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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. - PowerPoint PPT Presentation

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• Theoretical Modelling in Biology (G0G41A ) Pt I. Analytical Models

II. Difference and differential equation modelsTom WenseleersDept. of Biology, K.U.Leuven14 October 2008

• Recurrence equations and differential equationsdifferential 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 equationsmain applications in evolution & ecology:model increase or decrease of a genotype frequencymodel 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# mice n(t)mb.n(t)d.n(t)d(n(t))/dt=b.n(t)-d.n(t)+m 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 diagramn'(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

Dn=-d.n(t) + b.(1-d).n(t) + mcensus npredation n'births n''migration n'''order of events matters !

• 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 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

• How to make a discrete time model?census sinfection s'e.g. flu dynamics: make life-cycle diagram fraction of healthy people potentially exposed each day c prob. of transmission upon exposure a Susceptibles (healthy)census ninfection 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 equationsDn(t)=a.c.n(t).s(t) Ds(t)=-a.c.n(t).s(t)

• Furtherexamples

• Differential equationmodels

• 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) where n0=initial population sizeexponential growth with r > 0pop size n(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 case dn/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-(n1+g12.n2)/K1).n1dn2/dt=r2.(1-(n2+g21.n1)/K2).n2 ri=intrinsic growth rate of species i in optimal conditionsKi=carrying capacity for species in absence of other speciesgij=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 preydn1/dt=r1.n1-a1.n1.n2predatordn2/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 dependencen1 and n2=densities of prey and predator with density dependent growth in prey population: preydn1/dt=r1.(1-n1/K1).n1-a1.n1.n2predatordn2/dt=-r2.n2+a2.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)=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 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 selectionIf 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 selectionSingle-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

• Finding equilibria & conditions for gene spreadA allele will spread when p(t+1)>p(t) Equilibrium when p(t+1)=p(t) i.e. whenThree candidate equilibria : Stable or unstable depending on parameter values.

• Population ecologySingle species modelsAbundance 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 ratedensity dependent growthg = decreasing function of n(t)

• Single species modelsdensity 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 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 (Brnnstrm & Sumpter 2005)

• Single species modelsdensity 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 modelsdensity 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

• Two-species model e.g. Nicholson-Bailey host-parasitoid modeln and p=host and parasitoid densitymean 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 parasitoidthereforen(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...