Mathematical Modeling, Population Ecology, Population Models

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LURE 2009 SUMMER PROGRAM
John Alford
Sam Houston State University
Mathematical Modeling
Mathematical modeling is an area of
applied mathematics concerned with
describing and/or predicting real-world
system behavior
 Examples of real-world systems

–
–
–
–
an object moving in a gravitational field
stock market fluctuations
predator-prey interactions
nerve impluse propagation
Mathematical Modeling

Many familiar concepts in mathematics
evolved from the desire to answer basic
scientific questions using mathematical
models
– e.g. Newton used calculus to describe
and predict how an object of a certain
mass will move in response to applied
forces
Mathematical Modeling
Why model?
 Simplification of a complex system
 Ease of manipulation (simulation as
opposed to experimentation)
 Gains in understanding of the system

– helps to formulate new hypotheses
– aids in design of new experiments
Mathematical Modeling
 There
is no perfect model!
– a model should balance accuracy,
flexibility, and cost
A
general rule of thumb
– increasing accuracy decreases flexibility
and increases cost
 Goal
– construct a “sufficiently” accurate model
with high flexibility and low cost
Mathematical Modeling
 No
model is perfect
 The problem may never be
“completely” solved
– you need to get used to this if you are
going to be doing research
Mathematical Modeling
 Step
1: find a real-world problem
– obtain data and general knowledge
 perform
experiments
 search the literature (journals, books, etc.)
– make simplifying assumptions and
neglect some complexity
Mathematical Modeling
 Step
2: formulate the model
– research the literature for other models
(don’t re-invent the wheel)
– decide on the model type and form
 equations-
algebraic, differential, integral,
difference, etc.
 deterministic (non-random) or stochastic
Mathematical Modeling
 Step
2: formulate the model (cont.)
– model will include
 variables
represent system parts
 parameters influence variables but are not
influenced by them (typically)
 equations describe behavior and relate
variables and parameters
Mathematical Modeling

Step 3: analyze the model
– run computer simulations
– apply mathematical theories

Step 4: interpret and verify the results to
explain or predict
– iterate the model for improved accuracy
Mathematical Modeling Process
real world problem
formulate
modify
report, explain, predict
mathematical model
analyze
interpret
and verify
mathematical results
Population Ecology

Unstructured population models
– populations are treated as
“homogeneous green gunk” (Kot,
Elements of Mathematical Ecology)

Structured population models
– Include effects due to age, spatial
location, genetic variation, etc.
Population Ecology

Unless otherwise stated, we will
assume for this discussion that our
population models are
unstructured.
Population Ecology
 The
six axioms of Turchin
– Conservation
– Individualism
– Upper density bound
– Mass action
– Biomass conversion
– Max physiological rates
Population Ecology
 We
will now use the first two axioms
to construct the exponential growth
model for population growth
Exponential Growth Model

Population density

dN
Rate of change of population
dt

N (t )
Per capita rate of change of
population
1 dN
N dt
Exponential Growth Model
 Conservation
– The number of organisms in a
population can change only as a result
of births, deaths, immigrations, and
emigrations
Exponential Growth Model

Model Using Conservation
– The rate of change of a closed population
(i.e. no immigration or emigration) is the
number of births minus the number of
deaths
dN
 BD
dt
Exponential Growth Model
 Individualism
– Population mechanisms are individual
based. That is, all population processes
affecting population change (e.g. births,
deaths, movement) are a result of what
happens to individuals.
Exponential Growth Model

Model Using Conservation
– birth and death rates can be expressed
in per capita form
B  N  per capita birth rate
D  N  per capita death rate
Exponential Growth Model

Model using conservation
– Assume the per capita birth rate and
the per capita death rate are constant
equal to b and d respectively
dN
 B  D  bN  dN
dt
Exponential Growth Model

Define the intrinsic rate of growth
(net per capita growth rate)
r bd
so that
dN
 rN
dt
Exponential Growth Model

Some Mathematics
– Solution by separation of variables
Exponential Growth Model

Separate variables
dN
dN
 rN 
 rdt
dt
N
Exponential Growth Model

Integrate
1
 N dN   rdt
to get
ln N  rt  C
Exponential Growth Model

Malthus’ equation (1798)
N  N 0e
where
rt
N 0  N (t  0)
is the initial population density
Exponential Growth Model

Exponential population growth
b  d r bd  0

For positive initial populations,
there is no limit on population size
as time increases
lim N0e  
rt
t 
Exponential Growth Model

Exponential population decay
b  d r bd  0

All initial populations (eventually)
become extinct.
lim N 0e  0
rt
t 
Exponential Growth Model

per capita growth rate vs. N (r>0)
Exponential Growth Model

growth rate vs. N
Exponential Growth Model

population growth vs. t
Exponential Growth Model
Equilibrium solutions are constant
solutions

N  0 is an equilibrium for
*
dN
 rN
dt
check
d
d
*
*
[ N ]  rN  [0]  r[0]
dt
dt
Exponential Growth Model

Equilibria may be stable or unstable
– stability “means” any small
perturbation results in a return (over
time) to the equilibrium
– instability “means” some small
perturbation will not result in a return
(over time) to the equilibrium

Equilibria and stability may depend
on parameter(s) in the equation(s)
Exponential Growth Model
dN
rt
 rN  N  N 0 e
dt
– without emigration or immigration,
populations that start at 0 stay at 0
– if per capita growth rate is positive,
small perturbations from 0 result in
large population changes
r  0  N  0 is unstable
*
Exponential Growth Model
dN
rt
 rN  N  N 0 e
dt
– if per capita growth rate is negative,
small perturbations from 0 result in
population sizes returning to 0
r  0  N  0 is stable
*
Population Ecology
 The
six axioms of Turchin
– Conservation
– Individualism
– Upper density bound
– Mass action
– Biomass conversion
– Max physiological rates
Population Ecology

Limitations exponential growth
– Constant per capita growth rate yields
unlimited growth
– Deterministic nature of the model
ignores random (stochastic) effects
which are (particularly) important at
small population sizes
– Model is unstructured and ignores
temporal and spatial variability
Population Ecology

Factors that regulate growth of
populations: biotic vs. abiotic
– competition within and between
species (biotic)
– variation in the weather (abiotic)
Population Ecology

A.J. Nicholson, 1933, The Balance
of Animal Populations, Journal of
Animal Ecology
– population densities are regulated by
biotic factors such as competition and
disease which affect high-density
populations more than low-density
populations
Population Ecology
 The
six axioms of Turchin
– Conservation
– Individualism
– Upper density bound
– Mass action
– Biomass conversion
– Max physiological rates
Logistic Growth Model



Per capita growth rate is positive for
small population densities
Per capita growth rate is negative
for large population densities
Per capita growth rate decreases as
population increases (competition
for resources including food, space)
Logistic Growth Model

per capita growth rate vs. N
Logistic Growth Model

Per capita growth rate is a linear
function of the population density
1 dN
 N
 r 1  
N dt
 K

Here r>0 and K>0 are parameters
– r=intrinsic growth rate
– K=carrying capacity
Logistic Growth Model

Growth rate is quadratic function of
N
dN
 N
 rN 1  
dt
 K
Logistic Growth Model

growth rate vs. N
Logistic Growth Model

Exercise 1: solve to get the
Verhulst (1838) model of
population growth
N0 K
N
 rt
N 0  ( K  N 0 )e
where
N 0  N (t  0)
Logistic Growth Model

For positive initial populations, the
limiting population is carrying
capacity (K)
N0 K
lim
K

rt
t   N  ( K  N )e
0
0
WHY (mathematically and biologically)?
Logistic Growth Model

N vs. t
N0  K
Logistic Growth Model

N vs. t
N0  K
Logistic Growth Model Equilibria

There are two equilibria for the
logistic model
– carrying capacity
N K
*
– zero population density
N 0
*
Logistic Growth Model Equilibria

Multiply right side
dN
N
 N
 rN 1    rN  r
dt
K
 K
2

Assume 0  N / K  1
– to get
dN
 rN
dt
Logistic Growth Model Equilibria


Because r>0, previous slide shows
that a small perturbation of N away
from zero will grow exponentially
(approximately)
Zero equilibrium is unstable
Logistic Growth Model Equilibria

Let   N  K  N    K

Substitute and do algebra
dN
d
r 2
 N
 rN 1   
  r  
dt
dt
K
 K
Logistic Growth Model Equilibria

Assume
 very small ( N very close to K )
to get
d
  r
dt
Logistic Growth Model Equilibria



Because r>0, previous slide shows
that a small perturbation of  away
from zero will decay exponentially
back to zero
 decays to zero  N decays to K
K is a stable equilibrium
Population Ecology
Interacting Species: aphid infestation
Population Ecology
A
common way ecologists classify
species interactions between two
species is by denoting positive,
negative, or zero (neutral) pairings.
(+,+), (+,-), etc.
Population Ecology
A
consumer-resource or trophic
interaction is a (+,-) pairing between
two species
 Examples of consumer-resource
interactions
– predator-prey (e.g. fox and rabbit)
– herbivore-plant (e.g. leaf-mining fly and
hydrilla)
Population Ecology
 The
last three axioms of Turchin
– Mass action
– Biomass conversion
– Max physiological rates
 These
may be used to derive models
of consumer-resource interactions
Population Ecology
 Mass
action
– At low resource densities the
number of resource individuals
encountered and captured by a
single consumer is proportional to
resource density
capture rate  aN as N  0
(N is resource density, a is constant)
Population Ecology
 Biomass
conversion
– The amount of energy that an
individual consumer can derive
from captured resources to be used
for growth, maintenance, and
reproduction, is a function of the
amount of captured biomass
Population Ecology
 Maximum
physiological rate
– No matter how high the resource
density is, an individual consumer
can ingest resource biomass no
faster than some upper limit
imposed by its physiology (e.g. the
size of its gut)
Population Ecology

Mass action and max physiological
Population Ecology
Ecologists call the functional rate at which
each predator captures prey as it depends
on prey density the functional response.
 C.S. Holling (ca 1960) described three
types of functional response relations
 Each of Holling’s functional response
relations obey Turchin’s mass action and
max physiological axioms

Population Ecology
 Holling’s
functional response Type II
aN
f (N ) 
b N
(a, b constants)
Population Ecology
Population Ecology
 Holling’s
functional response Type II
aN
f (N ) 
b N
(a, b constants)
Population Ecology

Holling’s functional response Type III
2
aN
f (N )  2
2
b N
(a, b constants)
Population Ecology

LURE Students
– Analyze the following two models for a
predator-prey interaction. Treat the
number of predators P as a (constant)
bifurcation parameter (for now).
– Consider r and K to be fixed (for now)
– Interpret your results biologically
Population Ecology

Model 1
dN
 N
 rN 1    NP
dt
 K

Model 2
dN
N
 N
 rN 1   
P
dt
 K  1 N
Population Ecology
 The
number of predators is not
(typically) fixed but changes in time.
 This requires a separate differential
equation to describe predator
density.
Lotka-Volterra Model
dN
 rN  cNP
dt
dP
 bNP  mP
dt

A classic predator-prey model due to
Lotka and Volterra (ca 1925)
Lotka-Volterra Model
dN
 rN  cNP
dt
dP
 bNP  mP
dt
r, c, b, m are all positive constants
 N is prey (resource) density, P is
predator (consumer) density

Lotka-Volterra Model

Lotka and Volterra model has oscillatory
solutions (why biologically?)
Lotka-Volterra Model

The first equation describes the rate
of change of prey (resource) density
dN
 rN  cNP
dt

Let’s consider each term
Lotka-Volterra Model
 The
first term shows that in the
absence of predation the prey
grow exponentially
dN
 rN
dt
(if P=0)
Lotka-Volterra Model

The second term describes the loss
(minus sign) of prey due to predators
dN
 rN  cNP
dt

The loss is proportional to both the
number of prey and the number of
predators (linear consumption rate)
Lotka-Volterra Model

The second equation describes the
rate of change of predator
(consumer) density
dP
 bNP  mP
dt

Let’s consider each term
Lotka-Volterra Model


The first term describes the gain of predators
due to prey (equals the loss of prey due to
predators)
The second term shows that the predator
population decreases exponentially in the
absence of prey
dP
 mP
dt
(if N=0)
Rosenzweig MacArthur Model
dN
 N  cN
 rN 1   
P
dt
 K  aN
dP
bN

P  mP
dt a  N

A classic predator-prey model due to
Rosenzweig and MacArthur (ca 1963)
Population Ecology

Can you describe qualitative differences
in these two models??
Lotka-Volterra
dN
 rN  cNP
dt
dP
 bNP  mP
dt
Rosenzweig-MacArthur
dN
 N  cN
 rN 1   
P
dt
 K  aN
dP
bN

P  mP
dt a  N
Population Ecology
A plant-herbivore system can be thought
of as a type of predator (=herbivore)
and prey (=plant) system
 The Rosenzweig-MacArthur model has
been used to describe plant-herbivore*
dynamics where the variables become

V=vegetation biomass
N=herbivore density
* the herbivore here is assumed to be a mammalian grazer
Population Ecology

The Rosenzweig-MacArthur model for
plant-herbivore (mammal) system
dV
 V  cV
 rV 1   
N
dt
 K  a V
dV
bV

N  mN
dt a  V
Population Ecology

Stability analysis of the RosenzweigMacArthur model for plant-herbivore
system yields the paradox of enrichment
(Turchin):
As plant standing biomass (= K) is increased, the
dynamics of the system become increasingly less
stable (i.e. small parameter changes become
more likely to result in large qualitative changes
in the dynamics)
Population Ecology
 Other
models (Turchin) account
for plant re-growth as there is a
part of the plant that the
herbivore typically does not
consume (i.e. underground
biomass)
Population Ecology
 Still
other models (EdelsteinKeshet) use dependent
variables to describe the
system in terms of plant quality
(rather than plant density) and
herbivore density
Population Ecology
 LURE
students
 Research
(google) plantherbivore models
Population Ecology
 LURE
students
 Propose
a plant herbivore model
that will account for insect
herbivory and plant quality
variations (e.g. via fertilizer or
sunlight variation)
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