Theoretical Modelling in Biology (G0G41A )

Pt I. Analytical Models

II. Difference and differential equation models

Tom WenseleersDept. of Biology, K.U.Leuven

14 October 2008

Recurrence equations and differential equations

differential equation: rate of change of variable over timed(n(t))/dt = "some function of n(t)"

= rate of increase - rate of decreasecontinuous 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 unitn(t+1) = "some function of n(t)"

= n(t) + increase - decrease

or we can calculate the difference equationn = 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

# micen(t)

m

b.n(t)

d.n(t)

d(n(t))/dt=b.n(t)-d.n(t)+m

order of events doesn't matter !

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) after predationn''(t)=n'(t)+b.n'(t) after birthsn'''(t)=n''(t)+m after migration

n(t+1)=n'''(t) =n''(t)+m =n'(t)+b.n'(t)+m =n(t)(1-d)(1+b)+m

n=-d.n(t) + b.(1-d).n(t) + m

census n

predation n'

births n''

migration n'''

order of events matters !

How to make a discrete time model?

How to make a continuous time model?

e.g. flu dynamics: make flow diagramrate of exposure for each healthy individual per day cprob. of transmission upon exposure a

people with flu

n(t)a.c.s(t).n(t)

d(n(t))/dt=a.c.n(t).s(t)d(s(t))/dt=-a.c.n(t).s(t)

people without flu

s(t)

influences flow from other circle

census s

infection s'

How to make a discrete time model?

e.g. flu dynamics: make life-cycle diagram fraction of healthy people potentially exposed each day cprob. of transmission upon exposure a

Susceptibles(healthy)

census n

infection n'

Flu carriers(sick)

recurrence equationsn(t+1)=n(t)+a.c.n(t).s(t)s(t+1)=s(t)-a.c.n(t).s(t)

difference equationsn(t)=a.c.n(t).s(t)s(t)=-a.c.n(t).s(t)

Furtherexamples

Differential equationmodels

Exponential population growth(no density dependence)

• if per capita growth rate r is constant thendn/dt=r.n(t)solution is n(t)=n0.exp(r.t)where n0=initial population size

exponential growth with r > 0pop sizen(t)

r.n(t)

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 casedn/dt=r0.(1-n(t)/K) .n(t)

logistic growth up to carrying capacity K

Lotka-Volterra modeln1 and n2=densities of two competing species

dn1/dt=r1.(1-(nn)/K1).n1dn2/dt=r2.(1-(n2+21.n1)/K2).n2

ri=intrinsic growth rate of species i in optimal conditionsKi=carrying capacity for species in absence of other speciesij=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 dn1/dt=r1.n1-1.n1.n2predator dn2/dt=-r2.n2+2.n1.n2

prey species increases exponentially at rate r1 in absence of predator

predator decreases exponentially at rate r2 in absence of prey

2/1 = conversion factor for converting prey into new predators

...with density dependencen1 and n2=densities of prey and predator

with density dependent growth in prey population:

prey dn1/dt=r1.(1-n1/K1).n1-1.n1.n2predator dn2/dt=-r2.n2+2.n1.n2

...with other functional responsesn1 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)=1.n1.n2 (linear type I functional response)

other choices:

f(n1,n2)=.n1.n2/(+n1) (saturating type II functional response)

f(n1,n2)=.n1k.n2/(+n1k) (generalized type III functional response)

Solving differential equationsIn 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 equationmodels

Population genetic example:Haploid selection

aA

A

WtpWtptWtWWtp)p(t

)).(1().()( where)(

).(1

If relative fitness WA/Wa does not depend on population density, gene frequency change is unaffected by population density.

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)

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

aaAaAA

AaAA

WtpWtptpWtptW

tWWtptpWtptp

.))(1()).(1).((.2.)()( where

)()2/1.()).(1).((.21..)()1(

22

2

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

aaAaAA

AaAA

WpWppWpWppWpp

.)1().1.(.2.)2/1.().1.(.21..

22

2

Three candidate equilibria :

)2/()( ,1 ,0 AAAaaaAaaa WWWWWppp

Stable or unstable depending on parameter values.

Population ecologySingle 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 growthg = decreasing function of n(t)

• density dependent growth: discrete logistic modelg(n) = r.(1-n(t)/K) becomes 0 when n(t)=K - when n(t)>K

simplest possible model: linear decrease of growth rateas 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 diesnumber 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(-).1 / 1! = .exp(-) (Poisson distr.)where = 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 formg(n) = r / (1+n(t).(r-1)/k) becomes 0 when n(t)→k

Single species models

Two-species model e.g. Nicholson-Bailey host-parasitoid model

n and p=host and parasitoid densitymean number of encounters per host per unit time is = a.p(t) a = searching efficiency of parasitoid fraction of hosts that escape parasitism f = exp(-).0 / 0! = exp(-) = exp(-a.p(t)) (Poisson distribution)unparasitized host produces R offspringparasitized host produces 1 parasitoidtherefore 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 hostR = exp(r(1-n(t)/k)) (Ricker model)

More to come....

• when population contains different classes(sexes, age or stage categories...)

• stability criteria...