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

2. Australian rabbits crisis
Photo Selected from Wikipedia

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

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

5. 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

6. 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

7. 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 = rx --- basic one --- multiply
X large e^(-x) === close to too many indivi ---r sth tiny --- end up small ---too crowded – population not 生育

8. 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

9. 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)

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

11. 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)

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

13. 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

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

15. 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

16. 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

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

20. 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

21. 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

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

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

25. d = 0

26. d = 0.1

27. d = 0.2

28. 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

29. d = 0.4

30. d = 0.5

31. 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

32. 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

33. r = 20; d = 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

34. r = 20; d = r = 20; d= 0.4

35. r = 20; d = 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

36. 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

37. 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)

38. Question & Suggestion?