1 < 0
N2 < 0
K2
.
N
.
1
.
N
.
1
<0
N2 > 0
>0
N2 > 0
0
0
K1
Population Size N1
K2/b
Assess Entire Repertoire of Scenarios
Coexistence depends on location of nullclines (parameter values):
Population Size N2
K1/a
K2
0
0
K2/b
Population Size N1
K1
Previous Example: Actual Simulations
Population Size N2
K1/a
K2
0
0
K1
Population Size N1
K2/b
Population Size N2
Representation of a “Vector Field”
Population Size N1
Population Size N2
Comparison
Population Size N1
Note on Stochasticity
Coexistence, existence, and other features of populations can be very strongly
affected by stochastic events (e.g., perturbations in the environment).
These effects are particularly influential for small population sizes. Why?
This is a great concern for the survival of endangered species.
Larger Systems
Phase-plane analysis is not very helpful
n
Model format often Lotka-Volterra models (Chapter 4):
X i  ai X i   bij X i X j
j 1
Recall: strict format, lots of
complicated dynamics
possible nevertheless!
Applications
Traditional examples:
Predator-prey systems
Competing species
Modern examples at the forefront:
Microbiomes
Metapopulations
(see Book, pages 406-407)
Spatial Considerations
Example: Solenopsis invicta
http://pestcemetery.com/wp-content/uploads/
2009/04/fire-ant-spread-map-pestcemetery.jpg
Spatial Considerations
Example: Solenopsis invicta
http://www.ars.usda.gov/research/docs.htm?docid=9165
Spatial Considerations
Modeling Approaches:
PDEs
Simulation
“Agent-based models” (ABMs; Book pages 411-412)
Each “individual” is an agent
Agents are governed by rules of what they can do
In addition, there are rules for interactions between agents
Simulations are stochastic and very flexible
Simulations can yield very complicated results
Easy Implementation: Netlogo
Go to Netlogo (>> Download) >> Models Library >> Biology >> Wolf Sheep Predation
Set up with defaults
Emergency Break: Tools > Halt
Behind
the
Netlogo
Interface
Behind
the
Netlogo
Interface
Exactly the same settings as before
Same settings as before, but account for dynamics of grass
Summary
Population dynamics is of great academic and societal interest
Base models very simple (functions or simple ODEs)
Study:
Overall growth (or decline)
Carrying capacity
Subpopulations
Age classes
Different types of individuals
Spatial expansion
Stochastic effects
Agent-based models
Netlogo