Subjects see chapters n Basic about models n Discrete processes u Deterministic models u Stochastic models u Many equations F Linear algebra F Matrix, eigenvalues eigenvectors n Continuous processes u Deterministic models n (Stochastic models)

Stages, States and Classes n Can we always treat a population as a single entity? n Do we need to divide it into different stages or classes? u Age-classes u Size-classes u Subdivided in space u Morphological classes n The subpopulations (stages-classes) differ from each other in aspects important for the purpose and dynamics of the modell. For example in fecundity, survival, dispersal, or risk of predation, or environmental variation, or….. Specific example. Young individuals give birth to fewer than mean aged individuals

Stages, States and Classes n We can use linear algebra, matrix calculations to: n Determine equlibriums (eigenvectors) n Time to equilibrium. (eigenvalues) n Run simulations (matrix multiplication) n Calculate velocity constants (eigenvalues)

Distribution of the population n A population can be treated as one unity if only number of individuals define its property, for example if 50 individuals give birth to twice as many as 25 do. n If the population has a constant distribution of individuals in its relevant classes/stages, it can be treated as one unity. u For example if it’s always 30% newborns, 20% young, 20% newly reproductive, 20% highly reproductive and 10% postreproductive.

Distribution of the population n n If a population of 50 consist of 10 adults/reproductively mature/ the population will reproduce less then if it consists of 20 adults. If the population varies in proportion of adults it will reproduce differently per capita over time. n If the distribution (proportion in stages/classes) of the population varies over time the population either have to include stages/subpopulation or one have to show that it is reasonable to approximate with a simpler non-stage model.

Distribution of the population The dichotomy Stable proportions of classes/ subpopulations Stable per capita growth rates and dispersal rates etc Non-structured model Variation in proportion of individuals in stages/ subpopulations Variation in per capita growth rates and dispersal rates etc Structured model

Ageclasses method: structured population Three ageclasses, n 1, n 2 och n 3. n Next timestep is calculated as n 1 (t+1)= b 1 n 1 (t)+ b 2 n 2 (t)+ b 3 n 3 (t) n 2 (t+1)= s 12 n 1 (t) n 3 (t+1)= s 23 n 2 (t) Note, one time step correspondence to size/span of an ageclass. 123 b1b1 s 23 s ij = probability for an individual in age-class i to survive into the next age-class, j s 12 b2b2 b3b3 b i = how many newborns from ageclass i during one timestep (span of an ageclass)

Ageclasses method: structured population Three ageclasses, n 1, n 2 och n 3. n Next timestep is calculated as n 1 (t+1)= b 1 n 1 (t)+ b 2 n 2 (t)+ b 3 n 3 (t) n 2 (t+1)= s 12 n 1 (t) n 3 (t+1)= s 23 n 2 (t) this is a linear system of equations, one can use linear algebra. Matrix multiplication.

Ageclasses n Next timestep is calculated as n 1 (t+1)= b 1 n 1 (t)+ b 2 n 2 (t)+ b 3 n 3 (t) n 2 (t+1)= s 12 n 1 (t) n 3 (t+1)= s 23 n 2 (t)

Ageclasses n Next timestep is calculated as n 1 (t+1)= b 1 n 1 (t)+ b 2 n 2 (t)+ b 3 n 3 (t) n 2 (t+1)= s 12 n 1 (t) n 3 (t+1)= s 23 n 2 (t)

Ageclasses, an example n Ageclass 1 do not reproduce n Ageclass 2 give birth to 2 n Ageclass 3 give birth to 8 40% of individuals in ageclass 1 survives to ageclass 2 80% of individuals in ageclass 2 survives to ageclass 3 100% of the individuals in agecass 3 dies. Start population conisist of 10 young, 8 subadults and 6 adults..

Ageclasses, matrix multiplication – run a simulation One can calculate this for ever after a while a constant distribution will evolve The right hand side distributions will be the same for all following timesteps Note that the number of individuals may change (density) but the distribution over classes becomes stable densities proportions

Ageclasses, eigenvalues and eigenvectors If the distribution becomes stable then the per capita growth rate also stabilize and becomes a constant value If the per capita growth rate becomes stable/constant over time, one can use that instead of the matrix

Ageclasses, eigenvalues and eigenvectors The other two eigenvalues are complex values and generates the oscillations that occurs prior the stabilisation.

From the beginning again: Solution space and eigenvectors. The lefthand side, vector ( ), exist in a solution space spanned by the three eigenvectors. This means that you can reach the point ( ) in the 3D space by moving along the directions of the three vectors Mathematically this is expressed by: Assume nay population distribution (not an eigenvector)

Solution to n(t)=Atn(0) We know that matris*eigenvector equals eigenvalue*eigenvector Av 1 =λ 1 v 1. And that: Combine these two and x(t)=A t n(0) can be written as What happends at large t (long time???

Stage models n A stage model have classes of different time span, not equals the time step. Hence some of the individuals may stay in the original stage after a timestep. A proportion g i may stay. n Note one have to consider survíval, during one time step, in both p and g parameters. 123 g1g1 p 23 p 12 b2b2 b3b3 g2g2 g3g3

Simple Markov chains n Handles probabilities for an organism to change state, for example running to sleeping or standing or.., healthy to sick to recovered to.. n Can also deal with dispersal. A specific place/habitat is then a state n All numbers are then between 0 and 1 since probability to change from one state to another. n Closed systems, hence no losses or addition. More general model

Simple Markov chains n Handles probabilities for an organism to ‘move’ between different states n All numbers along the arrows have to be between 0 and 1 n All numbers out from a state have to sum up to 1. (otherwise a loss or addition)

Simple Markov chains n A row is the input to a state. n A column is the output of the state. n The row can sum to [0,>1] n The columns always sum to =1

Simple Markov chains, absorbing states n A state is absorbing if the probability is 1 to stay in the state. n With time the probability, where the individuals are will move, to this absorbing state.

Simple Markov chains, equilibriums n What happends over time? n x(t)=Atx(0)? n Is there any equilibrium, x’=Ax’? n If At after a time t only consist of positive elements (>0), the a equlibrium exists. This equilibrium is the eigenvector with eigenvalue 1 of matrix A. n This equilibrium is also a column in At, for large t. At is then the steady state matrix n This equilibrium is a kind of ultimate probability between the states. For example that that there is a 60% probability that an individual is in state 1,…

Simple Markov chains, eigenvalues eigenvectors n An equilibrium exists if all states are connected (direct or indirect). No state is completed isolated. No groups of stes ae isolated from the other. n Calculate eigenvectors and eigenvalues by matlab code, [x,y]=eig(A) n Several equilibriums may exists if there are several absorbing states

Absorbing state, equilibrium n It is possible to calculate the probability that a system reaches the different equilibriums n In the example the question is what the probability is to end up in state 2 or three? n More on page 126 and 127, yet this you can read briefly

Summarizing classes/stages/state-matrices n Ageclass/stages n Population growth – eigenvalue. n Population distribution eigenvector n The state of individuals and populations-Markov chains n Probability for the state of the individual n Equlibrium-eigenvector