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Theoretical Modelling in Biology (G0G41A )
Pt I. Analytical Models
II. Difference and differential
equation models
Tom Wenseleers
Dept. of Biology, K.U.Leuven
14 October 2008
Recurrence equations and
differential equations
differential equation: rate of change of variable over time
d(n(t))/dt
= "some function of n(t)"
= rate of increase - rate of decrease
continuous time, for continuously breeding organisms
recurrence equations: variable (n) in next time unit is written as
a function of the variable in the current time unit
n(t+1)
= "some function of n(t)"
= n(t) + increase - decrease
or we can calculate the difference equation
Dn = n(t+1) - n(t) = "some function of n(t)" = increase - decrease
discrete time steps, for seasonally breeding organisms
OR used to numerically approach differential equations
Recurrence, difference and
differential equations
• main applications in evolution & ecology:
- model increase or decrease of a genotype frequency
- model increase or decrease in species abundance
• but many other applications, e.g. in physiology &
medicine (tumor growth, blood flow, heartbeat, reaction
kinetics, neuronal excitation, circadian rhythms, gene
switches, growth & development, ...), self-organisation
(pattern formation, collective behaviour, ...)
How to make
a model?
How to make a
continuous time model?
e.g. how does the presence of a cat change
the number of mice in a yard?
flow diagram
b.n(t)
d(n(t))/dt=b.n(t)-d.n(t)+m
m
# mice
n(t)
d.n(t)
order of events
doesn't matter !
How to make a
discrete time model?
e.g. how does the presence of a cat change
the number of mice in a yard?
life-cycle diagram
n'(t)=n(t)-d.n(t)
n''(t)=n'(t)+b.n'(t)
n'''(t)=n''(t)+m
census
n
after predation
after births
after migration
n(t+1)=n'''(t)
=n''(t)+m
=n'(t)+b.n'(t)+m
order of events =n(t)(1-d)(1+b)+m
predation
n'
migration
n'''
births
n''
matters !
Dn=-d.n(t) + b.(1-d).n(t) + m
How to make a
continuous time model?
e.g. flu dynamics: make flow diagram
rate of exposure for each healthy individual per day c
prob. of transmission upon exposure a
influences flow from other circle
people
without flu
a.c.s(t).n(t)
s(t)
people
with flu
n(t)
d(n(t))/dt=a.c.n(t).s(t)
d(s(t))/dt=-a.c.n(t).s(t)
How to make a
discrete time model?
e.g. flu dynamics: make life-cycle diagram
fraction of healthy people potentially exposed each day c
prob. of transmission upon exposure a
census
n
census
s
Flu carriers
(sick)
Susceptibles
(healthy)
infection
n'
infection
s'
recurrence equations
n(t+1)=n(t)+a.c.n(t).s(t)
s(t+1)=s(t)-a.c.n(t).s(t)
difference equations
Dn(t)=a.c.n(t).s(t)
Ds(t)=-a.c.n(t).s(t)
Further
examples
Differential
equation
models
Exponential population growth
(no density dependence)
• if per capita growth rate r is
constant then
dn/dt=r.n(t)
solution is n(t)=n0.exp(r.t) r.n(t)
where n0=initial population
size
pop size
exponential growth with n(t)
r > 0
Logistic population growth
(density dependence)
• if per capita growth rate r
linearly declines with resource
level
r=r0.(1-n(t)/K)
approaches 0 when n(t)→K
in this case
dn/dt=r0.(1-n(t)/K) .n(t)
logistic growth up to carrying
Lotka-Volterra model
n1 and n2=densities of two competing species
dn1/dt=r1.(1-(n1+g12.n2)/K1).n1
dn2/dt=r2.(1-(n2+g21.n1)/K2).n2
ri=intrinsic growth rate of species i in optimal conditions
Ki=carrying capacity for species in absence of other species
gij=competitive coefficient that measures how members of
species j inhibit growth of species i relative to extent to
which they inhibit their own species' growth
Simple predator-prey model...
n1 and n2=densities of prey and predator
prey
predator
dn1/dt=r1.n1-a1.n1.n2
dn2/dt=-r2.n2+a2.n1.n2
prey species increases exponentially at rate r1 in absence of
predator
predator decreases exponentially at rate r2 in absence of prey
a2/a1 = conversion factor for converting prey into new predators
...with density dependence
n1 and n2=densities of prey and predator
with density dependent growth in prey population:
prey
predator
dn1/dt=r1.(1-n1/K1).n1-a1.n1.n2
dn2/dt=-r2.n2+a2.n1.n2
...with other functional responses
n1 and n2=densities of prey and predator
Other assumption in simple model:
number of prey eaten by each predator is proportional to the prey
abundance and increases without limit as the number of prey
increase, i.e. f(n1,n2)=a1.n1.n2 (linear type I functional response)
other choices:
f(n1,n2)=a.n1.n2/(b+n1) (saturating type II functional response)
f(n1,n2)=a.n1k.n2/(b+n1k) (generalized type III functional response)
Solving differential equations
In Mathematica differential equations can be
algebraically solved using DSolve[] or, if an
analytical solution cannot be obtained, they can be
numerically solved using NDSolve[].
Equilibria can be identified by checking when
dn/dt = 0 using Solve[]
(or dn1/dt and dn2/dt are both zero for a system of
differential equations).
Recurrence
equation
models
Population genetic example:
Haploid selection
Single-locus, diallelic model for a haploid species with
nonoverlapping generations :
nA(t+1)=WA.nA(t)
na(t+1)=Wa.na(t)
Frequency of A allele in next generation = p(t+1) = nA / (nA+na)
p(t ).WA
p(t + 1 ) 
W (t )
where W (t )  p(t ).WA + (1  p(t )).Wa
If relative fitness WA/Wa does not depend on population density,
gene frequency change is unaffected by population density.
Population genetic example:
Diploid selection
Single-locus, diallelic model (A/a) for a diploid
species with nonoverlapping generations :
Frequency of A allele in next generation
= A gametes produced /
total number of gametes produced
p(t ) 2 .WAA.1 + 2. p(t ).(1  p(t )).WAa .(1 / 2)
p(t + 1) 
W (t )
where W (t )  p(t ) 2 .WAA + 2. p(t ).(1  p(t )).WAa + (1  p(t )) 2 .Waa
Finding equilibria &
conditions for gene spread
A allele will spread when p(t+1)>p(t)
Equilibrium when p(t+1)=p(t)
i.e. when
p 2 .WAA.1 + 2. p.(1  p).WAa .(1 / 2)
p 2
p .WAA + 2. p.(1  p).WAa + (1  p) 2 .Waa
Three candidate equilibria :
p  0, p  1, p  (Waa  WAa ) /(Waa  2WAa + WAA )
Stable or unstable depending on parameter values.
Population ecology
Single species models
Abundance of species in next generation
n(t+1)=g(n).n(t)
g = growth rate
• no density dependence
(unlimited geometric growth)
g = constant = R = intrinsic growth rate
• density dependent growth
g = decreasing function of n(t)
Single species models
• density dependent growth:
discrete logistic model
g(n) = r.(1-n(t)/K)
becomes 0 when n(t)=K
- when n(t)>K
simplest possible model: linear decrease of growth rate
as a function of population size BUT UNREALISTIC!
- population size can become negative
- purely phenomenological or "top-down model", i.e. no clear
mechanistic interpretation at individual level (how individuals
compete) (bottom-up approach)
other models have either been fitted based on empirical data
or have been derived bottom-up, from first principles
(Brännström & Sumpter 2005)
Single species models
• density dependent growth:
Ricker model (scramble competition)
individuals randomly (Poisson) distributed over N resource sites
each resource site can only support 1 individual,
if a site contains more than 1 individual everybody dies
number of offspring produced at a site with 1 individual = b
n(t+1) = # sites N . prop sites with 1 individual at time t . b
prop sites with 1 individual = exp(-m).m1 / 1! = m.exp(-m) (Poisson distr.)
where m = mean number of individuals per site = n(t) / N
therefore
n(t+1) = N . (n(t) / N).exp(- n(t) / N) . b = b . exp(- n(t) / N) . n(t), so that
g(n) = b . exp(- n(t) / N)
never becomes negative !
Single species models
• density dependent growth:
Beverton-Holt model (contest competition)
individuals show clustered (neg. binom.) distribution over resource sites
engage in contest competition - if there are insufficient resources to
support two individuals one will "win"
resulting growth rate function can be shown to be of
the form
g(n) = r / (1+n(t).(r-1)/k) becomes 0 when n(t)→k
Two-species model
e.g. Nicholson-Bailey host-parasitoid model
n and p=host and parasitoid density
mean number of encounters per host per unit time is
m = a.p(t) a = searching efficiency of parasitoid
fraction of hosts that escape parasitism
f = exp(-m).m0 / 0! = exp(-m) = exp(-a.p(t)) (Poisson distribution)
unparasitized host produces R offspring
parasitized host produces 1 parasitoid
therefore
n(t+1) = R.n(t).f = R.n(t).exp(-a.p(t))
p(t+1) = n(t).(1-f) = n(t).(1-exp(-a.p(t)))
extension: density-dependent growth in host
R = exp(r(1-n(t)/k))
(Ricker model)
More to come....
• when population contains different classes
(sexes, age or stage categories...)
• stability criteria...