Mathematical model of Coupled-patch system Xinyao Wang 2018.07.19 Tuesday talk Some similar but diff Ode – iterated map ---chaotic behavior Third coupled --- coupled patch model Topo (spatial configuration)--- grid (space)

Australian rabbits crisis Photo Selected from Wikipedia

Coupled-patch model 2 Dimensional spatial model Model with 2 parameters Population:  1) growth rate                       2) dispersal rate

Significance I hope my research can be used to: keep track of the dynamics of species population preserve vulnerable population prevent overproduction

Coupled-patch model 2D grid of patches that experience growth and dispersal (1) equation regarding the growth rate (Ricker model) (2) equation regarding the dispersal between different areas

Growth function for coupled-patch model: The Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected  number X t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation, Xt. Discrete time model Map but not ODE (ordinary differential equation) He first used this model in his studies of stock and recruitment in fisheries. Different from former presentation Not a coupled-patch model, 1 D model Not spatial model Ricker has no spatial dimension

Population equation:X(t+1) = f (x(t)) Growth equation:        f(x) = r x e-x     T represents time X represents the population in the area we want to analyze Parameter r is the rapidity of population growth (growth rate) X =0 close to 0 ex = 1 ----- rx --- basic one --- multiply X large e^(-x) === close to 0 --- too many indivi ---r sth tiny --- end up small ---too crowded – population not 生育

Cobweb Compare with line f(x) = x --- no change in original population Compare with line f(x) = x --- no change in these two lines Compare with line f(x) = x --- no change in original population Fixed point --- where the two lines cross Cobweb diagram: a visual tool in dynamic systems to investigate the qualitative behavior of one-dimensional iterated function. Infer the long term behavior of the dynamic map by tracing these two lines *Cobweb diagram: a visual tool in dynamic systems to investigate the qualitative behavior of one-dimensional iterated function. Infer the long term behavior of the dynamic map by tracing these two lines

Example when r = 5 f(x) = 5 x e-x X(t+1) X(t) Say the two axes X Clear image Put ricker aside Show equation aside X(t)

Example: when r = 20 f(x) = 20 x e-x intercept : ln(r) ---equilibrium X(t+1) Say the two axes X(t)

Cobweb plot for Ricker map when r =5 X(t+1) f(x) = 5 x e-x Transition: cobweb shows us the dynamics for one parameter combination; Bifurcation diagram shows you dynamics over large parameter space (combinations) Explain --- why process that way? F(0.2) , Plug again --- horizontal move Progression Another slide with r = 20 Same initial condition F(0.2) ------ X(t)

Cobweb plot for Ricker map when r =20 f(x) = 20 x e-x X(t+1) X(t)

Bifurcation diagram In dynamical systems, a bifurcation diagram shows the values visited or approached like, fixed points of a system as a function of a bifurcation parameter in the system. Population values over 100 iterations overlapped for different growth rate Long term behavior over different parameter combination

At 5, one value overtime----stable At 20, 100 unique population value– unstable, chaos

Thought from Ricker map Stable and unstable fixed points --- change of population Now we can move to two dimensional map General idea about how population would change regarding to the first equation

Coupled patch model (2 dimensional map) Include second component --- dispersal rate x= (1-d12) r x e-x +d21 r y e-y y=(1-d21) r y e-y +d12 r x e-x x represents the population in area a y represents the population in area b d12 represents the dispersal rate from area a to area b d21 represents the dispersal rate from area b to area a D --- fraction? Bettween 0 1 D21 * x

Bifurcation for coupled patch model x = 0.5;      y = 0.7;      d12 = d21= 0.1

Bifurcation diagram does not give me information about space Now we are going to move on to a bigger model with more parameters and look at spatial features

Starting Model one Population from the middle point Only every other grid State the three different initial condition Words – 1: symm to four directions portions 2: test the model on three different initial condition Diagram provided by Laura

State start from iteration one One time step Explain this Population is low in the original place Regular Expect beta

Go extinct for three corners at iteration one Write the four corner popu Fourth corner --- invasion

d = 0

d = 0.1

d = 0.2

d = 0.3 Scale log(r) on the bar Fram of movie and mark where is log (r) Above blue Below white Why log(r)---fix point---flipping around log(r) in the map Why make sense? Big – small and small - big

d = 0.4

d = 0.5

For series of images for increasing dispersal rate: The spatial scale of the patterns increases with the increasing dispersal rates Need to involve topology ideas More quantatitive

β0 : connected component β1: holes Betti number: kth Betti number refers to the number of k-dimensional holes on a topological surface. We identity the first and the second Betti numbers for spatial populations time series. β0 : connected component β1: holes Beta –connect Beta = holes (in slides) Language - Examples provided by Laura

r = 20; d = 0.1 r = 20; d= 0.4 When we make the time series plot, we made images binary for Betti number calculation. (since matlab does not take grid as an input) For the audiences Contrast to right one

r = 20; d = 0.1 r = 20; d= 0.4

r = 20; d = 0.1 r = 20; d= 0.4 Low dispersal rate—almost periodic, flipping between two points and in the end settle down to the point High dispersal rate- not settle down-fluctuating-chaotic

Possible conclusion Smaller dispersal rate: β1> β0 --- almost periodic behavior Larger dispersal rate: β0> β1--- Fluctuating What implies: Besides dispersal rate, population pattern will matter

Further steps: Move to more complex model(dispersal scenario) Using actual data from literature to test the usefulness of conclusion Work with GIS (geospatial information system)

Question & Suggestion?