Lecture 5: Modelling Population Dynamics N(t)

Transcription

1 Lecture 5: Modelling Population Dynamics N( Part II: Stochastic limited growth Jürgen Groeneveld SGGES, University of Auckland, New Zealand

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3 How to deal with noise? Nt N t+ = RNt + Nt + Ntξ, K ( µ σ ) ξ a normally distributed random variable with µ = 0 and σ = 0.05 σ = 0 σ = 0.05 Extinctions more likely

4 How to deal with noise? Nt N t+ = RNt + Nt + Ntξ, K ( µ σ ) ξ a normally distributed random variable with µ = 0 and σ = 0.05 Are there continuous descriptions of this problem? dn( = rn( dt Langevin equation N( K + But only specialists really work with them ( η σ ) N( ξ, Stochastic differential equation: P t dn( = rn( dt ( N, N( K + ( η σ ) N( ξ, Partial differential equation (Fokker Planck) = N Ito s Lemma, Stratonovich N [ A ( N ) P( N, ] [ A ( N) P( N, ]

5 So, can we calculate P(N,? P t ( N, = N N [ A ( N ) P( N, ] [ A ( N) P( N, ] Well, I can't and most often it is not tractable, but if dynamics end in an equilibrium and the system has no memory than.. P r rn c exp ln( N ) σ Kσ * ( N) = Maximum of P*: N max σ = K r if σ = 0, than N max = K Mean of P*: N = K 3

6 P r rn c exp ln( N ) σ Kσ * ( N) = Good old Paramecium r = 0.8 K = 58.5 σ = 0.05 Software implemented by Lorenz Fahse But system also has to reach stationary stage: # Simulation example for comparison with Fokker-Planck result comp<-function(sigma,repe) { r<-exp(0.8)- N <- rep(, times = 000) K < All <- c(:repe) for(j in :repe) { for (i in :000) { N[i+] <- N[i]+(r*N[i]*(-N[i]/K) + N[i]*rnorm(,0,sigma)) if (N[i+]<0) N[i+] = 0 } All[j]=N[i] } hist(all,freq=t, xlim=c(0,500),breaks=30) } 4

7 Modelling population dynamics Unlimited growth Limited growth Stochastic limited growth N Time discrete t t + = R N 0 Simulations complex dynamics Simulations Extinction Time continuous N ( = N(0) exp( r dn ( N( = rn( dt K dn( N( = rn( dt K + Nξ (0, σ ) Example : How does individual behaviour influence population dynamics? Fahse et al. 998, AmNat Complex individual-based and spatially explicit simulation model for nomadic lark species (Alaudidae) in a heterogeneous, randomly varying landscape (Nama-Karoo, South Africa) Optimal flocking and searching strategy? 5

8 How does individual behaviour influence population dynamics? modified after Fahse et al. 998, AmNat Larks move in small flocks, searching for grass patches to breed Grass patches occur at random (rain fall) and last only weeks Grass patch filled square Flocks circles Breeding Flocks crosses Squares range of vision How does individual behaviour influence population dynamics? modified after Fahse et al. 998, AmNat Flock dynamics 6

9 How does individual behaviour influence population dynamics? Computation would be too time consuming Growth rate f is a complex function: dn/dt = f(nad,njuv,flock size distribution, da, psplit, landscape, mortality) Idea: Population dynamics and behaviour are operating on two time scales Separation of time scales (Haken 99) Separation of time scales: Fast variable: Slow variable: f = dn/dt N( That means there is a characteristic f for any given N Therefore it is possible to split the model: behavioural model population dynamics model modified after Fahse et al. 998, AmNat 7

10 Separation of time scales: Method: Switch of demographic processes in the simulation model and determine a Distribution of f(n) for several N s Regression: r(n) = r m (-N/K) modified after Fahse et al. 998, AmNat Comparison between full simulation model and the method of separated time scales: N ( N( = rn( dt K derive the parameters from bottom up models 8

11 Population Viability Analysis (PVA) Application of ecological theory Following: Grimm and Wissel 004, OIKOS The intrinsic mean time to extinction: a unifying approach to analysing persistence and viability of populations I Basics of Population Viability Analysis PVA Population models often aim to assess population viability Assessing the risk that a population goes extinct To compare the effect of different measures and scenarios, this risk has to be quantified. What would be good measures? 9

12 Simple deterministic case: Remember age structured models F P 0 F P 0 F3 n 0 n 0 n,t,t 3,t n = n n,t+,t+ 3,t+ The growth rate λ is a good measure If λ, population will persist forever If λ <, population will deterministically go extinct Stochastic case: f All populations have a certain risk of extinction! = λ exp( λ < f >=/ λ λ = 0.5 λ = 0. Mean ~ 3.9 Mean ~ 0 0

13 What is the currency of our PVA? Usually people are interested in the Risk P 0 that a population has gone extinct during a time t. t P =< >= 0( t, TM ) f exp TM 0 Relationship of P 0 ( and T m t T M dt So far, we assumed that extinction times are exponentially distributed. Lets analyse general demographic models

14 Master Equation Markov model Birth and death type dpn ( dt = bn Pn ( + dn+ Pn + ( bnpn ( dnpn ( P n (: Probability having n individuals at time t b n : birth rate d n : death rate Master Equation Markov model Birth and death type dpn ( dt = bn Pn ( + dn+ Pn + ( bnpn ( dnpn ( State of t does only depend on the state t- and not on previous time steps (no memory)

15 Equation from first principles dpn ( dt = bn Pn ( + dn+ Pn + ( bnpn ( dnpn ( Solution of P 0 ( can be approximated! No detailed description here see Grimm and Wissel 004 for further references. Main idea: Set of linear (differential) equations, i.e. Solution can be expressed by its eigenvectors and eigenvalues! Largest eigenvalue and eigenvector will dominate the solution and therefore the contribution of all other eigenvectors can be neglected. dpn ( dt = bn Pn ( + dn+ Pn + ( bnpn ( dnpn ( Note that b and d can be functions, e.g. logistic equation, Equation must be linear in P ( ω Pn ( = un, ici exp i i u n,i is the n-th component of the normalized i-th right hand side eigenvector ω is the i-th eigenvalue of the i-th eigenvector (right hand side) 3

16 P P0 ( = c exp( ωt ) What do the parameters mean? ω = Meaning of c? T M ( ω n ( = un, ici exp i i If ω >> ω i for all i > and using that all u i are normalized Grimm and Wissel 004 suggest a protocol to derive the intrinsic mean time T M to extinction from simulations Time to extinction modified after Grimm and Wissel 004 Time to extinction 4

17 modified after Grimm and Wissel 004 Time to extinction P0 ( = c exp( Tm log( P 0( ) = t + log( c ) Tm Theoretically it can be shown that c reflects the initial conditions Bad conditions Population much below its capacity at the start of the simulation Good conditions Population at its capacity at the start of the simulations modified after Grimm and Wissel 004 5

18 T m does not depend on the initial conditions! Bad conditions Population much below its capacity at the start of the simulation Good conditions Population at its capacity at the start of the simulations Modified after Grimm and Wissel 004 Modelling population dynamics Time discrete Time continuous Unlimited growth Limited growth Stochastic limited growth N t t + = R N 0 Simulations complex dynamics Simulations Extinction N ( = N(0) exp( r dn ( N( = rn( dt K dn( N( = rn( dt K + Nξ (0, σ ) Equation based modelling provides important ways to aggregate, understand and to communicate results of simulation models Simulation models might be needed to parameterize equation based models Ecological theory can provide robust measures (e.g. Tm) 6

19 But grey is all theory Thank you for your attention! 7