Tom Wenseleers Dept. of Biology, K.U.Leuven

1. Theoretical Modelling in Biology (G0G41A )Pt I. Analytical ModelsII. Difference and differential equation models Tom Wenseleers Dept. of Biology, K.U.Leuven 14 October 2008

2. 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 - decreaseor we can calculate the difference equationDn = n(t+1) - n(t) = "some function of n(t)" = increase - decrease discrete time steps, for seasonally breeding organismsOR used to numerically approachdifferential equations

3. 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, ...)

4. How to make a model?

5. How to make a continuous time model? e.g. how doesthe 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 # micen(t) order of events doesn't matter ! d.n(t)

6. How to make a discrete time model? e.g. how doesthe presence of a cat change the number of mice in a yard?life-cycle diagram n'(t)=n(t)-d.n(t) after predation n''(t)=n'(t)+b.n'(t) after births n'''(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) + m census n predation n' migration n''' order of events matters ! births n''

7. 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 influences flow from other circle d(n(t))/dt=a.c.n(t).s(t)d(s(t))/dt=-a.c.n(t).s(t) people with flun(t) people without flus(t) a.c.s(t).n(t)

8. How to make a discrete time model? e.g. flu dynamics: make life-cycle diagramfraction of healthy people potentially exposed each day cprob. of transmission upon exposure a 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) census n census s Flu carriers(sick) Susceptibles(healthy) infection n' infection s'

9. Furtherexamples

10. Differential equationmodels

11. 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 > 0 r.n(t) pop sizen(t)

12. 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

13. 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).n2ri=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

14. Simple predator-prey model... n1 and n2=densities of prey and predator prey dn1/dt=r1.n1-a1.n1.n2 predator dn2/dt=-r2.n2+a2.n1.n2prey species increases exponentially at rate r1 in absence of predatorpredator decreases exponentially at rate r2 in absence of prey a2/a1 = conversion factor for converting prey into new predators

15. ...with density dependence n1 and n2=densities of prey and predatorwith density dependent growth in prey population: prey dn1/dt=r1.(1-n1/K1).n1-a1.n1.n2 predator dn2/dt=-r2.n2+a2.n1.n2

16. ...with other functional responses n1 and n2=densities of prey and predatorOther 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) (saturatingtype II functional response) f(n1,n2)=a.n1k.n2/(b+n1k) (generalized type III functional response)

17. 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).

18. Recurrence equationmodels

19. 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) If relative fitness WA/Wadoes not depend on population density, gene frequency change is unaffected by population density.

20. 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

21. 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 Three candidate equilibria : Stable or unstable depending on parameter values.

22. 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 growth g = decreasing function of n(t)

23. 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 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)

24. Single species models • density dependent growth: Ricker model (scramble competition)individuals randomly (Poisson) distributed over N resource siteseach 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 = bn(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 !

25. Single species models • density dependent growth: Beverton-Holt model (contest competition)individuals show clustered (neg. binom.) distribution over resource sitesengage 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

26. Two-species modele.g. Nicholson-Bailey host-parasitoid model n and p=host and parasitoid density mean number of encounters per host per unit time ism = 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 offspringparasitized 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 hostR = exp(r(1-n(t)/k)) (Ricker model)

27. More to come.... • when population contains different classes(sexes, age or stage categories...) • stability criteria...